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3 results

  • Generalized divisor functions in arithmetic progressions: II

  • Part of
  • D. T. Nguyen
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 67 / Issue 1 / February 2024
    Published online by Cambridge University Press:
    16 November 2023, pp. 28-56
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  • We obtain a new bound on the second moment of modified shifted convolutions of the generalized threefold divisor function and show that, for applications, the modified version is sufficient.

  • The Convolution Sum Σm<n/16σ(m)σ(n – 16m)

  • Ayşe Alaca, Şaban Alaca, Kenneth S. Williams
  • Journal:
    Canadian Mathematical Bulletin / Volume 51 / Issue 1 / 01 March 2008
    Published online by Cambridge University Press:
    20 November 2018, pp. 3-14
    Print publication:
    01 March 2008
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  • The convolution sum $\sum{_{m<n/16}\,\sigma (m)\sigma (n\,}-16m)$ is evaluated for all $n\,\in \,\mathbb{N}$ . This evaluation is used to determine the number of representations of $n$ by the quadratic form $x_{1}^{2}\,+\,x_{2}^{2}\,+\,x_{3}^{2}\,+\,x_{4}^{2}\,+\,4x_{5}^{2}\,+\,4x_{6}^{2}\,+\,4x_{7}^{2}\,+\,4x_{8}^{2}$ .