A Proof of Merca's Conjectures on Sums of Odd Divisor Functions

TL;DR

This paper completes the proof of three conjectures related to congruences involving sums of odd divisor functions and generalized m-gonal numbers, extending previous work in number theory.

Contribution

It provides the first complete proof of Merca's conjectures on convolutions of odd divisor sums and m-gonal number generating functions.

Findings

Confirmed the conjectures on congruences for odd divisor sums

Extended the understanding of convolutions with m-gonal number generating functions

Established new results in number theory related to divisor functions

Abstract

In a recent paper, Merca posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalized $m$-gonal numbers. Extending Merca's work, we complete the proof of these conjectures.