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

Abstract

Let $$\sum _{\begin{array}{c}d|n\\ d\equiv 1(2)\end{array}}\frac{1}{d}$$ 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: $$\sum _{\begin{array}{c}d|n\\ d\equiv 1(2)\end{array}}\frac{1}{d}=\frac{1}{2}\sum _{r=1}^n\frac{(-1{)}^{n+r}}{r}\left(\genfrac{}{}{0ex}{}{n}{r}\right)c_r(n).$$

arXiv comment4 pages; To appear in the Rocky Mountain J. Math