Convolution Sums of Some Functions on Divisors

Abstract

One of the main goals in this paper is to establish convolution sums of functions for the divisor sums σ s (n) = d|n (−1) d−1 d s and σ s (n) = d|n (−1) n d −1 d s , for certain s, which were first defined by Glaisher. We first introduce three functions P(q), E(q), and Q(q) related to σ(n), σ(n), and σ 3 (n), respectively, and then we evaluate them in terms of two parameters x and z in Ramanujan's theory of elliptic functions. Using these formulas, we derive some identities from which we can deduce convolution sum identities. We discuss some formulae for determining r s (n) and δ s (n), s = 4, 8, in terms of σ(n), σ(n), and σ 3 (n), where r s (n) denotes the number of representations of n as a sum of s squares and δ s (n) denotes the number of representations of n as a sum of s triangular numbers. Finally, we find some partition congruences by using the notion of colored partitions.

CONGRUENCES OF THE WEIERSTRASS <TEX>${\wp}(x)$</TEX> AND <TEX>${\wp}^{{\prime}{\prime}}(x)$</TEX>(<TEX>$x=\frac{1}{2}$</TEX>, <TEX>$\frac{\tau}{2}$</TEX>, <TEX>$\frac{\tau+1}{2}$</TEX>)-FUNCTIONS ON DIVISORS

“…Remark 3.12. µ(3n − 1) ≡ 0 (mod 6) shown by us induces that µ(3n − 1) ≡ 0 (mod 3) which is also the Hahn's result in [11,Theorem 6.1].…”

Convolution Sums and Their Relations to Eisenstein Series

“…Equating the coefficients of q, we get d = − 762048 691 . Now, equating the coefficients of q n (n ∈ N), indeed we obtain the identity in (4).…”

Certain Combinatoric Convolution Sums and Their Relations to Bernoulli and Euler Polynomials

“…Many recent works on convolution formulas for divisor functions can be found in B. C. Berndt [2], J. W. L. Glaisher [7], H. Hahn [8], J. G. Huard et al [9], D. Kim et al [11], G. Melfi [16] and K. S. Williams [25,24,26]. In particular, the problem of convolution sums of the divisor function σ 1 (n) and the theory of Eisenstein series has recently attracted considerable interest with the emergence of quasimodular tools.…”