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Czechoslovak Mathematical Journal

, Volume 59, Issue 3, pp 847–859 | Cite as

Evaluation of the sums \( \sum\limits_{\mathop {m = 1}\limits_{m \equiv a(\bmod 4)} }^{n - 1} {\sigma (m)\sigma (n - m)} \)

  • Ayşe Alaca
  • Şaban Alaca
  • Kenneth S. Williams
Article
  • 47 Downloads

Abstract

The convolution sum
$$ \sum\limits_{\mathop {m = 1}\limits_{m \equiv a(\bmod 4)} }^{n - 1} {\sigma (m)\sigma (n - m)} $$
is evaluated for a ∈ {0, 1, 2, 3} and all n ∈ ℕ. This completes the partial evaluation given in the paper of J.G. Huard, Z.M. Ou, B.K. Spearman, K. S. Williams.

Keywords

convolution sums sum of divisors function theta functions 

MSC 2000

11A25 11F27 

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References

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    A. Alaca, S. Alaca, K. S. Williams: Seven octonary quadratic form. Acta Arith. 135 (2008), 339–350.zbMATHCrossRefMathSciNetGoogle Scholar
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    N. Cheng: Convolution sums involving divisor functions. M.Sc. thesis. Carleton University, Ottawa, 2003.Google Scholar
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    J. G. Huard, Z. M. Ou, B. K. Spearman, K. S. Williams: Elementary evaluation of certain convolution sums involving divisor functions. Number Theory for the Millenium II (Urbana, IL, 2000). A.K. Peters, Natick, 2002, pp. 229–274.Google Scholar
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    K. S. Williams: The convolution sum \( \sum\limits_{m < n/8} {\sigma (m)\sigma (n - 8m)} \). Pac. J. Math. 228 (2006), 387–396.zbMATHCrossRefGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2009

Authors and Affiliations

  • Ayşe Alaca
    • 1
  • Şaban Alaca
    • 1
  • Kenneth S. Williams
    • 1
  1. 1.Centre for Research in Algebra and Number Theory, School of Mathematics and StatisticsCarleton UniversityOttawaCanada

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