Multiplicity one for $L$-functions and applications

TL;DR

This paper establishes conditions under which two Euler products are identical, leading to multiplicity one results for associated number-theoretic objects, with stronger conclusions when the L-function is automorphic.

Contribution

It provides new, weaker hypotheses for multiplicity one theorems for L-functions and their related objects, extending previous results and leveraging automorphy.

Findings

Conditions for equality of Euler products with functional equations

New multiplicity one theorems for L-functions and related objects

Stronger results when L-functions are automorphic

Abstract

We give conditions for when two Euler products are the same given that they satisfy a functional equation and their coefficients are not too large and do not differ from each other by too much. Additionally, we prove a number of multiplicity one type results for the number-theoretic objects attached to $L$-functions. These results follow from our main result, which has slightly weaker hypotheses than previous multiplicity one theorems for $L$-functions. Significantly stronger results are available when the L-function is known to be automorphic.