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Scalar Products of Certain Hecke L-Series and Moments of Weighted Norm-Counting Functions

Published online by Cambridge University Press:  20 November 2018

R. W. K. Odoni
R. W. K. Odoni*
Affiliation:
Department of Mathematics, University of Exeter Exeter Ex4 4qe, England
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Abstract

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We consider Dirichlet series R(s), constructed by taking scalar products of Hecke L-series with ray-class characters. Using a theorem of G. W. Mackey on tensor products of representations of finite groups we show that R(s) has a meromorphic continuation into Re(s) > 1/2 (obtained by more sophisticated methods in [l]-[5]); we then obtain estimates for the growth of R(s) on vertical lines. Via the Mellin transformation we deduce asymptotics for various weighted moment sums involving ideals of given ray-class and norm, in one or several fields simultaneously.

Keywords

12A7012A55

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 3 , 01 September 1985 , pp. 272 - 279
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Chandrasekharan, K. and Good, A., On the number of integral ideals in Galois extensions, Monatshefte fur Math. 95 (1983), pp. 99–109.Google Scholar
2. Chandrasekharan, K. and Narasimhan, R., The approximate functional equation for a class of zeta functions, Math. Annalen 152 (1963), pp. 30–64.Google Scholar
3. Curtis, C.W., and Reiner, I., Representation theory of finite groups and associative algebras, (Wiley-Interscience, New York, 1962).Google Scholar
4. Draxl, P.K.J., L-Funktionen algebraischer Tori, J. Number Theory 3 (1971), pp. 444–467.Google Scholar
5. Kurokawa, N., On the meromorphic continuation of Euler products — I-Artin type, (preprint) Tokyo Inst. Techn., 1977.Google Scholar
6. Kurokawa, N., On the meromorphy of Euler products, Proc. Japan Acad. Sci. 54A (1978), pp. 193–196.Google Scholar
7. Lang, S., Algebraic Number Theory, Addison-Wesley, Reading, Mass. 1970.Google Scholar
8. Moroz, B.Z., On the convolution of Hecke L-series, Mathematika 27 (1980), pp. 312–320.Google Scholar
9. Moroz, B.Z., Scalar products of L-functions with Grössencharacters … , J. für reine und angew. Math. 332(1982), pp. 99–117.Google Scholar
10. Moroz, B.Z., Three lectures on the distribution of integral and prime divisors, (preprint), Univ. de Paris-Sud, Orsay, 1983.Google Scholar
11. Narkiewicz, W., Elementary and analytic theory of algebraic numbers (PWN-Warsaw, 1974), p. 317.Google Scholar
12. Odoni, R.W.K., Representations of algebraic integers by binary quadratic forms …, J. Number Theory 10 (1978), pp. 324–333.Google Scholar
13. Serre, J.-P., Divisibilité de certaines fonctions arithmétiques, L'Enseignement Math. 22 (1976), pp. 227–260.Google Scholar