Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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REMARKS OF CONGRUENT ARITHMETIC SUMS OF THETA FUNCTIONS DERIVED FROM DIVISOR FUNCTIONS

  • Kim, Aeran (Department of Mathematics and Institute of Pure and Applied Mathematics, Chonbuk National University) ;
  • Kim, Daeyeoul (National Institute for Mathematical Sciences) ;
  • Ikikardes, Nazli Yildiz (Department of Elementary Mathematics Education, Necatibey Faculty of Education, Balikesir University)
  • Received : 2013.03.20
  • Accepted : 2013.06.11
  • Published : 2013.09.25
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Abstract

In this paper, we study a distinction the two generating functions : ${\varphi}^k(q)=\sum_{n=0}^{\infty}r_k(n)q^n$ and ${\varphi}^{*,k}(q)={\varphi}^k(q)-{\varphi}^k(q^2)$ ($k$ = 2, 4, 6, 8, 10, 12, 16), where $r_k(n)$ is the number of representations of $n$ as the sum of $k$ squares. We also obtain some congruences of representation numbers and divisor function.

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References

  1. N. Cheng and K. S. Williams, Evaluation of some convolution sums involving the sum of divisors functions, Yokohama Mathematical J., 52 (2005), 39-57.
  2. J. G. Huard, Z. M. Ou, B. K. Spearman, and K. S. Williams, Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions, Number theory for the millennium, II, (2002), 229-274.
  3. K. S. Williams, Number Theory in the Spirit of Liouville, London Mathematical Society, Student Texts 76, Cambridge, (2011).

Cited by

  1. Bernoulli numbers and certain convolution sums with divisor functions vol.2013, pp.1, 2013, https://doi.org/10.1186/1687-1847-2013-277
  2. THE RELATION PROPERTY BETWEEN THE DIVISOR FUNCTION AND INFINITE PRODUCT SUMS vol.38, pp.3, 2016, https://doi.org/10.5831/HMJ.2016.38.3.507