Bailey-Zeta Limits: A $q$-Series Bridge to Dirichlet $L$-Functions and the Riemann Zeta Function
Mahipal Gurram
TL;DR
The paper builds a bridge between $q$-series combinatorics and analytic number theory by introducing Bailey--Zeta pairs and a two-step limit that converts carefully weighted $q$-series into Dirichlet $L$-functions scaled by $1/\sqrt{\pi}$. It proves a general theorem that, under bounded arithmetic weights $χ$ and suitable $q\to1^-$ asymptotics of $\alpha_r(s,q)$, the limit $\lim_{n\to\infty}\lim_{q\to1^-} T_n(s,q)$ equals $L(s,χ)/\sqrt{\pi}$ with $T_n(s,q)$ defined in terms of $\beta_n(s,q)$. The framework unifies multiple classical functions (Riemann zeta, Dirichlet eta, Dirichlet beta) and even a regularized Euler–Mascheroni constant through concrete $\alpha_r(s,q)$ choices, showcasing the deep interplay between combinatorial identities and analytic number theory. This provides new q-series representations for special $L$-values and constants with potential implications for both theory and computation.
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 $1/\sqrtπ$. This construction generalizes to arbitrary bounded arithmetic progressions via character weights, providing a unified $q$-series asymptotic for $L(s,χ)$. 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.