Sums of squares and sequences of modular forms

TL;DR

This paper explores the relationship between sums of squares, modular forms, and rational functions, establishing new criteria for Lehmer's conjecture through residues and poles linked to number-theoretic properties.

Contribution

It introduces a novel connection between poles of rational functions and sums of two squares, extending to modular forms and providing a new perspective on Lehmer's conjecture.

Findings

Pole at v=1/n iff n is a sum of two squares

Explicit residue formula involving r_2(n)

Generalization to modular forms and Lehmer's conjecture

Abstract

Let $h_n(v)$ be the sequence of rational functions with $$ \frac{h_n(v)}{v}-nh_n(v)+(n-1)h_{n-1}(v)-vh_{n-1}'(v)+\frac{v(v(vh_{n-1}(v))')'}{4}=0 $$ for $n>0$ and $h_0(v)=1$. We prove that $h_n(v)$ has a pole at $v=\frac{1}{n}$ if and only if $n$ is a sum of two squares of integers. Moreover, if $r_2(n)=\#\{(a,b)\in \mathbb Z^2: a^2+b^2=n\}$, then we derive the formula $$ \underset{v=1/n}{\mathrm{Res}}h_n(v)=\frac{(-1)^{n-1}r_2(n)}{n16^n}. $$ The results are then generalized to arbitrary modular forms with respect to $\Gamma(2)$ and as a consequence we obtain a new criterion for Lehmer's conjecture for Ramanujan's $\tau$-function.