Multiplicity one for $L$-functions and applications

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.