$L$-function invariants for 3-manifolds and relations between generalized Bernoulli polynomials

TL;DR

This paper introduces $L$-functions for negative definite plumbed 3-manifolds, showing their entire nature, relation to Witten--Reshetikhin--Turaev invariants, and explores relations between special values of generalized Bernoulli-related functions.

Contribution

It establishes the entire property of these $L$-functions, connects their values at zero to topological invariants, and uncovers linear relations among their special values, generalizing classical zeta functions.

Findings

$L$-functions are entire functions.

Values at $s=0$ equal Witten--Reshetikhin--Turaev invariants.

Linear relations among special values generalize classical zeta functions.

Abstract

We introduce $ L $-functions attached to negative definite plumbed manifolds as the Mellin transforms of homological blocks. We prove that they are entire functions and their values at $ s=0 $ are equal to the Witten--Reshetikhin--Turaev invariants by using asymptotic techniques developed by the author in the previous papers. We also prove that linear relations between special values at negative integers of some $ L $-functions, which are common generalizations of Hurwitz zeta functions, Barnes zeta functions and Epstein zeta functions.