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Published online by Cambridge University Press:  12 August 2025

Yoichi Motohashi
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Finnish Academy of Science and Letters
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References

Abel, N. H. (1828): Aufgabe. J. reine angew Math., 3, 212.Google Scholar
Abel, N. H. (1829): Mémoire sur une classe particulière d’équations résolubles algébriquement. J. reine angew Math., 4, 131156.Google Scholar
Agărgün, A. G. and Özkan, E. M. (2001): A historical survey of the fundamental theorem of arithmetic. Historia Math., 28, 207214.10.1006/hmat.2001.2318CrossRefGoogle Scholar
Agrawal, M., Kayal, N., and Saxena, N. (2004): PRIMES is in P Annals of Math., 160, 781793.CrossRefGoogle Scholar
Ahmes, (ca 1650 BCE): The Rhind mathematical papyrus (British Museum 10057 and 10058). Free translation and commentary by Chace, A.B. (1927). Math. Ass. America, 1.Google Scholar
Alford, W. R., Granville, A., and Pomerance, C. (1994): There are infinitely many Carmichael numbers. Annals of Math., 140, 703722.10.2307/2118576CrossRefGoogle Scholar
al-Khwarizmi, (ca 820a): The algebra. Translation by Rosen, F. from an Arabic edition. The Oriental Translation Fund, London, 1831.Google Scholar
al-Khwarizmi, (ca 820b): Thus spake al-Khwārismī. A translation of the text of Cambridge University Library ms. Ii.vi.5. Translation by Clossrey, J. N. and Henry, A. S.. Historia Math., 17 (1990), 103131.Google Scholar
Ankeny, N. C. (1952): The least quadratic non residue. Annals of Math., 55, 6572.CrossRefGoogle Scholar
Anonymous, (1864): On primes and proper primes. The Oxford, Cambridge, and Dublin Messenger of Math., 2, 16.Google Scholar
Arcozzi, N. (2012): Beltrami’s models of non-Euclidean geometry. In: Mathematics in Bolonia 1861–1960, Birkhäuser, Basel, pp.130.Google Scholar
Arndt, A. F (1846): Disquisitiones de residuis cujusvis ordinis. J. reine angew. Math., 31, 333342.Google Scholar
Arndt, A. F (1859a): Auflösung einer Aufgabe in der Composition der quadratischen Formen. J. reine angew Math., 56, 6471.Google Scholar
Arndt, A. F (1859b): Ueber die Anzahl der Genera der quadratischen Formen. J. reine angew Math., 56, 7278.Google Scholar
Arndt, A. F (1859c): Einfacher Beweis für die Irreductibilität einer Gleichung in der Kreistheilung. J. reine angew Math., 56, 178181.Google Scholar
Artin, E. (1971): Galois theory. University of Notre Dame Press, London.Google Scholar
Aryabhata, I (499 CE): The āryabhatīya. An ancient Indian work on mathematics and astronomy. Translated with notes by Clark, W. E., The University of Chicago Press, Chicago, 1930.Google Scholar
Atkinson, F. V. (1949). The mean-value of the Riemann zeta function. Acta Math., 81, 353376.10.1007/BF02395027CrossRefGoogle Scholar
Ayyangar, A. A. K. (1941): Theory of the nearest square continued fraction. Jo. of the Mysore Univ., 1, 2132, 97–117.Google Scholar
Bach, E. (1990): Explicit bounds for primality testing and related problems. Math. Comp., 55, 355380.10.1090/S0025-5718-1990-1023756-8CrossRefGoogle Scholar
Bachmann, P (1872): Die Lehre von der Kreistheilung und ihre Beziehungen zur Zahlentheorie. B.G. Teubner, Leipzig.Google Scholar
Bachmann, P (1894): Die analytische Zahlentheorie. B.G. Teubner, Leipzig.Google Scholar
Bachmann, P (1898): Die Arithmetik der quadratischen Formen. Erste Abt. B.G. Teubner, Leipzig; Zweite Abt. 1923.Google Scholar
Bachmann, P (1902): Niedere Zahlentheorie. Erster Teil. B.G. Teubner, Leipzig; Zweiter Teil. 1910.Google Scholar
Bachmann, P (1911): Über Gauβ' zahlentheoretische Arbeiten. Reprinted in: Gauss Werke X-2, pp.169.Google Scholar
Baker, R. C., Harman, G., and Pintz, J. (2001): The difference between consecutive primes. II. Proc. London Math. Soc., 83, 532562.10.1112/plms/83.3.532CrossRefGoogle Scholar
Balasubramanian, R. and Ramachandra, K. (1989): A lemma in complex function theory I. Hardy-Ramanujan J., 12, 15.Google Scholar
Balazard, E., Naimi, M., and Pétermann, Y.-F S. (2008): Étude d’une somme arithmétique multiple liée à la fonction de Möbius. Acta Arith., 132, 245298.10.4064/aa132-3-4CrossRefGoogle Scholar
Barban, M. B. (1966): The large sieve method and its application to number theory. Uspehi Mat. Nauk, 21, 51102. (Russian)Google Scholar
Barenco, A. C., Bennett, H., Cleve, R., Di Vincenzo, D. P., Margolus, N., Shor, P., Sleator, T., Smolin, J. A., and Weinfurter, H. (1995): Elementary gates for quantum computation. Phys. Rev. A, 31 (5), 34573467.10.1103/PhysRevA.52.3457CrossRefGoogle Scholar
Bateman, P. T. (1951): On the representations of a number as the sum of three squares. Trans. Amer. Math. Soc., 71, 70101.10.1090/S0002-9947-1951-0042438-4CrossRefGoogle Scholar
Baumgart, O. (1885): Über das quadratische Reciprocitätsgesetz.. B.G. Teubner, Leipzig.Google Scholar
Beeger, N. G. W. H. (1922): On a new case of the congruence 2p-1 = 1 mod p2. Messenger of mathematics, 51, 149150.Google Scholar
Beltrami, E. (1868): Teoria fondamentale degli spazii di curvatura costante. Ann. Mat. Pura App., 2, 232255. (Translated: Ann. Sci. de l'É.N.S., 6 (1869), 347–375)10.1007/BF02419615CrossRefGoogle Scholar
Bennett, C. H. (1973): Logical reversibility of computation. IBM J. Res. Develop., 17, 525532.10.1147/rd.176.0525CrossRefGoogle Scholar
Berndt, B. C. and Evans, R. J. (1981): The determination of Gauss Sums. Bull. Amer. Math. Soc., 5, 107129. Corrigendum. 7 (1982), 441.10.1090/S0273-0979-1981-14930-2CrossRefGoogle Scholar
Bhargava, M. (2001): Higher Composition Laws. Ph.D. Thesis, Princeton University.Google Scholar
Bhaskara, II (1150): Bija Ganita. In: Algebra of the Hindus from a Persian manuscript of 1634, translated by Strachey, E. (1813). The Honourable East Indian Company, London; Lilavati, Vijaganita. In: Algebra with arithmetic and mensuration from the Sanscrit. Translated by Colebrooke, H. T. (1817). J. Murray, London, pp.1276.Google Scholar
Binet, J. P. M. (1831): Sur la résolution des équations indéterminées du premier degrée en nombres entirers. J. École Polyt., 13, 289296.Google Scholar
Binet, J. P. M. (1914a): Ein Satz über Dirichletsche Reihen mit Anwendung auf die ζ -Funktion und die L-Funktionen. Rend. di Palermo, 37, 269272.Google Scholar
Bohr, H. und, Landau E. (1914b): Sur les zèros de la fonction ζ(s) de Riemann. Comptes rendus, 158, 106110.Google Scholar
Bombelli, R. (1579): L’Algebra. G. Rossi, Bologna.Google Scholar
Bombieri, E. (1965). On the large sieve. Mathematika, 12, 201225.10.1112/S0025579300005313CrossRefGoogle Scholar
Bombieri, E. (1971): A note on the large sieve. Acta arith., 18, 401404.10.4064/aa-18-1-401-404CrossRefGoogle Scholar
Bombieri, E. (1987): Le grand crible dan la théorie analytique des nombres. Seconde édition. Astérisque, 18, Paris.Google Scholar
Bombieri, E., Friedlander, J. B., and Iwaniec, H. (1986): Primes in arithmetic progressions to large moduli. Acta Math., 156, 203251; Part II. Math. Ann., 277 (1987), 361–393; Part III. J. Amer. Math. Soc., 2 (1989), 215–224.10.1007/BF02399204CrossRefGoogle Scholar
Bombieri, E. and Iwaniec, H. (1986): On the order of . Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13, 449486.Google Scholar
Borwein, J., Bradley, D., and Crandall, R. (2000): Computational strategies for the Riemann zeta function. J. Comp. App. Math., 121, 247296.10.1016/S0377-0427(00)00336-8CrossRefGoogle Scholar
Botts, T. (1967): A chain reaction process in number theory. Math. Magazine, 40,5565.10.1080/0025570X.1967.11975767CrossRefGoogle Scholar
Bouniakowsky, V. (1857): Sur les diviseurs numériqes invariables des fonctions rationnelles entières. Mém. Acad. Impér. Sci. Saint-Pétersbourg, sixème série, Sci. Math. Phys., VI, 306329.Google Scholar
Bourgain, J. (2017): Decoupling, exponential sums and the Riemann zeta function. J. Amer. Math. Soc., 30, 205224.CrossRefGoogle Scholar
Brahmagupta, (628): Ganita, Cuttaca. In: Algebra with arithmetic and mensuration from the Sanscrit, translated by Colebrooke, H. T. (1817). J. Murray, London, pp.277378.Google Scholar
Brillhart, J. (1972): Note on representing a prime as a sum of two squares. Math. Computation, 26, 10111013.10.1090/S0025-5718-1972-0314745-6CrossRefGoogle Scholar
Broughan, K. (2021): Bounded gaps between primes. The epic breakthroughs of the early twenty-first century. Cambridge Univ. Press, Cambridge.10.1017/9781108872201CrossRefGoogle Scholar
Bruggeman, R. W. (1978): Fourier coefficients of cusp forms. Invent. math., 45, 118.CrossRefGoogle Scholar
Brun, V. (1915): Über das Goldbasche Gesetz und die Anzahl der Primzahlpaare. Arkiv for Math. og Natur., 34, 819.Google Scholar
Brun, V. (1919): î a série ou les dénominateurs sont “nombres premiers jumeaux” est convergente ou finie. Bull. Sci. Math., (2) 43, 100104, 124–128.Google Scholar
Brun, V. (1920). Le crible d’Eratosthéne et le théorème de Goldbach. Videnskaps. Skr., Mat. Natur. Kl. Kristiana, No. 3.Google Scholar
Brun, V. (1925): Untersuchungen über das Siebverfahren des Eratosthenes. Jahresbericht der Deutschen Math. Verein., 33, 8196.Google Scholar
Buchstab, A. A. (1937): Asymptotische Abschätzung einer allgemeinen zahlentheoretischen Funktion. Mat. Sbornik, (2) 44, 12391246. (Russian with German résumé)Google Scholar
Caldwell, C. K., Reddick, A., Xiong, Y., and Keller, W. (2012): The history of the primality of one: a selection of sources. J. Integer Sequences, 15, article 12.9.8.Google Scholar
Cardano, G. (1545): Artis magnæ, sive de regulis algebraicis, liber unus. Petreius, Nürnberg. (Ars Magna. Dover Publ., New York, 1993.)Google Scholar
