Arithmetic convolution sums derived from eta quotients related to divisors of 6

Abstract

Abstract The aim of this paper is to find arithmetic convolution sums of some restricted divisor functions. When divisors of a certain natural number satisfy a suitable condition for modulo 12, those restricted divisor functions are expressed by the coefficients of certain eta quotients. The coefficients of eta quotients are expressed by the sine function and cosine function, and this fact is used to derive formulas for the convolution sums of restricted divisor functions and of the number of divisors. In the sine function used to find the coefficients of eta quotients, the result is obtained by utilizing a feature with symmetry between the divisor and the corresponding divisor. Let N , r N,r be positive integers and d d be a positive divisor of N N . Let e r ( N ; 12 ) {e}_{r}\left(N;\hspace{0.33em}12) denote the difference between the number of 2 N d − d \frac{2N}{d}-d congruent to r r modulo 12 and the number of those congruent to − r -r modulo 12. The main results of this article are to find the arithmetic convolution identities for ∑ a 1 + ⋯ + a j = N ( ∏ i = 1 j e ˆ ( a i ) ) {\sum }_{{a}_{1}+\cdots +{a}_{j}=N}({\prod }_{i=1}^{j}\hat{e}\left({a}_{i})) with e ˆ ( a i ) = e 1 ( a i ; 12 ) + 2 e 3 ( a i ; 12 ) + e 5 ( a i ; 12 ) \hat{e}\left({a}_{i})={e}_{1}\left({a}_{i};\hspace{0.33em}12)+2{e}_{3}\left({a}_{i};\hspace{0.33em}12)+{e}_{5}\left({a}_{i};\hspace{0.33em}12) and j = 1 , 2 , 3 , 4 j=1,2,3,4 . All results are obtained using elementary number theory and modular form theory.