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

TL;DR

This paper introduces a novel theory of L-series using symmetric functions mod N, enabling explicit calculations of series related to odd zeta values like ζ(3) and ζ(5), extending Dirichlet character methods.

Contribution

It generalizes the classical L-series theory by replacing Dirichlet characters with symmetric functions, facilitating explicit evaluations of series connected to odd zeta values.

Findings

Developed a new framework for L-series with symmetric functions

Explicitly calculated series related to ζ(3) and ζ(5)

Extended classical Dirichlet character-based methods

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.