Biases in gaps between zeros of Dirichlet L-functions

Abstract

We describe a new family of Dirichlet L-functions that provably have an unusual value distribution. Experimentally, we find that the L-functions in this family have a significant and previously undetected bias in the distribution of gaps between their zeros. These phenomena appear to have an arithmetic explanation that corresponds to the nonvanishing of a certain Gauss-type sum. We give a complete classification of the characters for which this sum is nonzero and count them. This sum vanishes for 100% of primitive Dirichlet characters, so L-functions in our newly discovered family are rare (a set of density zero among all primitive characters). The nonvanishing of this sum implies that there is a main term when one integrates the Z-function associated to a Dirichlet L-function over a long interval. We prove that the L-functions in this family are the only GL(1)/Q or GL(2)/Q cuspidal automorphic L-functions whose Z-functions have a "large" mean value when integrated. We discuss analogous results for the Riemann zeta-function.