Carlson, F. (1921): Über die Nullstellen der Dirichletschen Reihen und der Riemannschen ζ-Funktion. Arkiv for Mat. Ast. och Fysik., 15, No. 20.Google Scholar
Carmichael, R. D. (1907): On Euler’s 𝜙-function. Bull. Amer. Math. Soc., 13, 241243; Note on Euler’s 𝜙-function. 28 (1922), 109–110.10.1090/S0002-9904-1907-01453-2CrossRefGoogle Scholar
Carmichael, R. D. (1910): Note on a new number theory function. Bull. Amer. Math. Soc., 16, 232238.10.1090/S0002-9904-1910-01892-9CrossRefGoogle Scholar
Cataldi, P A. (1613): Trattato del modo brevissimo di trouare la radice quadra delli numeri. Bartolomeo Cochi, Bologna.Google Scholar
Cauchy, A.-L. (1816): Démonstration d’un théorème curieux sur les nombres. Bull. Sci. Soc. Philom. Paris, année 1816, 133135.Google Scholar
Cauchy, A.-L. (1825): Mémoire sur les intégrales définies, prises entre des limites imaginaires. Chez de Bure Frères, Libraires du doi et de la bibliothèque du roi, Paris.Google Scholar
Cauchy, A.-L. (1829): Exercices de mathématiques. Quartiéme année. Chez de Bure Frère, Libraires du doi et de la bibliothèque du roi, Paris.Google Scholar
Cauchy, A.-L. (1840): Méthode simple et nouvelle pour la détermination complète des sommes alternees formées avec les racines primitives des equations binomes. J. math. pures et appliq., 5, 154168.Google Scholar
Charves, (L. Charve) (1877): Démonstration de la périodiché des fractions continues, engendrées par les racines d’une équation du deuxème degré. Bull. Sci. Math. Astron., (2) 1, 4143.Google Scholar
Chebyshev, P. L. (1848): Teoria sravneny. St. Petersburg Univ. (Russian): German translation. Theorie der Congruenzen. Mayer & Müller, Berlin 1889.Google Scholar
Chebyshev, P. L. (1851): Sur la fonction qui détemine la totalité des nombres premiers inférieur à une limite donnée. Mémoire présentes à la. Acad. Impériale de St. Pétersbourg par divers savants, VI, 141157. Also: J. math. pures et appliq., XVII, 1852, 341–365. (Œuvres I, pp.29–48)Google Scholar
Chebyshev, P. L. (1854): Sur nombres premiers. Mémoire présentés à la Acad. Impériale de St. Pétersbourg par divers savants, VII, 1733. Also: J. math. pures et appliq., XVII, 1852, 366–390. (Œuvres I, pp.51–70)Google Scholar
Cipolla, M. (1903): Un metodo per la risolutione della congruenza di secondo grado. Rendiconto dell’ Accademia delle Scienze Fisiche e Matematiche, Napoli, Ser. 3, IX, 154163.Google Scholar
Cipolla, M. (1907): Sulla risoluzione apiristica delle congruenze binomie secondo un modulo primo. Math. Ann., 63, 5461.10.1007/BF01448423CrossRefGoogle Scholar
Chen, J. R. (1973): On the representation of a larger even integer as the sum of a prime and the product of at most two primes. Sci. Sinica, 16, 157176.Google Scholar
Conrey, B. (1989): More than two fifths of the zeros of the Riemann zeta function are on the critical line. J. reine angew. Math., 399, 126.Google Scholar
Coppersmith, D. (1994): An approximate Fourier transform useful in quantum factoring. IBM Research Report, RC 19642 (07/12/94), Mathematics. IBM Research Division, T. J. Watson Research Center, Yorktown Heights, New York.Google Scholar
Cornacchia, G. (1908): Su di un metodo per la risoluzione in numeri interi dell’equazione . Giornale di matematiche di Battaglini, 46, 3390.Google Scholar
Crandall, R. and Pomerance, C. (2005): Prime numbers. A computational perspective. Second edition. Springer-Verlag, New York.Google Scholar
Datta, B. and Singh, A. N. (1935): History of Hindu mathematics. A source book. Part I. Motilal Banarasidas, Lahore, and Part II. 1938; Single volume edition. Asia Publishing House, Bombay, 1962.Google Scholar
Dedekind, R. (1857a): Abriss einer Theorie der höhern Congruenzen in Bezug auf einen reellen Primzahl-Modulus. J. reine angew. Math., 54, 126.Google Scholar
Dedekind, R. (1857b): Beweis für die Irreductibilität der Kreistheilungs-Gleichungen. J. reine angew. Math., 54, 2730.Google Scholar
Dedekind, R. (1873): Die Lehre von der Kreistheilung und ihre Beziehungen zur Zahlentheorie. Akad. Vorlesungen von Dr. Paul Bachmann. Literaturzeitung. Zeits. für Math. Physik, 18, 1424.Google Scholar
Dedekind, R. (1876): Sur la théorie des nombres entiers algébrique. Bull. Sci. Math. Astron., (1) II, 278288; (2) 1 (1877a), 17–41, 69–92, 144–164, 207–248.Google Scholar
Dedekind, R. (1877b): Schreiben an Herrn Borchardt über die Theorie der elliptischen Modul-Functionen. J. reine angew. Math., 83, 265292.Google Scholar
Dedekind, R. (1877c): Über die Anzahl der Ideal-Klassen in den verschiedenen Ordnungen eines endlichen Zahlkörpers. In: Festschrift der Tech. Hochsch. Braunschweig zur Säkularfeier des Gebrustages von C. F. Gauss, Braunschweig, pp.151.Google Scholar
Dedekind, R. (1892): Stetigkeit und irrationale Zahlen. Friedrich Biewerg, Braunschweig. (Continuity and irrational numbers. In: Essays on the theory of numbers, pp.113, The Open Court Publ., Chicago 1901)Google Scholar
de, la Vallée Poussin C.-J. (1896): Recherches analytiques sur la théorie des nombres premiers. Ann. Soc. Sci. Bruxelles, 20, 183256; Reprinted by Acad. Royal de Belgique, Bruxelles, 1897.Google Scholar
de, la Vallee Poussin C.-J. (1900): Sur la fonction 𝜁(s) de Riemann et le nombres des nombres premiers inferieurs à une limite donnée. Mém. Courronnés et Autres Mém. Publ. Acad. Roy. Sci., des Lettres Beaux-Arts de Belgique, 59, Nr.1.Google Scholar
Desmarest, E. (1852): Théorie des nombres. Traité de l’analyse indéeterminée du second degreé a deux inconnues. Librairie de L. Hachette, Paris.Google Scholar
Dickson, L. E. (1911): Notes on the theory of numbers. Amer. Math. Monthly, 18, 109111.10.1080/00029890.1911.11997618CrossRefGoogle Scholar
Dickson, L. E. (1919): History of number theory. I. Carnegie Inst., Washington; II. 1920; III. 1923.Google Scholar
Dickson, L. E. (1929): Introduction to the theory of numbers. The Univ. Chicago Press, Chicago.Google Scholar
Diffie, W. and Hellman, M. E. (1976): New directions in cryptography. IEEE Trans. Inf. Theory, 22, 644654.10.1109/TIT.1976.1055638CrossRefGoogle Scholar
Diophantus (ca 250): Arithmetica.Google Scholar
Méziriac, C.G. Bachet de (1621): Diophanti Alexandrini arithmeticorum libri sex et de numeris multangulis liber unus. Lutetia Parisiorum.Google Scholar
Heath, T. L. (1910): Diophantus of Alexandria. A study in the history of Greek algebra. Second edition. Cambridge Univ. Press, Cambridge.Google Scholar
Dirichlet, P. G. L. (1828): Démonstrations nouvelles de quelques thèorèmes relatifs aux nombres. J. reine angew Math., 3, 390393. (Werke I, pp.99–104)Google Scholar
Dirichlet, P. G. L. (1834): Einige neue Sätze über unbestimmte Gleichungen. Abh. Königl. Preuss Akad. Wissens., 649664. (Werke I, pp.219–236)Google Scholar
Dirichlet, P. G. L. (1835): Über eine neue Anwendung bestimmter Integrale auf die Summation endlicher order unendlicher Reihen. Abh. Königl. Preuss Akad. Wissens., 391407. (Werke I, pp.237–256)Google Scholar
Dirichlet, P. G. L. (1837a): Beweis des Sätzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abh. Königl. Preuss Akad. Wissens., 4581. (Werke I, pp.313–342)10.1017/CBO9781139237321.012CrossRefGoogle Scholar
Dirichlet, P. G. L. (1837b): Sur la manière de résoudre lréquation t2 — pu2 = 1 au moyen des fonctions circulaires. J. reine angew. Math., 17, 286290. (Werke I, pp.343–350)Google Scholar
Dirichlet, P. G. L. (1838): Sur l’usage des séries infinies dans la théorie des nombres. J. reine angew. Math., 18, 259274. (Werke I, pp.357–374)Google Scholar
Dirichlet, P. G. L. (1839): Recherches sur diverses applications de l’analyse infinitésimale à la théorie des nombres. Premièr part. J. reine angew. Math.,19, 324369; (1840a) Seconde partier. 21, 1–12 and 134–155. (Werke I, pp.411–496)Google Scholar
Dirichlet, P. G. L. (1840b): Auszug aus einer der Akademie der Wissenschaften zu Berlin am 5ten März 1840 vorgelesenen Abhandlung. J. reine angew. Math., 21, 98100; Extrait d’une lettere de M. Lejeune-Dirichlet a M. Liouville. C.R. Acad. Sci., 10 (1840), 285–288.Google Scholar
Dirichlet, P. G. L. (1842): Recherches sur les formes quadratiques à coefficients et à indéterminées complexes. J. reine angew. Math., 24, 291371. (Werke I, pp.533–618)Google Scholar
Dirichlet, P. G. L. (1846): Zur Theorie der complexen Einheiten. Abh. König. Preuss. Akad. Wiss., 103107. (Werke I, pp.640–644)Google Scholar
Dirichlet, P. G. L. (1849): Über die Bestimmung der mittleren Werthe in der Zhalenthéorie. Abh. König. Preuss. Akad. Wiss., 6983. (Werke II, pp.49–66)Google Scholar
Dirichlet, P. G. L. (1851): De formarum binariarum secundi gradus compositione. Commentatio qua ad audiendam orationem pro loco in facultate philosophica. Berolini Typis Academicis. (Werke II, pp.105–114; Traduit: J. math. pures et appliq., 4 (1859), 389–398)Google Scholar
Dirichlet, P. G. L. (1854a): Über den ersten der von Gauss gegebenen Beweise des Reciprocitäts-gesetzes in der Theorie der quadratischen Reste. J. reine angew. Math., 47, 139150. (Werke II, pp.121–138)Google Scholar
Dirichlet, P. G. L. (1854b): Vereinfachung der Theorie der binären quadratischen Formen von positiver Determinante. Abh. König. Preuss. Akad. Wiss., 99115. (Werke II, pp.139–158)Google Scholar
Dirichlet, P. G. L. (1857): Démonstration nouvelle d’une proposition relative à la théorie des formes quadratiques. J. math. pures et appliq., sér. II, 1, 273276. (Werke II, pp.209–214)Google Scholar
Dirichlet, P. G. L. (1863): Vorlesungen uber Zahlenthéorie. von, Herausgegeben Dedekind, R.. Sohn, Friedrich Vieweg und, Braunschweig; Zweite Auflage. 1871; Dritte Auflage. 1879; Vierte Auflage. 1894.Google Scholar
Dress, F. and Olivier, M. (1999): Polynômes prenant des valeurs premières. Experiment. Math., 8, 319338.10.1080/10586458.1999.10504622CrossRefGoogle Scholar
