On Dirichlet series satisfying Riemann's functional equation

Summary According to convention, Hamburger's theorem (1921) says-roughly-that Riemann's (s) is uniquely determined by its functional equation. In 1944 Hecke pointed out that there are two distinct versions of Hamburger's theorem. Hecke's remark has led me, in examining just how rough the convention is, to prove that, with a weakening of certain auxiliary conditions, there are infinitely many linearly independent solutions of Riemann's functional equation (Theorem 1). In Theorem 1, as in Hamburger's theorem, the weight parameter is 1/2. In Theorem 2 we obtain stronger results when this parameter is 2: a Mittag-Leffler theorem for Dirichlet series with functional equations.Oblatum 23-XII-1992 & 9-IX-1993Research supported in part by NSA/MSP Grant MDA 90-H-4025 To the memory of Martin Eichler