Bernoulli numbers and certain convolution sums with divisor functions

Abstract

In this paper, we investigate the convolution sums

Sigma((a+b+c)x=n) a, Sigma(ax+by=n) ab, Sigma((ax+by+cz)x=n) abc, Sigma((ax+by+cz+du)x=n) abcd,

where a, b, c, d, x, y, z, u, n is an element of N. Many new equalities and inequalities involving convolution sums, Bernoulli numbers and divisor functions have also been given.