du, Bois-Reymond P. (1883): Ueber den Gültigkeitsbereich der Taylor’schen Reihenentwickelung. Math. Ann., 21, 109117.Google Scholar
Dunnington, G. W. (2004): Carl Friedrich Gauss. Titan of Science. Second edition. The Math. Assoc. America, Washington, DC.Google Scholar
Edwards, H. M. (1974): Riemann's zeta-function. Academic Press, New York; Reprint: Dover Publ., Inc., Mineola, New York, 2001.Google Scholar
Ellis, J. H. (1987): The story of non-secret encryption. (private document)Google Scholar
Elsholtz, C. and Tao, T. (2013): Counting the number of solutions to the Erdős-Straus equation on unit fractions. J. Austral. Math. Soc., 94, 50105.10.1017/S1446788712000468CrossRefGoogle Scholar
Elstrodt, J. (2007): The life and work of Gustav Lejeune Dirichlet (1805–1859). Clay Math. Proc., 7, 137.Google Scholar
Erdős, P. (1950): Az 1/x1 + 1/x2 + ... + 1/xn = a/b egyenlet egész számú megoldásairól. Mat. Lapok, 1, 192210.Google Scholar
Estermann, T. (1931): Über die Darstellung einer Zahl als Differenz von zwei Produkten. J. reine angew. Math., 164, 173182.10.1515/crll.1931.164.173CrossRefGoogle Scholar
Estermann, T. (1948): On Dirichlet’s L-functions. J. London Math. Soc., 23, 275279.10.1112/jlms/s1-23.4.275CrossRefGoogle Scholar
Euclid (ca 300 BCE): STOIXEIA (the Elements).Google Scholar
Novara, Campano da (1482): Opus elementorum Euclidis megarensis in geometriam artem in id quoque Campani perspicacissimi commentationes finiunt. E. Randolt, Venetiis.Google Scholar
Billingsley, H. (1570): The elements of geometrie of the most auncient philosopher Euclide of Megara. J. Daye, London.Google Scholar
Heath, T. L. (1956): The thirteen books of Euclid’s Elements translated from the text of Heiberg. Second edition. Vols. I-III. Dover, New York.Google Scholar
Euler, L. (1729): De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt. Comm. Acad. Sci. Petropolitanae, 5 (1738), 3657.Google Scholar
Eúler, L. (1732a): Observationes de theoremate quodam Fermatiano, aliisque ad numeros primos spectantibus. (Comm. Arith. Collect., I, pp.13)Google Scholar
Eúler, L. (1732b): Methodus generalis summandi progressiones. Comm. Acad. Sci. Petropoli-tanae, 6 (1738), pp.6897.Google Scholar
Eúler, L. (1733a): De solutione problematum Diophanteorum per numeros integros. (Comm. Arith. Collect., I, pp.410)Google Scholar
Eúler, L. (1733b): Solutio problematis arithmetici de inveniendo numero, qui per datos numeros divisus, relinquat data residua. (Comm. Arith. Collect., I, pp.1120)Google Scholar
Eúler, L. (1735): De summis serierum reciprocarum. Comm. Acad. Sci. Petropolitanae, 7 (1740), 123134.Google Scholar
Eúler, L. (1736): Theorematum quorundam ad numeros primos spectantium demonstratio. (Comm. Arith. Collect., I, pp.2123)Google Scholar
Eúler, L. (1737a): De fractionibus continuis dissertatio. Comm. Acad. Sci. Petropolitanae, 9 (1744), 98137.Google Scholar
Eúler, L. (1737b): Variae observationes circa séries infinitas. Comm. Acad. Sci. Petropolitanae, 9 (1744), 160188.Google Scholar
Eúler, L. (1740): De extractione radicum ex quantitatibus irrationalibus. Comm. Acad. Sci. Petropolitanae, 13 (1751), 1660.Google Scholar
Eúler, L. (1747): Theoremata circa divisores numerorum. (Comm. Arith. Collect., I, pp.5061)Google Scholar
Eúler, L. (1748a): Theoremata circa divisores numerorum in hac forma paa ± qbb contentorum. (Comm. Arith. Collect., I, pp.3549)Google Scholar
Eúler, L. (1748b): Introductio in analysin infinitorum. M.M. Bousquet & Socios, Lausannæ.Google Scholar
Eúler, L. (1749a): De numeris, qui sunt aggregata duorum quadratorum. (Comm. Arith. Collect., I, pp.155173)Google Scholar
Eúler, L. (1749b): Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques. Mem. Acad. Sci. Berlin, 17 (1768), 83106.Google Scholar
Eúler, L. (1751): Demonstratio theorematis Fermatiani, omnem numerum primum formae 4n + 1 esse summam duorum quadratorum. (Comm. Arith. Collect., I, pp.210233)Google Scholar
Eúler, L. (1752): Observatio de summis divisorum. (Comm. Arith. Collect., I, pp.146154)Google Scholar
Eúler, L. (1755): Theoremata circa residua ex divisione potestatum relicta. (Comm. Arith. Collect., I, pp.260273)Google Scholar
Eúler, L. (1758a): Theoremata arithmetica nova methodo demonstrata. (Comm. Arith. Collect., I, pp.274286)Google Scholar
Eúler, L. (1758b): De resolutione formularum quadraticarum indeterminatarum per numeros integros. (Comm. Arith. Collect., I, pp.297315)Google Scholar
Eúler, L. (1759): De usu novi algorithmi in problemate Pelliano solvendo. (Comm. Arith. Collect., I, pp.316336)Google Scholar
Eúler, L. (1760): De numeris primis valde magnis. (Comm. Arith. Collect., I, pp.356378)Google Scholar
Eúler, L. (1765): Quomodo numeri praemagni sint explorandi, utrum sint primi nec ne? (Comm. Arith. Collect., I, pp.379390)Google Scholar
Eúler, L. (1771): Vollständige Anleitung zur Algebra. Kaiser. Akad. Wiss., St. Petersburg. (English translation: Elements of algebra. Third edition. Longman, Rees, Omre, and Co., London, 1822)Google Scholar
Eúler, L. (1772a): Observationes circa divisionem quadratorum per numeros primos. (Comm. Arith. Collect., I, pp.477486; Opusc. Analy., I, pp.6484)Google Scholar
Eúler, L. (1772b): Disquisitio accuratior circa residua ex divisione quadratorum altiorumque potestatum per numeros primos relicta. (Comm. Arith. Collect., I, pp.487506; Opusc. Analy., I, pp.121–156)Google Scholar
Eúler, L. (1772c): Démonstrationes circa residua ex divisione potestatum per numeros primos resultantia. (Comm. Arith. Collect., I, pp.516537)Google Scholar
Eúler, L. (1772d): Novae demonstrationes circa resolutionem numerorum in quadrata. (Comm. Arith. Collect., I, pp.538548)Google Scholar
Eúler, L. (1772e): De criteriis aequationis fxx + gyy = hzz, utrum ea resolutionem admittat, nec ne? (Comm. Arith. Collect., I, pp.556569; Opusc. Analy., I, pp.211–241)Google Scholar
Eúler, L. (1772f): Extrait d’une letter a M. Bernoulli. (Comm. Arith. Collect., I, p.584)Google Scholar
Eúler, L. (1773a): De quibusdam eximiis proprietatibus circa divisores potestatum occurrentibus. (Comm. Arith. Collect., II, pp.126; Opusc. Analy., I, pp.242–267)Google Scholar
Eúler, L. (1773b): Nova subsidia pro resolutione formulae axx + 1 = yy. (Comm. Arith. Collect., II, pp.35–43; Opusc. Analy., I, pp.310–328)Google Scholar
Eúler, L. (1773c): Miscellanea analytica. (Comm. Arith. Collect., II, pp.44–52; Opusc. Analy., I, pp.329344)Google Scholar
Eúler, L. (1774): De tabula numerorum primorum, usque ad millionem et ultra continuanda; in qua simul omnium numerorum non primorum minimi divisores exprimantur. (Comm. Arith. Collect., II, pp.6491)Google Scholar
Eúler, L. (1775): Speculationes circa quasdam insignes proprietates numerorum. (Comm. Arith. Collect., II, pp.127133)Google Scholar
Eúler, L. (1778a): De variis modis numeros praegrandes examinandi, utrum sint primi nec ne? (Comm. Arith. Collect., II, pp.198214)Google Scholar
Eúler, L. (1778b): Utrum hic numerus: 1000009 sit primus, nec ne, inquiritur. (Comm. Arith. Collect., II, pp.243248)Google Scholar
Eúler, L. (1783): Opuscula analytica. Tomus primus. Acad. Imper. Sci., Petropoli; Tomus secundus. 1785.Google Scholar
Eúler, L. (1849a): Tractatus de numerorum doctrina capita XVI, quae supersunt. (Comm. Arith. Collect., II, pp.503575)Google Scholar
Eúler, L. (1849b): De numeris amicabilibus. (Comm. Arith. Collect., II, pp.627636)Google Scholar
Eúler, L. (1849c): Commentationes arithmeticae collectae. Tomus prior. Acad. Imper. Sci., Petropoli; Tomus posterior. (Ed. Fuss, P. H. and Fuss, N.; Bouniakowsky, V. and Tchébychew, P.)Google Scholar
Farey, J. (1816): On a curious property of vulgar fractions. Philos. Mag. J., 47, 385386.Google Scholar
Fermat, P. (1679): Varia opera mathematca. Joannem Pech, Tolosæ. (Edited by author’s Fermat, son S.)10.5479/sil.128299.39088002705879CrossRefGoogle Scholar
Fermat, P. (1891): Œuvres. Tome premier. Gauthier-Villars, Paris; Tome deuxième. 1894; Tome troisième. 1896; Tome quatrième. 1912.Google Scholar
Fibonacci (1202): Liber abbaci. In: Scritti di Leonardo Pisano publ. da B. Boncom-pagni. Vol. I. Tipografia delle scienze matematiche e fisiche, Roma 1857.Google Scholar
Fogels, E. (1965): On the zeros of L-functions. Acta Arith., 11, 6796.10.4064/aa-11-1-67-96CrossRefGoogle Scholar
Fowler, D. H. (1979): Ratio in early Greek mathematics. Bull. Amer. Math. Soc., 1, 807846.10.1090/S0273-0979-1979-14684-6CrossRefGoogle Scholar
Franel, J. (1925): Les suites de Farey et le problème des nombres premiers. Nachr Ges. Wiss. Göttingen Math.-Phys. Kl., J. 1924, 198201.Google Scholar
Friedlander, J. B. and Iwaniec, H. (2010): Opera de cribro. Amer. Math. Society, Providence, RI.Google Scholar
Frobenius, G. (1879): Theorie der linearen Formen mit ganzen Coefficienten. J. reine angew. Math., 86, 146208.Google Scholar
Fuss, P. H. (1843): Correspondance mathématique et physique de quelques célèbres géomètres du XVIIiéme siècle. Tomes I et II. Académie Impériale des Sciences, St.-Pétersbourg.Google Scholar
Gallagher, P. X. (1968): Bombieri’s mean value theorem. Mathematika, 15, 16.10.1112/S002557930000231XCrossRefGoogle Scholar
Gallagher, P. X. (1970): A large sieve density estimate near σ = 1. Invent. math., 11, 329339.10.1007/BF01403187CrossRefGoogle Scholar
Galois, E. (1828): Démonstration d’un théorème sur les fractions continues périodiques. (Œuvres, pp.385392)Google Scholar
Galois, E. (1830): Sur la théorie des nombres. (Œuvres, pp.398407)Google Scholar
Galois, E. (1831): Sur les conditions de résolubilité des equations par radicaux. (Œuvres, pp.417433)Google Scholar
