Five families of rapidly convergent evaluations of zeta values

Abstract

This work derives 5 methods to evaluate families of odd zeta values by combining a power of $\pi$ with Lambert series whose ratios of successive terms tend to $e^{-\pi\sqrt{a}}$ with integers $a\ge7$, outperforming Ramanujan's results with merit $a=4$. Families with $a=7$ and $a=8$ evaluate $\zeta(2n+1)$. Families with $a=9$ and $a=16$ evaluate $\zeta(4n+1)$ with faster convergence. A fifth family with $a=12$ evaluates $\zeta(6n+1)$ and gives the fastest convergence for $\zeta(6n+7)$. Members of three of the families were discovered empirically by Simon Plouffe. An intensive new search strongly suggests that there are no more than 5 families with integers $a\ge7$. There are at least 20 families that involve Lambert series with rational $a>4$. Quasi-modular transformations of Lambert series resolve rational sequences that were discovered empirically. Expansions of Lambert series in polylogarithms, familiar from quantum field theory, provide proofs of all known evaluations with rational merit $a>4$.