converse theorem

He also proved a converse theorem for the Riemann zeta function.

The theory was completely resolved by Moeglin-Waldspurger, and was reverse-engineered to establish the "converse theorem".

One of his major works at Yale dealt with the converse theorem which establishes a link between automorphic forms on n by n matrix groups and zeta functions.

G.H. Hardy and J.E. Littlewood coined the term Tauberian to describe converse theorems like that proved by Tauber.

The first converse theorems were proved by who characterized the Riemann zeta function by its functional equation, and by who showed that if a Dirichlet series satisfied a certain functional equation and some growth conditions then it was the Mellin transform of a modular form of level 1.