Gandz, S. (1937): The origin and development of the quadratic equations in Babylonian, Greek, and early Arabic Algebra. Osiris, 3, 405557.10.1086/368481CrossRefGoogle Scholar
Gantmacher, F. R. (1959): The theory of matrices. Chelsea, New York.Google Scholar
Gauss, C. F. (1801): Disquisitiones arithmeticae. Fleischer, Lipsiae. (Werke I, pp.1478);10.5479/sil.324926.39088000932822CrossRefGoogle Scholar
Gauss, C. F. French edition. Courcier, Paris 1807.Google Scholar
Gauss, C. F. German edition. Julius Springer, Berlin 1889.Google Scholar
Gauss, C. F. English edition. Springer Verlag, New York 1986.Google Scholar
Gauss, C. F. The shaping of arithmetic after C. F. Gauss's Disquisitiones Arithmeticae. Edited by Goldstein, C., Schappacher, N., and Schwermer, J.. Springer Verlag, Berlin 2007.Google Scholar
Gauss, C. F. (1808): Theorematis arithmetici demonstratio nova. (Werke II-1, pp.18)Google Scholar
Gauss, C. F. (1811): Summatio quarumdam serierum singularium. (Werke II-1, pp.945)Google Scholar
Gauss, C. F. (1818): Theorematis fundamentalis in doctrina de residuis quadraticis demonstrationes et ampliationes novae.Google Scholar
Gauss, C. F. Demonstratio quinta. (Werke II-1, pp.4954);Google Scholar
Gauss, C. F. Demonstratio sexta. (Werke II-1, pp.5559)Google Scholar
Gauss, C. F. (1828): Theoria residuorum biquadraticorum. Commentatio prima. (Werke II-1, pp.6592); Commentatio secunda. (1832) (Werke II-1, pp.93–148).Google Scholar
Gauss, C. F. (1863a): Analysis residuorum.Google Scholar
Gauss, C. F. Caput sextum. Pars prior. Solutio congruentiae xm 1 = 0. (Werke II-1, pp.199211)Google Scholar
Gauss, C. F. Caput octavum. Disquisitiones generales de congruentiis. (Werke II-1, pp.212242)Google Scholar
Gauss, C. F. (1863b): Disquisitionum circa aequationes puras ulterior evolutio. (Werke II-1, pp.243265)Google Scholar
Gauss, C. F. (1863c): De nexu inter multitudinem classium, in quas formae binariae secundi gradus distribuuntur, earumque determinantem. (Werke II-1, pp.269303)Google Scholar
Goldberg, K. (1953): A table of Wilson quotients and the third Wilson prime. J. London Math. Soc., (1) 28, 252256.10.1112/jlms/s1-28.2.252CrossRefGoogle Scholar
Goldfeld, D. (1976): The class number of quadratic fields and the conjecture of Birch and Swinnerton-Dyer. Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 3, 624663.Google Scholar
Goldfeld, D. (1985): Gauss’ class number problem for imaginary quadratic number fields. Bull. Amer. Math. Soc., 13, 2337.10.1090/S0273-0979-1985-15352-2CrossRefGoogle Scholar
Goldston, D. A., Pintz, J., and Yildirim, C. Y. (2005): Primes in tuples. I. arXiv: 0508185 v1. (Annals of Math., (2) 170 (2009), 819862)Google Scholar
Gradshtein, I. S. and Ryzhik, I. M. (2007): Tables of integrals, séries and products. Seventh edition. Academic Press, Elsevier, London.Google Scholar
Graham, S. W. and Kolesnik, G. (1991): van der Corput’s method of exponential sums. London Mathematikal Society Lecture Note Series, Vol. 126, Cambridge University Press, Cambridge.10.1017/CBO9780511661976CrossRefGoogle Scholar
Grandi, A. (1883): Dimostrazione di un teorema della teoria dei numeri. Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti, ser. 6, 1, 809812.Google Scholar
Granville, A. (2008): Smooth numbers: computational number theory and beyond. Algorithmic Number Theory. MSRI Publications, 44, 267323.Google Scholar
Granville, A. and Mollin, R. A. (2000): Rabinowitsch revisited. Acta Arith., 96, 139153. Greaves, G. (2001). Sieves in number theory. Springer-Verlag, Berlin.Google Scholar
Hadamard, J. (1896): Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques. Bull. Soc. Math. France, 24, 199220.10.24033/bsmf.545CrossRefGoogle Scholar
Hadamard, J. (1954): An essay on the psychology of invention in the mathematical field. Dover, New York.Google Scholar
Halász, G. (1968): Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen. Acta Math. Acad. Sci. Hungar., 19, 365403.10.1007/BF01894515CrossRefGoogle Scholar
Halász, G. and Turán, P. (1969): On the distribution of roots of Riemann zeta and allied functions. J. Number Theory, 1, 121137.10.1016/0022-314X(69)90031-6CrossRefGoogle Scholar
Halberstam, H. and Richert, H.-E. (1974): Sieve methods. Academic Press, London.Google Scholar
Haneke, W. (1963): Verschärfung der Abschätzung von . Acta Arith., 8, 357430.10.4064/aa-8-4-357-430CrossRefGoogle Scholar
Hardy, G. H. (1914): Sur les zéros de la fonction ζ(s). Comptes rendus de l’académie des sciences Paris, 158, 10121014.Google Scholar
Hardy, G. H. and Littlewood, J. E. (1923): Some problems of ‘Partitio Numerorum’; III: On the expression of a number as a sum of primes. Acta Math., 44, 170.10.1007/BF02403921CrossRefGoogle Scholar
Hardy, K., Muskat, J. B., and Williams, K. S. (1990): A deterministic algorithm for solving n = fu2 + gv2 in coprime integers u and v. Math. Computation, 55, 327343.Google Scholar
Haros, C. (1802): Tables pour évaluer une fraction ordinaire avec autant de decimales qu’on voudra; et pour trouver la fraction ordinaire la plus simple, et qui approche sensiblement d’une fraction décimale. J. de l'Ecole Polytech., 4, 364368.Google Scholar
Hasse, H. (1964): Vorlesungen über Zahlenthéorie. Grundl. Math. Wiss., 59, Springer-Verlag, Berlin.10.1007/978-3-642-88678-2CrossRefGoogle Scholar
Heath-Brown, D. R. (1978): The twelfth power moment of the Riemann zeta-function. Quart. J. Math. Oxford, 29, 443462.10.1093/qmath/29.4.443CrossRefGoogle Scholar
Hecke, E. (1917a): Über die Kroneckersche Grenzformel für reelle quadratische Körper und die Klassenzahl relativ-Abelscher Körper. Verhandl. der Natur. Gesell. Basel, 28, 363372.Google Scholar
Hecke, E. (1917b): Über die Zetafunktion beliebiger algebraischer Zahlkörper. Nachr Gesell. Wiss. Göttingen, Math.-Phy. Klasse, J., 1917, 7789.Google Scholar
Hecke, E. (1923): Vorlesung über die Theorie der algebraischen Zahlen. Akademische Verlagsgesellschaft, Leipzig; English edition. Springer, New York, 1981.Google Scholar
Hecke, E. (1937): Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I. Math. Ann., 114, 128; II. 316–351.10.1007/BF01594160CrossRefGoogle Scholar
Heilbronn, H. (1934): On the class number in imaginary quadratic fields. Quart. J. Math., 5, 150160.10.1093/qmath/os-5.1.150CrossRefGoogle Scholar
Hejahl, D. A. (1983): The Selberg trace formula for PSL(2, ). Vol. 2. Lecture Notes in Math., 1001, Springer Berlin, Heidelberg.10.1007/BFb0061302CrossRefGoogle Scholar
Hensel, K. (1901): Üeber die Entwickelung der algebraischen Zahlen in Potenzreihen. Math. Ann., 55, 301336.10.1007/BF01444976CrossRefGoogle Scholar
Hertzer, H. (1908): Über die Zahlen der Form ap1 — 1, wenn p eine Primzahl. Archiv Math. Phys., (3) 13, 107.Google Scholar
Hilbert, D. (1897): Die Theorie der algebraischen Zahlkörper. Jahresbericht der Deutschen Math., 4, 175546.Google Scholar
Hoheisel, G. (1930): Primzahlprobleme in der Analysis. Sitz. Preuss. Akad. Wiss., 33, 311.Google Scholar
Holst, E., Strømer, C., and Sylow, L. (1902): Niels Henrik Abel. Memorial publié à l’occsion du centenaire de sa naissance. J. Dybwad, Christiania.Google Scholar
Hooley, C. (1967): Artin’s conjecture for primitive roots. J. reine angew. Math., 225, 209220.Google Scholar
Hôlder, O. (1936): Zur Theorie der Kreisteilungsgleichung Km(x) = 0. Prace Matematyczno-Fizyczyne, 43, 1323.Google Scholar
Hua, L.-K. (1942): On the least solution of Pell’s equation. Bull. Amer. Math. Soc., 42, 731735.10.1090/S0002-9904-1942-07768-8CrossRefGoogle Scholar
Humbert, G. (1915): Sur les formes quadratiques binaires positives. C. R. Acad. Sci., 160, 647650.Google Scholar
Huxley, M. N. (1972): On the difference between consecutive primes. Invent. math., 15, 164170.10.1007/BF01418933CrossRefGoogle Scholar
Huygens, C. (1728): Automati planetarii. In: Opuscula posthuma. Tomus secundus. Janssonio-Waesbergios, Amstelodami, pp.155184.Google Scholar
Hyde, A. M., Lee P, D., and Spearman, B. K. (2014): Polynomials (x3 — n)(x3 + 3) solvable modulo any integer. Amer. Math. Monthly, 121, 355358.10.4169/amer.math.monthly.121.04.355CrossRefGoogle Scholar
Iamblichus (ca 300 CE): Life of Pythagoras. Translated from the Greek by Taylor, T.. Valpy, A.J., , London, 1818.Google Scholar
Ingham, A. E. (1930): Note on Riemann’s ζ-function and Dirichlet’s L-functions. J. London Math. Soc., 5, 107112.10.1112/jlms/s1-5.2.107CrossRefGoogle Scholar
Ingham, A. E. (1937): On the difference between consecutive primes. Q. J. Math. Oxford, 8, 255266.10.1093/qmath/os-8.1.255CrossRefGoogle Scholar
Ingham, A. E. (1940): On the estimation of N(σ, T). Q. J. Math. Oxford, 11, 291292.Google Scholar
Ivic, A. (1985): The Riemann zeta-function. Theory and applications. John Wiley & Sons, New York; Reprint: Dover Publ., Inc., Mineola, New York, 2003.Google Scholar
Ivic, A. (1991): Mean values of the Riemann zeta-function. Tata Inst. Fund. Res. Lect. Math. Phy., 82, TIFR, Bombay.Google Scholar
Ivic, A. (2001): On sums of Hecke séries in short intervals. J. Théor. Nombres Bordeaux, 13, 453468.10.5802/jtnb.333CrossRefGoogle Scholar
Ivic-, A. and Motohashi, Y. (1994): The mean square of the error term for the fourth power moment of the zeta-function. Proc. London Math. Soc., 69, 309329.10.1112/plms/s3-69.2.309CrossRefGoogle Scholar
Ivory, J. (1806): Démonstration of a theorem respecting prime numbers. New Series of Math. Repository, I, Part II, 68.Google Scholar
Iwaniec, H. (1971). On the error term in the linear sieve. Acta Arith., 19, 130.10.4064/aa-19-1-1-30CrossRefGoogle Scholar
Iwaniec, H. (1980a): Sieve methods. Intern. Congress of Math. Proc., Helsinki 1978, Acad. Sci. Fennica, Helsinki, pp.357364.Google Scholar
