An identity for the sum of inverses of odd divisors of $n$ in terms of the number of representations of $n$ as a sum of $r$ squares

TL;DR

This paper establishes a novel combinatorial identity linking the sum of inverses of odd divisors of a positive integer to the number of its representations as a sum of squares, revealing a deep connection between divisor sums and quadratic representations.

Contribution

It introduces a new identity connecting odd divisor inverses with sum of squares representations, expanding understanding of divisor functions and quadratic forms.

Findings

Proves the identity relating odd divisors and sum of squares representations.

Provides a combinatorial formula involving binomial coefficients and alternating signs.

Establishes a link between divisor sums and quadratic form counts.

Abstract

Let $$\sum_{\substack{d|n\\ d\equiv 1 (2)}}\frac{1}{d}$$ denote the sum of inverses of odd divisors of a positive integer $n$, and let $c_{r}(n)$ be the number of representations of $n$ as a sum of $r$ squares where representations with different orders and different signs are counted as distinct. The aim is of this note is to prove the following interesting combinatorial identity: $$ \sum_{\substack{d|n\\ d\equiv 1 (2)}}\frac{1}{d}=\frac{1}{2}\,\sum_{r=1}^{n}\frac{(-1)^{n+r}}{r}\,\binom{n}{r}\, c_{r}(n). $$