Zero Spacing Distributions for Differenced L-Functions

Abstract

This paper studies the local spacings of deformations of the Riemann zeta function under certain averaging and differencing operations. For real h it considers A_h(s)= 1/2(xi(s+h)+ xi(s-h)) and B_h(s)=1/(2i)(xi(s+h)-xi(s-h)), where xi(s) is the Riemann xi-function. For |h|\ge 1/2 all zeros of these functions lie on the critical line and are simple zeros. Assuming the Riemann hypothesis, the same holds for all nonzero h. The number of zeros to height T has the same asymptotics as for the zeta function in its main terms, so one can define normalized zero spacings as for the xi-function. It is shown that the first k normalized zero spacings have a limiting distribution as T goes to infinity, and it is the "trivial" distribution with all spacings equal to 1. This is shown unconditionally for |h|> 1/2 and conditionally on RH for 0< |h|< 1/2. Similar results are indicated for Dirichlet L-functions. The results are interpreted in terms of de Branges's theory of Hilbert spaces of entire functions.