The Riemann Hypothesis, the Generalized Riemann Hypothesis, and the Cesáro Operator

Abstract.

A result due to Nyman establishes the equivalence of the Riemann hypothesis with the density of a set of functions in L ²[0, 1]. Here a large class of analytic functions is considered, which includes the Riemann zeta function and the Dirichlet L-functions as well as functions not given by a Dirichlet series. For each such function $\phi(s)$ there is an associated integral operator T on L ²[0, 1] such that $\phi(s)$ has no zeros in Re(s) > 1/2 iff the operator T has dense range iff a specified set of functions is dense in L ²[0, 1].