CONVOLUTION SUMS OF ODD AND EVEN DIVISOR FUNCTIONS

• Journal title : Honam Mathematical Journal
• Volume 35, Issue 3,  2013, pp.445-506
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2013.35.3.445
Title & Authors
CONVOLUTION SUMS OF ODD AND EVEN DIVISOR FUNCTIONS
Kim, Daeyeoul;

Abstract
Let $\small{{\sigma}_s(N)}$ denote the sum of the s-th power of the positive divisors of N and $\small{{\sigma}_{s,r}(N;m)={\sum_{d{\mid}N\\d{\equiv}r\;mod\;m}}\;d^s}$ with $\small{N,m,r,s,d{\in}\mathbb{Z}}$, $\small{d,s}$ > 0 and $\small{r{\geq}0}$. In a celebrated paper [33], Ramanuja proved $\small{\sum_{k=1}^{N-1}{\sigma}_1(k){\sigma}_1(N-k)=\frac{5}{12}{\sigma}_3(N)+\frac{1}{12}{\sigma}_1(N)-\frac{6}{12}N{\sigma}_1(N)}$ using elementary arguments. The coefficients' relation in this identity ($\small{\frac{5}{12}+\frac{1}{12}-\frac{6}{12}=0}$) motivated us to write this article. In this article, we found the convolution sums $\small{\sum_{k}$$\small{&}$$\small{lt;N/m}{\sigma}_{1,i}(dk;2){\sigma}_{1,j}(N-mk;2)}$ for odd and even divisor functions with $\small{i,j=0,1}$, $\small{m=1,2,4}$, and $\small{d{\mid}m}$. If N is an odd positive integer, $\small{i,j=0,1}$, $\small{m=1,2,4}$, $\small{s=0,1,2}$, and $\small{d{\mid}m{\mid}2^s}$, then there exist $\small{u,a,b,c{\in}\mathbb{Z}}$ satisfying $\small{\sum_{k&}$ $\small{lt;2^sN/m}{\sigma}_{1,i}(dk;2){\sigma}_{1,j}(2^sN-mk;2)=\frac{1}{u}[a{\sigma}_3(N)+bN{\sigma}_1(N)+c{\sigma}_1(N)]}$ with $\small{a+b+c=0}$ and ($\small{u,a,b,c}$) = 1(Theorem 1.1). We also give an elementary problem (O) and solve special cases of them in (O) (Corollary 3.27).
Keywords
Divisor functions;Convolution sums;
Language
English
Cited by
1.
CONVOLUTION SUMS OF ODD AND EVEN DIVISOR FUNCTIONS II,;

호남수학학술지, 2015. vol.37. 2, pp.149-185
1.
CONVOLUTION SUMS OF ODD AND EVEN DIVISOR FUNCTIONS II, Honam Mathematical Journal, 2015, 37, 2, 149
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