Iwaniec, H. (1980b): Rosser’s sieve. Acta Arith., 36, 171202.10.4064/aa-36-2-171-202CrossRefGoogle Scholar
Iwaniec, H. (1980c): A new form of the error term in the linear sieve. Acta Arith., 37, 307320.10.4064/aa-37-1-307-320CrossRefGoogle Scholar
Iwaniec, H. (1981): Rosser’s sieve - bilinear forms of the remainder terms - some applications. In: Recent progress in analytic number theory. Vol. 1. Acad. Press, London, pp.203230.Google Scholar
Iwaniec, H. (1982): On the Brun-Titchmarsh theorem. J. Math. Soc., Japan, 34, 95123. Iwaniec, H. and Jutila, M. (1979): Primes in short intervals. Arkiv för mathematik, 17, 167–176.10.2969/jmsj/03410095CrossRefGoogle Scholar
Iwaniec, H. and Kowalski, E. (2004): Analytic number theory. Amer. Math. Society, Providence, RI.10.1090/coll/053CrossRefGoogle Scholar
Jacobi, C. G. J. (1828): Beantwortung der Aufgabe S. 212. J. reine angew. Math., 3, 301302.Google Scholar
Jacobi, C. G. J. (1837): Über die Kreistheilung und ihre Anwendung auf die Zahlenthéorie. Monatsbericht Akad. Wiss. Berlin, 127136. (J. reine angew. Math., 30, 166–182)Google Scholar
Jacobson, M. J. Jr. and Williams, H. C. (2000): The Size of the fundamental solutions of consecutive Pell equations. Experimental Math., 9, 631-640.10.1080/10586458.2000.10504666CrossRefGoogle Scholar
Jones, J. P., Sato, D., Wada, H., and Wiens, D. (1976): Diophantine representation of the set of prime numbers. Amer. Math. Monthly, 83, 449464.10.1080/00029890.1976.11994142CrossRefGoogle Scholar
Jurkat, W. B. and Richert, H.-E. (1965): An improvement of Selberg’s sieve method. I. Acta Arith., 11, 217240.10.4064/aa-11-2-217-240CrossRefGoogle Scholar
Jutila, M. (1977a): Zero-density estimates for L-functions. Acta Arith., 32, 5662.10.4064/aa-32-1-55-62CrossRefGoogle Scholar
Jutila, M. (1977b): Linnik constant. Math. Scand., 41, 4562.10.7146/math.scand.a-11701CrossRefGoogle Scholar
Jutila, M. and Motohashi, Y. (2005): Uniform bounds for Hecke L-functions. Acta Math., 195, 61115.10.1007/BF02588051CrossRefGoogle Scholar
Kaczorowski, J. and Szydlo, B. (1997): Some Ω-results related to the fourth power moment of the Riemann zeta-function and to the additive divisor problem. J. Théor. Nombres Bordeaux, 9, 4150.10.5802/jtnb.188CrossRefGoogle Scholar
Karatsuba, A. A. (1975): Elements of analytic number theory. Nauka, Moscow; Second edition. Fizmatlit, Moscow 1983. (Russian)Google Scholar
Kinkelin, H. (1862): Allgemeine Theorie der harmonischen Reihen mit Angwendung auf die Zahlenthéorie. Schweighauserische Buchdruckerei, Basel.Google Scholar
Klein, F. (1890): Vorlesungen über die Theorie der elliptischen Modulfunctionen. Ausgearbeitet und vervollständigt von R. Fricke. Erster Band. B.G. Teubner, Leipzig; Zweiter Band. 1892.Google Scholar
Kloosterman, H. D. (1926): On the representation of numbers in the form ax2 + by2 + cz2 + dt2. Acta Math., 49, 407464.10.1007/BF02564120CrossRefGoogle Scholar
Knapowski, S. (1962): On Linnik’s theorem concerning exceptional L-zeros. Publ. Math. Debrecen, 9, 168178.10.5486/PMD.1962.9.1-2.18CrossRefGoogle Scholar
Kobayashi, I. (1973): A note on the Selberg sieve and the large sieve. Proc. Japan Acad., 49, 15.Google Scholar
Korselt, A. R. (1899): Probléme chinois. L’Intermédiaire des Mathématiciens, 6, 142143.Google Scholar
Kraïtchik, M. (1922): Théorie des nombres. Tome, I. Gauthier-Villars, Paris; Tome II (1926).Google Scholar
Kronecker, L. (1845) : Beweis dass für jede Primzahl p die Gleichung 1 + x + x2 +... + xp1 = 0 irreductibel ist. J. reine angew. Math., 29, 280.Google Scholar
Kronecker, L. (1854): Mémoire sur les facteurs irreductibles de l’expression xn 1. J. math. pures et appliq., 19, 177192.Google Scholar
Kronecker, L. (1863): Über die Auflösung der Pell’schen Gleichung mittels elliptischer Functionen. Monats. Königl. Preuss. Akad. Wiss. Berlin., a.d.J. 1863, 4450.Google Scholar
Kronecker, L. (1865): Über den Gebrauch der Dirichletischen Methoden in der Theorie der quadatischen Formen. Monats. Königl. Preuss. Akad. Wiss. Berlin., a.d.J. 1864, 285303.Google Scholar
Kronecker, L. (1876): Zur Geschichte des Reciprocitätsgesetzes. Monats. Königl. Preuss. Akad. Wiss. Berlin., a.d.J. 1875, 267274.Google Scholar
Kronecker, L. (1883): Zur Theorie der elliptischen Functionen. Sitz. König. Preuss. Akad. Wiss. Berlin, 497506, 525–530; (1885), 761–784; (1886), 701–780; (1889), 53–63, 123–135.Google Scholar
Kronecker, L. (1889): Summirung der Gaussschen Reihen . J. reine angew. Math., 105, 267268.10.1515/crll.1889.105.267CrossRefGoogle Scholar
Krumbiegel, B. and Amthor, A. (1880): Das Problema bovinum des Archimedes. Zeitschrift Math. Phys., 25, Historische-liter. Abt., 121136 and 153171.Google Scholar
Kummer, E. E. (1860): Über die allgemeinen Reciprocitaïtsgesetze unter den Resten und Nichtresten der Potenzen, deren Grad eine Primzahl ist. Abh. Königl. Akad. Wiss. Berlin, J., 1859, Math. Abh., 19159.Google Scholar
Kummer, E. E. (1861): Gedächtnissrede auf Gustav Peter Lejeune Dirichlet. Abh. Königl. Akad. Wiss. Berlin, J., 1860, Historische Einleitung, 136. (Dirichlet Werke II, pp.309–344)Google Scholar
Kuznetsov, N. V. (1977): Petersson hypothesis for forms of weight zero and Linnik hypothesis. Khabarovsk Complex Res. Inst. Acad. Sci. USSR, Preprint. (Russian); also in Math. USSR-Sb., 39 (1981), 299–342.Google Scholar
Lagrange, J. L. (1768): Solution d’un problème d’arithmétique. (Œuvres 1, pp.671–731)Google Scholar
Lagrange, J. L. (1769): Sur la solution des problèmes indetérminés du second degré. (Œuvres 2, pp.377535)Google Scholar
Lagrange, J. L. (1770a): Additions au mémoire sur la résolution des equations numériques. (Œuvres 2, pp.581652)Google Scholar
Lagrange, J. L. (1770b): Nouvelle méthode pour résoudre les problèmes indéterminée en nombres entiers. (Œuvres 2, pp.655726)Google Scholar
Lagrange, J. L. (1770c): Démonstration d’un theorème d’arithmétique. (Œuvres 3, pp.189201)Google Scholar
Lagrange, J. L. (1771a): Réflexions sur la résolution algébrique des équations. (Œuvres 3, pp.205-421)Google Scholar
Lagrange, J. L. (1771b): Démonstration d’un théoréme nouveau concernant les nombres premiers. (Œuvres 3, pp.425438)Google Scholar
Lagrange, J. L. (1773): Recherches d’arithmétique. Premiére partie. (Œuvres 3, pp.695758); Seconde partie. (1775: Œuvres 3, pp.759–795)Google Scholar
Lagrange, J. L. (1798): Additions aux élémants d’algébre d’Euler. Analyse indéterminée. (Œuvres 7, pp.5180; English translation in Euler (1771: 1822))Google Scholar
Lagrange, J. L. (1808): Traité de la résolution des équations numériques de tous les degrés, avec des notes sur plusieurs points de la théorie des équations algébriques. Quatrième édition. (Œuvres 8, pp.9369)Google Scholar
Lambert, J. H. (1770): Zusätze zu den logaritmischen und trigonometrischen Tabellen zur Erleichterung und Abkürzung der bey Anwendung der Mathematik vorfallenden Berechnungen. Haude und Spener, Berlin.Google Scholar
Lamé, G. (1844): Note sur la limite du nombre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers. C.R. Acad. Sci., 19, 867870.Google Scholar
Landau, E. (1903a): Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes. Math. Ann., 56, 645670.10.1007/BF01444310CrossRefGoogle Scholar
Landau, E. (1903b): Über die Klassenzahl der binaren quadratischen Fromen von negativer Discriminante. Math. Ann., 56, 671676.10.1007/BF01444311CrossRefGoogle Scholar
Landau, E. (1908a): Nouvelle démonstation pour la formule de Riemann sur le nombre des nombres premiers inférieurs a une limite donnée et démonstation d’une formule plus geéneérale pour le cas des nombres premiers d’une progression arithmeétique. Ann. Sci. l’ École Norm. Supér., Sér. 3, 25, 399442.10.24033/asens.595CrossRefGoogle Scholar
Landau, E. (1908b): Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erfolderichen Quadrate. Archiv der Math. Phy., (3) 13, 305312.Google Scholar
Landau, E. (1909): Handbuch der Lehre von der Verteilung der Primzahlen. Erster Band und Zweiter Band. B.G. Teubner, Leipzig and Berlin.Google Scholar
Landau, E. (1911): Über die Nullstellen der Zetafunktion. Math. Ann., 71, 548564.10.1007/BF01456808CrossRefGoogle Scholar
Landau, E. (1918a): Über imaginär-quadratische Zahlkörper mit gleicher Klassenzahl. Nachr. Akad. Wiss. Göttingen, 277284.Google Scholar
Landau, E. (1918b): Über die Klassenzahl imaginär-quadratischer Zahlkörper. Nachr. Akad. Wiss. Göttingen, 285295.Google Scholar
Landau, E. (1921): Über die Nullstellen der Dirichletschen Reihen und der Riemannschen ζ -Funktion. Arkiv för Mat. Astr och Fys., 16, 118.Google Scholar
Landau, E. (1924): Über die Wurzeln der Zetafunktion. Math. Zeit., 20, 98104.10.1007/BF01188073CrossRefGoogle Scholar
Landau, E. (1925): Bemerkungen zu der vorstehenden Abhandlung von Herrn Franel. Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl., J., 1924, 202206.Google Scholar
Landau, E. (1927): Vorlesungen uber Zahlenthéorie. Erster Band. Erster Teil. Aus der elementaren Zahlenthéorie. S. Hirzel, Leibzig. (Reprinted by Chelsea, New York, 1950)Google Scholar
Landau, E. (1929): Über die Irreduzibilität der Kreisteilungsgleichung. Math. Z., 29, 462.10.1007/BF01180543CrossRefGoogle Scholar
Landauer, R. (1961): Irreversibility and heat generation in the computing process. IBM Journal of Res. Develop., 5, 183191.10.1147/rd.53.0183CrossRefGoogle Scholar
Lau, Y.-K. and Tsang, K.-M. (2009): On the mean square formula of the error term in the Dirichlet divisor problem. Math. Proc. Cambridge Phil. Soc., 146, 277287.10.1017/S0305004108001874CrossRefGoogle Scholar
Lebedev, N. N. (1972): Special Functions and Their Applications. Dover Publ. Inc., Mineola, New York.Google Scholar
