# On the Selberg class of Dirichlet series: small weights

Research paper by J. Brian Conrey, Amit Ghosh

Indexed on: 31 Mar '92Published on: 31 Mar '92Published in: Mathematics - Number Theory

#### Abstract

In the study of Dirichlet series with arithmetic significance there has appeared (through the study of known examples) certain expectations, namely (i) if a functional equation and Euler product exists, then it is likely that a type of Riemann hypothesis will hold, (ii) that if in addition the function has a simple pole at the point s=1, then it must be a product of the Riemann zeta-function and another Dirichlet series with similar properties, and (iii) that a type of converse theorem holds, namely that all such Dirichlet series can be obtained by considering Mellin transforms of automorphic forms associated with arithmetic groups.