Generalized L-functions for meromorphic modular forms and their relation to the Riemann zeta function

TL;DR

This paper introduces a family of generalized L-functions associated with points in the upper half-plane, showing their convergence to a form related to the Riemann zeta function as the point approaches infinity.

Contribution

It constructs a new family of L-functions for each point in the upper half-plane and establishes their convergence to a zeta-related L-function at infinity.

Findings

Generalized L-functions are defined for each point in the upper half-plane.

As the point approaches infinity, these L-functions converge to a form involving the Riemann zeta function.

The work links modular forms, L-functions, and the Riemann zeta function in a novel way.

Abstract

In this paper, we construct a family of generalized $L$-functions, one for each point $z$ in the upper half-plane. We prove that as $z$ approaches $i\infty$, these generalized $L$-functions converge to an $L$-function which can be written in terms of the Riemann zeta function.