Lecerf, Y. (1963): Machines de Turing réversibles. Récursive insolubilité en n ∈ ℕ de l’équation u = θnu, éù θ est un ⪡isomorphisme de codes⪢. C.R. Acad. Sci., 257, 25972600.Google Scholar
Lee, M. A. (1969): Some irreducible polynomials which are reducible mod p for all p. Amer. Math. Monthly, 76, 1125.10.1080/00029890.1969.12000428CrossRefGoogle Scholar
Legendre, A.-M. (1785): Recherches d’analyse indéterminée. Histoire de l’Académie Royale des Sciences, 465559.Google Scholar
Legendre, A.-M. (1798 (An VI)): Essai sur la théorie des nombres. Duprat, Paris; Seconde edition. Courcier, Paris, 1808.10.5962/bhl.title.18546CrossRefGoogle Scholar
Legendre, A.-M. (1830): Théorie des nombres. Tome I et Tome II. Firmin Didot Frères, Paris.Google Scholar
Legendre, A.-M. and Jacobi, C. G. J. (1875): Correspondance mathématique entre Legendre et Jacobi. J. reine angew. Math., 80, 205279.Google Scholar
Lehmer, D. H. (1930): An extended theory of Lucas’ functions. Ann. Math., 31, 419443.10.2307/1968235CrossRefGoogle Scholar
Lehmer, D. H. (1932): Euler’s totient function. Bull. Amer. Math. Soc., 38, 745751.10.1090/S0002-9904-1932-05521-5CrossRefGoogle Scholar
Lehmer, D. N. (1914): List of prime numbers from 1 to 10, 006, 721. Carnegie Institution of Washington, Washington, D.C.Google Scholar
Levinson, N. (1974): More than one third of the zeros of Riemann’s zeta-function are on σ = 1 /2. Adv. Math., 13, 383436.Google Scholar
Lindelöf, E. (1908): Quelques remarques sur la croissance de la fonction ζ(s). Bull. Sci. Math., 32, 341356.Google Scholar
Linnik, U. V. (1941): The large sieve. C.R. Acad. Sci. URSS (N.S.), 30, 292294.Google Scholar
Linnik, U. V. (1943): On Weyl’s sums. Rec. Math. (Mat. Sbornik), 12, 2839.Google Scholar
Linnik, U. V. (1944): On the least prime in an arithmetic progression. I. The basic theorem. Rec. Math. (Mat. Sbornik), 5, 139178; II. The Deuring–Heilbronn phenomenon. 347–368.Google Scholar
Linnik, U. V. (1962): Additive problems and eigenvalues of the modular operators. Proc. Internat. Congress Math., Stockholm, pp.270284.Google Scholar
Linnik, U. V. (1963): Dispersion method in binary additive problems. Translations Math. Monographs. Vol. 4. Amer. Math. Soc., Providence.Google Scholar
Liouville, J. (1844): Remarques sur des classes trés etendues de quantités dont la valeur n’est ni rationnelle ni même réductible à des irrationnelles algébriques. Comptes Rendus, 18, 883885.Google Scholar
Lipschitz, R. (1857): Einige Sätze aus der Theorie der quadratischen Fromen. J. reine angew. Math., 53, 238259.Google Scholar
Littlewood, J.-E. (1914): Sur la distribution des nombres premiers. C.R. Acad. Sci., 158, 18691872.Google Scholar
Littlewood, J.-E. (1924): On the zeros of Riemann’s zeta function. Proc. Camb. Phil. Soc., 22, 295318.10.1017/S0305004100014225CrossRefGoogle Scholar
Lucas, E. (1867): Application de l’arithmétique a la construction de l’armure des satins réguliers. G. Retaux, Paris.Google Scholar
Littlewood, J.-E. (1878a): Théorèmes d’arithmétique. Atti della Reale Accademia delle Scienze di Torino, 13, 271284.Google Scholar
Littlewood, J.-E. (1878b): Théorie des fonctions numériques simplement périodiques. Amer. J. Math., 1, 184321.Google Scholar
Littlewood, J.-E. (1891): Théorie des nombres. Tome premier. Gauthier–Villars et Fils, Paris.Google Scholar
von Mangoldt, H. (1895): Zu Riemanns Abhandlung “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse”. J. reine angew. Math., 114, 255305.Google Scholar
Mathews, G. B. (1892): Theory of numbers, Part I. Deighton, Bell and Co., Cambridge.Google Scholar
Matthews, C. R. (1979): Gauss sums and elliptic functions. I. Invent. math., 52, 163185; II. 54, 23–52.10.1007/BF01403063CrossRefGoogle Scholar
Matthews, K. R. (2002): The Diophantine equation ax2 + bxy + cy2 = N, D = b2 — 4ac > 0. J. Théorie des Nombres de Bordeaux, 14, 257270.10.5802/jtnb.358CrossRefGoogle Scholar
Märcker, G. (1840): Ueber Primzahlen. J. reine angew. Math., 20, 350359.Google Scholar
Maynard, J. (2013): Small gaps between primes. arXiv:1311.4600v2. (Annals of Math., 181 (2015), 383413)Google Scholar
McCurley, K. S. (1984): Prime values of polynomials and irreducibility testing. Bull. Amer. Math. Soc., 11, 155158.10.1090/S0273-0979-1984-15247-9CrossRefGoogle Scholar
Meissner, W. (1913): Über die Teilbarkeit von 2p – 2 durch das Quadrat der Primzahl p = 1093. Sitz. König. Preuss. Akad. Wiss. Akad. Berlin, J. 1913, 663667.Google Scholar
Mersenne, F. M. (1636): Harmonicorum libri. G. Baudy, Lutetiæ Parisiorum.Google Scholar
Mersenne, F M. (1644): Cogitata physico mathematica. A. Bertier, Parisiis.Google Scholar
Mertens, F. (1874): Ein Beitrag zur analytischen Zahlenthéorie. J. reine angew. Math., 78, 4662.Google Scholar
Mertens, F. (1896): Über die Gaussischen Summen. Sitz. König. Preuss. Akad. Wiss. Akad. Berlin, 217219.Google Scholar
Mertens, F. (1897a): Über eine zahltheoretische Aufgabe. Sitz. Kaiser. Akad. Wiss. Wien, math.-natur. Classe, 106-2a, 132133.Google Scholar
Mertens, F. (1897b): Über eine zahlentheoretische Funktion. Sitz. Kaiser. Akad. Wiss. Wien, math.-natur. Classe, 106-2a, 761830.Google Scholar
Meurman, T. (1992): A simple proof of Voronoi’s identity. Astérisque, 209, 265274.Google Scholar
Michel, P. (2022): Recent progresses on the subconvexity problem. Sémi. Bourbaki, 2021é2022, no1190.Google Scholar
Milnor, J. (1982): Hyperbolic geometry: The first 150 years. Bull. Amer. Math. Soc., 6, 924.10.1090/S0273-0979-1982-14958-8CrossRefGoogle Scholar
Monier, L. (1980): Evaluation and comparison of two efficient probabilistic primality testing algorithms. Theoret. Comput. Sci., 12, 97108.10.1016/0304-3975(80)90007-9CrossRefGoogle Scholar
Montgomery, H. L. (1968): A note on the large sieve. J. London Math. Soc., 43, 9398.Google Scholar
Montgomery, H. L. (1969a): Mean and large values of Dirichlet polynomials. Invent. math., 8, 334345. (1969b): Zeros of L-functions. Invent. math., 8, 346–354.10.1007/BF01404637CrossRefGoogle Scholar
Montgomery, H. L. (1971): Topics in multiplicative number theory. Lect. Notes in Math., 227, Springer-Verlag, Berlin.10.1007/BFb0060851CrossRefGoogle Scholar
Montgomery, H. L. (1978): The analytic principle of the large sieve. Bull. Amer. Math. Soc., 84, 547567.10.1090/S0002-9904-1978-14497-8CrossRefGoogle Scholar
Montgomery, H. L. and Vaughan, R. C. (1973): The large sieve. Mathematika, 20, 119134.10.1112/S0025579300004708CrossRefGoogle Scholar
Montgomery, H. L. and Vaughan, R. C. (1974): Hilbert’s inequality. J. London Math. Soc., 8, 7382.10.1112/jlms/s2-8.1.73CrossRefGoogle Scholar
Montgomery, H. L. and Vaughan, R. C. (2006): Multiplicative number theory. I. Cambridge Univ. Press, Cambridge.10.1017/CBO9780511618314CrossRefGoogle Scholar
Mordell, L. J. (1961): The congruence (p – 1/2)! ≡ ± 1 (modp) [sic]. Amer. Math. Monthly, 68, 145146.10.2307/2312481CrossRefGoogle Scholar
Morrison, M. A. and Brillhart, J. (1975): A method of factoring and the factorization of F7. Math. Comp., 29, 183205.Google Scholar
Motohashi, Y. (1974): On some improvements of the Brun–Titchmarsh theorem. J. Math. Soc. Japan, 26, 306323.10.2969/jmsj/02620306CrossRefGoogle Scholar
Motohashi, Y. (1975): On a density theorem of Linnik. Proc. Japan Acad., 51, 815817.10.2183/pjab1945.51.Supplemnt_815CrossRefGoogle Scholar
Motohashi, Y. (1976): An induction principle for the generalization of Bombieri’s prime number theorem. Proc. Japan Acad., 52, 273275.Google Scholar
Motohashi, Y. (1977a): On the Deuring–Heilbronn phenomenon. Part I. Proc. Japan Acad., 53, 12; Part II. 2527.Google Scholar
Motohashi, Y. (1977b). A note on the large sieve. Proc. Japan Acad., 53, 1719; Part II. 122124.Google Scholar
Motohashi, Y. (1978): Primes in arithmetic progressions. Invent. math., 44, 163178.10.1007/BF01390349CrossRefGoogle Scholar
Motohashi, Y. (1983): Sieve methods and prime number theory. Tata Inst. Fund. Res. Lect. Math. Phy., 72, Tata IFR, Bombay.Google Scholar
Motohashi, Y. (1987): Riemann–Siegel Formula. Lect. Notes, Ulam Chair Seminar, Colorado Univ., Boulder.Google Scholar
Motohashi, Y. (1993): An explicit formula for the fourth power mean of the Riemann zeta-function. Acta Math., 170, 181220.10.1007/BF02392785CrossRefGoogle Scholar
Motohashi, Y. (1994): The binary additive divisor problem. Ann. Sci. École Norm. Sup., 4e ser., 27, 529572.10.24033/asens.1700CrossRefGoogle Scholar
Motohashi, Y. (1997): Spectral theory of the Riemann zeta-function. Cambridge Tracts in Math., 127, Cambridge Univ. Press, Cambridge.10.1017/CBO9780511983399CrossRefGoogle Scholar
Motohashi, Y. (1999): On the error term in the Selberg sieve. In: Number Theory in Progress. A. Schinzel Festschrift. Walter de Gruyter, Berlin, pp.10531064.10.1515/9783110285581.1053CrossRefGoogle Scholar
Motohashi, Y. (2007). Sums of Kloosterman sums revisited. In: The Conference on L-Functions, Fukuoka 2006, World Scientific, Singapore, pp.141163.Google Scholar
Motohashi, Y. (2015): On sums of Hecke–Maass eigenvalues squared over primes in short intervals. J. London Math. Soc., 91, 367382.10.1112/jlms/jdu079CrossRefGoogle Scholar
Motohashi, Y. and Pintz, J. (2006): A smoothed GPY sieve. arXiv: 0602599v2. (Bull. London Math. Soc., 40 (2008), 298310)10.1112/blms/bdn023CrossRefGoogle Scholar
Möbius, A. F. (1832): Über eine besondere Art von Umkehrung der Reihen. J. reine angew. Math., 9, 105123.Google Scholar
Murty, M. R. (1988): Primes in certain arithmetic progressions. J. Madras Univ., Section B, 51, 161169. Included in: Murty, M. R. and Thain, N. (2006): Prime numbers in certain arithmetic progressions. Functiones et Approximatio, 35 (2006), 249–259.Google Scholar
