대한수학회지 (Journal of the Korean Mathematical Society)

  • 제50권2호
  • /
  • Pages.331-360
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  • 2013
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  • 0304-9914(pISSN)
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  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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CONVOLUTION SUMS ARISING 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) ;
  • Yan, Li (Department of Applied Mathematics China Agriculture University)
  • 투고 : 2012.03.26
  • 발행 : 2013.03.01
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초록

Let ${\sigma}_s(N)$ denote the sum of the sth powers of the positive divisors of a positive integer N and let $\tilde{\sigma}_s(N)={\sum}_{d|N}(-1)^{d-1}d^s$ with $d$, N, and s positive integers. Hahn [12] proved that $$16\sum_{k. In this paper, we give a generalization of Hahn's result. Furthermore, we find the formula ${\sum}_{k=1}^{N-1}\tilde{\sigma}_1(2^{n-m}k)\tilde{\sigma}_3(2^nN-2^nk)$ for $m(0{\leq}m{\leq}n)$.

키워드

참고문헌

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피인용 문헌

  1. CONVOLUTION SUMS AND THEIR RELATIONS TO EISENSTEIN SERIES vol.50, pp.4, 2013, https://doi.org/10.4134/BKMS.2013.50.4.1389
  2. A REMARK OF ODD DIVISOR FUNCTIONS AND WEIERSTRASS ℘-FUNCTIONS vol.36, pp.1, 2014, https://doi.org/10.5831/HMJ.2014.36.1.55
  3. ON THE CONVOLUTION SUMS OF CERTAIN RESTRICTED DIVISOR FUNCTIONS vol.35, pp.2, 2013, https://doi.org/10.5831/HMJ.2013.35.2.251
  4. Combinatorial convolution sums derived from divisor functions and Faulhaber sums vol.49, pp.2, 2014, https://doi.org/10.3336/gm.49.2.09
  5. CONVOLUTION SUMS OF ODD AND EVEN DIVISOR FUNCTIONS vol.35, pp.3, 2013, https://doi.org/10.5831/HMJ.2013.35.3.445
  6. CERTAIN COMBINATORIC CONVOLUTION SUMS AND THEIR RELATIONS TO BERNOULLI AND EULER POLYNOMIALS vol.52, pp.3, 2015, https://doi.org/10.4134/JKMS.2015.52.3.537
  7. Convolution identities for twisted Eisenstein series and twisted divisor functions vol.2013, pp.1, 2013, https://doi.org/10.1186/1687-1812-2013-81
  8. Eisenstein series and their applications to some arithmetic identities and congruences vol.2013, pp.1, 2013, https://doi.org/10.1186/1687-1847-2013-84