$L$-series associated to symmetric functions mod $N$ with applications related to $\zeta(3)$, $\zeta(5)$

Abstract

We develop a new theory of $L$-series based on replacing Dirichlet characters mod $N$ by symmetric functions mod $N$ in order to calculate explicitly the sums of infinite series more closely related to $\zeta(2n+1)$, specifically $\zeta(3)$, $\zeta(5)$. This generalizes the corresponding theory of sums of $L$-series associated to Dirichlet characters.