Sums of squares and sequences of modular forms

Alexander Kalmynin

paper

Sums of squares and sequences of modular forms

arXiv:2407.03002

Abstract

Let $h_{n} (v)$ be the sequence of rational functions with $\frac{h _{n} ( v )}{v} - n h_{n} (v) + (n - 1) h_{n - 1} (v) - v h_{n - 1}^{'} (v) + \frac{v ( v ( v h _{n - 1} ( v ) ) ^{'} ) ^{'}}{4} = 0$ 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 $r_{2} (n) = # {(a, b) \in Z^{2} : a^{2} + b^{2} = n}$ , then we derive the formula $v = 1/ n Res h_{n} (v) = \frac{( - 1 ) ^{n - 1} r _{2} ( n )}{n 1 6 ^{n}} .$ The results are then generalized to arbitrary modular forms with respect to $Γ (2)$ and as a consequence we obtain a new criterion for Lehmer's conjecture for Ramanujan's $τ$ -function.