Proof of Sun's conjectures on hyperbolic cosine series via the Eisenstein--Lambert method

Abstract

We prove two conjectures of Zhi-Wei Sun concerning hyperbolic cosine Lambert series. The first one is the evaluation, for integers $m\geq 0$, of \[ S_m=\sum_{n=1}^\infty\left( \frac{n^{2m}}{\cosh(\pi n)-1} -\frac{(2^{2m+1}-(-1)^{m(m+1)/2}2^{m+1}+4)n^{2m}}{\cosh(2\pi n)-1} +\frac{2^{2m+2}n^{2m}}{\cosh(4\pi n)-1} \right). \] We prove that \[ S_0=\frac1{12},\qquad S_1=\frac1{2\pi^2},\qquad S_m=0\quad (m>1). \] The second one is the quadratic identity \[ \sum_{n=1}^\infty \left( \frac{4}{(\cosh(\pi n)-1)^2} -\frac{55}{(\cosh(2\pi n)-1)^2} +\frac{16}{(\cosh(4\pi n)-1)^2} \right) = \frac{77-234/\pi}{72}. \] The proof uses an elementary level-four identity for Lambert series and its consequences for Eisenstein series. After differentiating this identity and evaluating it at $i/2$, the first conjecture follows from the modular transformation law for $E_{2m}$, with the cases $m=0$ and $m=1$ treated separately by the quasimodular transformation law for $E_2$. The second conjecture follows by rewriting the corresponding squared-kernel series as $(E_4+10E_2-11)/360$ and evaluating only the resulting linear combinations of $E_2$ and $E_4$ at $i/2$, $i$, and $2i$.