Nair, M. and Tenenbaum, G. (1998): Short sums of certain arithmetic functions. Acta Math., 180, 119144.10.1007/BF02392880CrossRefGoogle Scholar
Nemet-Nejat, K. R. (2002): Daily life in ancient Mesopotamia. Hendrickson Publishers, Inc., Peabody.Google Scholar
Nesselmann, G. H. F. (1842): Die Algebra der Griechen. Verlag von G. Reimer, Berlin.Google Scholar
Nicomachus (ca 100 CE): Arithmetike eisagoge. (A.M.T.S. Boetii (Boethius) (ca 500): De institutione arithmetica libri duo, De institutione musica libri quinque. Teubner B.G., Lipsiæ 1867; D’Ooge, M. L. (1926): Introduction to arithmetic. Macmillan, New York and London)Google Scholar
Niven, I., Zuckerman, H. S., and Montgomery, H. L. (1991): An introduction to the theory of numbers. Fifth edition. John Wiley & Sons, New York.Google Scholar
Odlyzko, A. M. and te Riele, H. J. J. (1984): Disproof of the Mertens conjecture. J. reine angew. Math., 357, 138160.Google Scholar
Onishi, H. (1973): A Tauberian theorem on Dirichlet séries. J. Number Theory, 5, 5557.10.1016/0022-314X(73)90057-7CrossRefGoogle Scholar
Page, A. (1935): On the number of primes in an arithmetic progression. Proc. London Math. Soc., 39, 116141.10.1112/plms/s2-39.1.116CrossRefGoogle Scholar
Pengelley, D. and Richman, F. (2006): Did Euclid need the Euclidean algorithm to prove unique factorization? Amer. Math. Monthly, 113, 196205.10.1080/00029890.2006.11920298CrossRefGoogle Scholar
Penrose, R. (2007): The road to reality. Vintage Books, New York.Google Scholar
Pepin, T. (1877): Sur la formule 22n +1. Comptes Rendus Acad. Sci. Paris, 85, 329333.Google Scholar
Perron, O. (1908): Zur Theorie der Dirichletschen Reihen. J. reine angew. Math., 134, 95143.10.1515/crll.1908.134.95CrossRefGoogle Scholar
Pieper, H. (1997): Über Legendres Versuche, das der quadratische Reziprozitätsgesetz zu beweisen. Acta hist. Leopoldina, 27, 223237.Google Scholar
Platt, D. J. and Trudgian, T. S. (2016): Zeros of partial sums of the zeta-function. London Math. Soc., J. Comput. Math., 19, 3741.Google Scholar
Secundus, Plinius G. (ca 77 CE): Libros Naturalis Historiae. Johannes de Spira, Venezia, 1469.Google Scholar
Poincaré, H. (1880): Sur un mode nouveau de représentation géométrique des forms quadratiques définies ou indéfinies. J. l’École Polyt., 28, 177245.Google Scholar
Poinsot, L. (1824): Mémoire sur l’applications de l’algébre à la théorie des nombres. Mémoire de l’Acad. Roy. Sci. l’Inst. France (année 1819 et 1820), 4, 99183.Google Scholar
Poinsot, L. (1845): Réflexions sur les principes fondamentaux de la théorie des nombres. J. math. pures et appliq., 10, 193.Google Scholar
Poisson, S. D. (1827): Mémoire sur le calcul numérique des intégrales définies. Mémoire de l’Acad. Sci. l’Inst. France (annee 1823), 6, 571602.Google Scholar
Pollard, J. M. (1974): Theorems on factorization and primality testing. Math. Proc. Cambridge Phil. Soc., 76, 521528.10.1017/S0305004100049252CrossRefGoogle Scholar
Pollard, J. M. (1975): A Monte Carlo method for factorization. BIT, 15, 331334.10.1007/BF01933667CrossRefGoogle Scholar
Pollard, J. M. (1978): Monte Carlo methods for index computation (modp). Math. Comp., 32, 918924.Google Scholar
Polymath, D. H. J. (2014a): New equidistribution estimates of Zhang type. arXiv:1402.0811v3.Google Scholar
Polymath, D. H. J. (2014b): Variants of the Selberg sieve, and bounded intervals containing many primes. arXiv:1407.4897v4.10.1186/s40687-014-0012-7CrossRefGoogle Scholar
Pomerance, C. (1985): The quadratic sieve factoring algorithm. Lect. Notes in Comput. Sci., 209, 169182.10.1007/3-540-39757-4_17CrossRefGoogle Scholar
Pomerance, C., Selfridge, J. L., and Wagstaff, S. S. Jr. (1980): The pseudoprimes to 25 ·; 109. Math. Comp., 35, 10031026.Google Scholar
Pracher, K. (1957): Primzahlverteilung. Springer Verlag, Berlin.Google Scholar
Prestet, J. (1689): Nouvaux élémens des mathématiques. Premier et Second vol. A. Pralard, Paris.Google Scholar
Rabin, M. O. (1980): Probabilistic algorithm for testing primality. J. Number Theory, 12, 128138.10.1016/0022-314X(80)90084-0CrossRefGoogle Scholar
Rabinowicz, G. (1913): Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern. J. reine angew. Math., 142, 153164.Google Scholar
Ramachandra, K. (1974): A simple proof of the mean fourth power estimate for and . Ann. Scuola Norm. Sup. Pisa, (4) 1, 8197.Google Scholar
Ramanujan, S. (1916): Some formulae in the analytic theory of numbers. Mess. Math., 45, 8184.Google Scholar
Ramanujan, S. (1918): On certain trigonometrical sums and their applications in the theory of numbers. Trans. Cambridge Phil. Soc., 22, 259276.Google Scholar
Rashed, R. (1980): Ibn al-Haytham et le thérorème de Wilson. Arch. History of Exact Sci., 22, 305315.10.1007/BF00717654CrossRefGoogle Scholar
Rényi, A. (1948): On the representation of an even number as the sum of a prime and an almost prime. Izv. Akad. Nauk SSSR Ser. Mat., 12, 5778. (Russian)Google Scholar
Riemann, B. (1860): Über die Anzahl der Primzahlen under einer gegebenen Grösse. Monatsber. Königl. Preuss. Akad. Wiss. Berlin, J., 1859, 671680.Google Scholar
Rivest, R. L., Shamir, A., and Adleman, L. (1978): A method of obtaining digital signatures and public-key cryptosystems. Comm. Assoc. Comput. Mach., 21, 120126.Google Scholar
Rodosskii, K. A. (1954): On the least prime number in an arithmetic progression. Mat. Sb. (N.S.), 34, 331356. (Russian)Google Scholar
Rogers, K. (1974): Legendre’s theorem and quadratic reciprocity. J. Number Theory, 6, 339344.10.1016/0022-314X(74)90030-4CrossRefGoogle Scholar
Roth, K. F. (1955): Rational approximations to algebraic numbers. Mathematika, 2, 120.10.1112/S0025579300000644CrossRefGoogle Scholar
Sardi, C. (1869): Teoremi di aritmetica. Giornale di matematiche di Battaglini, 7, 2427.Google Scholar
Sarnak, P. (1982): Class numbers of indefinite quadratic forms. J. Number Theory, 15, 229247.10.1016/0022-314X(82)90028-2CrossRefGoogle Scholar
Schaar, M. (1850): Recherches sur la théorie des résidus quadratiques. Mém. Acad. Roy. Sci. Lettres et Beaux Arts Belgique, 25, 20 pp.Google Scholar
Schlesinger, L. (1912): Über Gauss’ Arbeiten zur Funktionenthéorie. (Gauss Werke X-2, Abhandlung 2)Google Scholar
Schmidt, W. M. (2004): Equations over finite fields. An elementary approach. Second edition. Kendrick Press, Herber City, UT.Google Scholar
Schoof, R. (1985): Elliptic curves over finite fields and the computation of square roots mod p. Math. Computation, 44, 483494.Google Scholar
Schönemann, T. (1846a): Grundzüge einer allgemeinen Theorie der höhern Cogruenzen, deren Modul eine reelle Primzahl ist. J. reine angew. Math., 31, 269325; Von denjenigen Moduln, welche Potenzen von Primzahlen sind. 32 (1846b), 93–105.Google Scholar
Schur, I. (1921): Über die Gaussschen Summen. Nachr Gesell. Wiss. Göttingen, Math.-Phys. Kl., 147153.Google Scholar
Schur, I. (1929): Zur Irreduzibilität der Kreisteilungsgleichung. Math. Z., 29, 463.10.1007/BF01180544CrossRefGoogle Scholar
Selberg, A. (1942): On the zeros of Riemann’s zeta-function. Skr. Norske Vid. Akad. Oslo, No. 10, 159. (Collected papers I, pp.85141)Google Scholar
Selberg, A. (1943): On the normal density of primes in small intervals, and the difference between consecutive primes. Arkiv for Math. og Naturv., 47, No.6, 87105. (Collected papers I, pp.160–178)Google Scholar
Selberg, A. (1946a): Contribution to the theory of the Riemann zeta-function. Arch. för Math. og Naturv., 48, 89155. (Collected papers I, pp.214–280)Google Scholar
Selberg, A. (1946b): Contributions to the theory of Dirichlet’s L-functions. Skr. Norske Vid. Akad. Oslo, No. 3, 162. (Collected papers I, pp.281–340)Google Scholar
Selberg, A. (1946c): The zeta-function and the Riemann hypothesis. 10th. Skand. Math. Köngr., 187200. (Collected papers I, pp.341–355)Google Scholar
Selberg, A. (1947): On an elementary method in the theory of primes. Det Kong. Norske Vid. Selsk. Forh., Trondhjem, 19, 6467. (Collected papers I, pp.363–366)Google Scholar
Selberg, A. (1949): On elementary methods in prime number theory and their limitations. 11th. Skand. Math. Kongr., Trondhjem, 1322. (Collected papers I, pp.388–397)Google Scholar
Selberg, A. (1950a): An elementary proof of the prime-number theorem for arithmetic progressions. Canadian J. Math., 2, 6678. (Collected papers I, pp.398–410)10.4153/CJM-1950-007-5CrossRefGoogle Scholar
Selberg, A. (1950b): The general sieve-method and its place in prime number theory. Proc. Intern. Cong. Math., Cambridge, Mass., 1, 286292. (Collected papers I, pp.411–417)Google Scholar
Selberg, A. (1956): Harmonic analysis and discontinuous groups in weakly symmetric Rieman-nian spaces with applications to Dirichlet séries. J. Indian Math. Soc., 20, 4787. (Collected papers I, pp.423–463)Google Scholar
Selberg, A. (1965): On the estimation of Fourier coefficients of modular forms. Proc. Symp. Pure Math., 8, 115. (Collected papers I, pp.506–520)10.1090/pspum/008/0182610CrossRefGoogle Scholar
Selberg, A. (1972): Remarks on sieves. Proc. 1972 Number Theory Conf., Boulder, pp. 205216. (Collected papers I, pp.609–615)Google Scholar
Selberg, A. (1977): Remarks on multiplicative functions. Springer Lect. Notes in Math., 626, 232241. (Collected papers I, pp.616–625)10.1007/BFb0063067CrossRefGoogle Scholar
Selberg, A. (1991): Lectures on sieves. In: Collected Papers, II, pp.65247.Google Scholar
Serret, J.-A. (1849a): Cours d’algébre supérieure. Bachelier, Paris; Deuxieme edition. 1854.Google Scholar
Serret, J.-A. (1849b): Sur un théorème relatif aux nombres entiers. J. math. pures et appliq., 13, 1214; Note de C. Hermite. p.15.Google Scholar
