Converse theorems: from the Riemann zeta function to the Selberg class

Abstract

Abstract. This is an expanded version of the author's lecture at the XX Congresso U.M.I., held in Siena in September 2015. After a brief review of L-functions, we turn to the classical converse theorems of H.Hamburger, E.Hecke and A.Weil, and to some later developments. Finally we present several results on converse theorems in the framework of the Selberg class of L-functions.Mathematics Subject Classification (2000): 11F66, 11M41

Forbidden conductors of $L$-functions and continued fractions of particular form

“…Note that the conjugate function F has conjugate coefficients a(n), and clearly F ∈ S ♯ . We refer to the survey papers [2,4,[10][11][12][13] for further definitions, examples and the basic theory of the classes S ♯ and S.…”

Classification of L-functions of degree 2 and conductor 1

“…The Selberg class S is, roughly speaking, the subclass of S ♯ of the functions with Euler product and satisfying the Ramanujan conjecture a(n) ≪ n ε . We refer to Selberg [27], Conrey-Ghosh [2] and to our survey papers [6], [10], [21], [22], [23], [24] for further definitions, examples and the basic theory of the Selberg class.…”

On the Rankin–Selberg convolution of degree 2 functions from the extended Selberg class

“…say; here we deal only with functions of degree d F = 2, and for simplicity we denote the conductor simply by q. We refer to our surveys [4], [5], [11], [12], [13] and [14] for definitions and the basic theory of the Selberg class. For F ∈ S and an integer k ≥ 0 we define…”