Bailey-Zeta Limits: A $q$-Series Bridge to Dirichlet $L$-Functions and the Riemann Zeta Function

TL;DR

This paper develops a new family of $q$-series derived from Bailey pairs that converge to Dirichlet $L$-functions and the Riemann zeta function, revealing deep links between combinatorics and number theory.

Contribution

It introduces a novel $q$-series framework that generalizes to arbitrary arithmetic progressions and connects Bailey chains with analytic number theory.

Findings

$q$-series converge to Dirichlet $L$-functions as $q o 1^-$ and $n o fty$

Unified asymptotic for $L(s, ext{character})$ in bounded progressions

Applications to special constants like Euler-Mascheroni constant

Abstract

We introduce a family of deformed Bailey pairs whose $q$-series, which converge in a two-step limit ($q \to 1^-$ followed by $n \to \infty$) to Dirichlet $L$-functions scaled by $1/\sqrt{\pi}$. This construction generalizes to arbitrary bounded arithmetic progressions via character weights, providing a unified $q$-series asymptotic for $L(s,\chi)$. Our approach unveils deep connections between the combinatorial machinery of Bailey chains and analytic number theory, with applications to special values like Euler-Mascheroni constant.