Shiu, P. (1980): A Brun–Titchmarsh theorem for multiplicative functions. J. reine angew. Math., 313, 161170.Google Scholar
Shor, P. W. (1994): Algorithms for quantum computation: Discrete logarithms and factoring. Proc. 35th Ann. Symp. Found. Comp. Sci., IEEE Comp. Soc. Press, pp.124134.10.1109/SFCS.1994.365700CrossRefGoogle Scholar
Shor, P. W. (1997): Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput., 26, 14841509.10.1137/S0097539795293172CrossRefGoogle Scholar
Siegel, C. L. (1932): Über Riemanns Nachlass zur analytischen Zahlenthéorie. Quellen und Studien zur Geschichte der Math. Astr. und Physik, Abt. B: Studien, 2, 4580.Google Scholar
Siegel, C. L. (1935): Über die Klassenzahl quadratischer Zahlkörper. Acta Arith., 1, 8386.10.4064/aa-1-1-83-86CrossRefGoogle Scholar
Siegel, C. L. (1957): Lectures on quadratic forms. Tata Inst. Fund. Res. Lect. Math. Phy., 7, TIER, Bombay.Google Scholar
Siegel, C. L. (1961): Lectures on advanced analytic number theory. Tata Inst. Fund. Res. Lect. Math. Phy., 23, TIFR, Bombay.Google Scholar
Simerka, W. (1858): Die Perioden der quadratischen Zahlformen bei negativen Determinanten. Sitzungsber Kaiserl. Akad. Wiss., Math.-Nat. Wiss., 31, 3367.Google Scholar
Simerka, W. (1885): Zbytky z arithmetické posloupnosti. Časopis pro péstování mathematiky a fysiky, 14, 221225.Google Scholar
Singh, P. (1985): The so-called Fibonacci numbers in ancient and medieval India. Historia Math., 12, 229244.10.1016/0315-0860(85)90021-7CrossRefGoogle Scholar
Skewes, S. (1933): On the difference π(x) – li(x). J. London Math. Soc., 8, 277283; II. 5 (1955), 48–70.10.1112/jlms/s1-8.4.277CrossRefGoogle Scholar
Smith, H. J. S. (1855): De compositione numerorum primorum formae 4λ +1 ex duobus quadratis. J. reine angew. Math., 50, 9192. (Collected papers I, pp.33–34)Google Scholar
Smith, H. J. S. (1859/1869): Report on the theory of numbers. (Collected papers I, pp.38364)Google Scholar
Smith, H. J. S. (1861): On systems of linear indeterminate equations and congruences. Phil. Trans. Royal Soc. London, 151, 293326. (Collected papers I, pp.367–409)Google Scholar
Smith, H. J. S. (1876): On the value of a certain arithmetical determinant. Proc. London Math. Soc., 7, 208212. (Collected papers II, pp.161–165)Google Scholar
Smith, H. J. S. (1877): Mémoire sur les équations modulaires. Abstract presented by Cremona: Atti della R. Accad. Lincei, Transunti, Ser. III, 1, 6869. (Collected papers II, p. 240)Google Scholar
Sorenson, J. and Webster, J. (2017): Strong pseudoprimes to twelve prime bases. Math. Computation, 86, 9851003.10.1090/mcom/3134CrossRefGoogle Scholar
Srinivasan, S.(1997). A weak Brun–Titchmarsh theorem for multiplicative functions. Proc. Indian Acad. Sci., (Math. Sci.), 107, 387389.10.1007/BF02837222CrossRefGoogle Scholar
Stepanov, S. A. (1969): On the number of points of a hyperelliptic curve over a finite prime field. Izv. Akad. Nauk SSSR, ser. Mat., 33, 11711181. (Russian)Google Scholar
Stepanov, S. A. (1994): Arithmetic of algebraic curves. Consultants Bureau, New York and London.Google Scholar
Stevin, S. (1625): L’arithmetique de Simon Stevin de Bruges. Annotations par A. Girard. Elzeviers, Leide.Google Scholar
Stieltjes, T.-J. (1894): Recherches sur les fractions continues. Ann. Facult. Sci. Toulouse, 8, no. 4, 1122; 9, no. 1 (1895), 5–47.10.5802/afst.108CrossRefGoogle Scholar
Sylvester, J. J. (1879): On certain ternary cubic-form equations. American J. Math., 2, 357393.10.2307/2369490CrossRefGoogle Scholar
Takhtajan, L. A. and Vinogradov, A. I. (1982): The Gauss–Hasse hypothesis on real quadratic fields with class number one. J. reine angew. Math., 335, 4086.Google Scholar
Tatuzawa, T. (1951): On a theorem of Siegel. Japanese J. Math., 21, 163178.10.4099/jjm1924.21.0_163CrossRefGoogle Scholar
Thue, A. (1902): Et par antydninger til en taltheoretisk metode. Kra. Vidensk. Selsk. Skrifter. I. Mat. Nat. Kl., 7, 5775. (Selected mathematical papers of Axel Thue. Universitetsforlaget, Oslo, 1977, pp.57–75).Google Scholar
Titchmarsh, E. C. (1951): The theory of the Riemann zeta-function. Oxford Univ. Press, Oxford.Google Scholar
Toffoli, T. (1980): Reversible computing. Technical Report, MIT Laboratory for computing science, TM-51.10.21236/ADA082021CrossRefGoogle Scholar
Tonelli, A. (1891): Bemerkung über die Auflösung quadratischer Congruenzen. Nachrichten Königl. Gesell. Wiss. Georg-Augusts-Univ. Göttingen, 344346.Google Scholar
Tsirelson, B. (1997): Quantum information processing. Lecture notes. Tel Aviv Univ., Tel Aviv.Google Scholar
Turán, P (1953): Eine Neue Method in der Analysis und deren Anwendungen. Akademiai Kiado, Budapest.Google Scholar
Turan, P (1961): On a density theorem of Yu.V. Linnik. Magyár Tud. Akad. Mat. Kutató Intz. (= Publ. Math. Inst. Hung. Acad. Sci.), (2) 6, 165179.Google Scholar
van, der Corput J. G. (1921): Zhalentheoretische Abschátzungen. Math. Ann., 84, 5379.Google Scholar
Vandermonde, A.-T. (1774): Mémoire sur la résolution des équations. Histoire de l’Académie Royale des Sciences, anné 1771, 365416.Google Scholar
Vaughan, R. C. (1977): Sommes trigonometriques sur les nombres premiers. C. R. Acad. Sci. Paris, Sér. A, 258, 981983.Google Scholar
Vinogradov, A. I. (1965). The density hypothesis for Dirichlet L-séries. Izv. Akad. Nauk SSSR Ser. Mat., 29, 903934; Corrigendum. 30 (1966), 719–720. (Russian)Google Scholar
Vinogradov, I. M. (1935): On Weyl’s sums. Rec. Math., (Mat. Sbornik), 42, 521530.Google Scholar
Vinogradov, I. M. (1936a): A new method of resolving of certain general questions of the theory of numbers. Rec. Math., (Mat. Sbornik), 43, 920.Google Scholar
Vinogradov, I. M. (1936b): A new method of estimation of trigonometrical sums. Rec. Math., (Mat. Sbornik), 43, 175188.Google Scholar
Vinogradov, I. M. (1937): A new method in analytic number theory. Akad. Nauk SSSR, Leningrad and Moscow. (Russian)Google Scholar
Vinogradov, I. M. (1958): A new estimate of the function ζ(1 + it). Izv. Akad. Nauk SSSR, Ser. Mat., 22, 161164. (Russian)Google Scholar
Voronin, S. M. (1976): The zeros of zeta-functions of quadratic forms. Trudy Mat. Inst. Steklov, 142, 135147. (Russian)Google Scholar
Voronoï, G. F. (1904): Sur une fonction transcendante et ses applications á la sommation de quelques séries. Ann. Écloe Norm., 21, 207267, 459–533.10.24033/asens.539CrossRefGoogle Scholar
Walfisz, A. (1936): Zur additiven Zahlenthéorie. II. Math. Z., 40, 592607.10.1007/BF01218882CrossRefGoogle Scholar
Wallis, J. (1656): Arithmetica infinitorum. L. Lichfield, Oxonii.Google Scholar
Wallis, J. (1685): A treatise of algebra both historical and practical. Shewing, the original, progress, and advancement thereof, from time to time; and by what steps it hath attained to the height at which now it is. Printed by J. Playford for R. Davis, London.Google Scholar
Waring, E. (1782): Meditationes algebricæ. J. Archdeacon, Cantabrigiæ.Google Scholar
Watt, N. (1995): Kloosterman sums and a mean value for Dirichlet polynomials. J. Number Theory, 53, 179210.10.1006/jnth.1995.1086CrossRefGoogle Scholar
Weber, H. (1882): Beweis des Sätzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen faïhig ist. Math. Ann., 20, 301329.10.1007/BF01443599CrossRefGoogle Scholar
Weber, H. (1895): Lehrbuch der Algebra. Erster Bd. Friedrich Vieweg und Sohn, Braunschweig; Zweite Bd. 1896; Dritter Bd. 1908.Google Scholar
Weinberger, P. J. (1973): Exponents of the class groups of complex quadratic fields. Acta Arith., 22, 117124.10.4064/aa-22-2-117-124CrossRefGoogle Scholar
Weyl, H. (1916): Über die Gleichverteilung von Zahlen mod.Eins. Math. Ann., 77, 313352.10.1007/BF01475864CrossRefGoogle Scholar
Weyl, H. (1921): Zur Abschätzung von ζ(1 + it). Math. Zeitt., 10, 88101.10.1007/BF02102307CrossRefGoogle Scholar
Whittaker, E. T. and Watson, G. N. (1969): A course of modern analysis. Cambridge Univ. Press, Cambridge.Google Scholar
Wieferich, A. (1909): Zum letzten Fermatschen Theorem. J. reine angew. Math., 136, 293302.10.1515/crll.1909.136.293CrossRefGoogle Scholar
Wiener, M. J. (1990): Cryptanalysis of short RSA secret exponents. IEEE Trans. Information Theory, 36, 553558.10.1109/18.54902CrossRefGoogle Scholar
Wolke, D. (1973): Über die mittlere Verteilung der Werte zahlertheoretischer Funktionen auf Restklassen. I. Math. Annalen, 202, 125.10.1007/BF01351202CrossRefGoogle Scholar
Wolstenholme, J. (1862): On certain properties of prime numbers. Q. J. Pure Appl. Math., 5, 3539.Google Scholar
Xylouris, T. (2011): Über die Nullstellen der Dirichletschen L-Funktionen und die kleinste Primzahl in einer arithmetischen Progression. Dissertation. Rheinischen Friedrich–Wilhelms–Universitat, Bonn.Google Scholar
Zhang, Y. (2013): Bounded gaps between primes. Draft. (Annals of Math., 179 (2014), 11211174)Google Scholar

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  • Bibliography
  • Yoichi Motohashi, Finnish Academy of Science and Letters
  • Book: Essays in Classical Number Theory
  • Online publication: 12 August 2025
  • Chapter DOI: https://doi.org/10.1017/9781009504522.007
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  • Bibliography
  • Yoichi Motohashi, Finnish Academy of Science and Letters
  • Book: Essays in Classical Number Theory
  • Online publication: 12 August 2025
  • Chapter DOI: https://doi.org/10.1017/9781009504522.007
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  • Bibliography
  • Yoichi Motohashi, Finnish Academy of Science and Letters
  • Book: Essays in Classical Number Theory
  • Online publication: 12 August 2025
  • Chapter DOI: https://doi.org/10.1017/9781009504522.007
Available formats
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