Numerical computations concerning the ERH

Abstract

This paper describes a computation which established the ERH to height 10000 for all primitive Dirichlet L-series with conductor Q ≤ 13 Q \leq 13 , and to height 2500 for all Q ≤ 72 Q \leq 72 , all composite Q ≤ 112 Q \leq 112 , and other moduli. The computations were based on Euler-Maclaurin summation. Care was taken to obtain mathematically rigorous results: the zeros were first located within 10 − 12 {10^{ - 12}} , then rigorously separated using an interval arithmetic package. A generalized Turing Criterion was used to show there were no zeros off the critical line. Statistics about the spacings between zeros were compiled to test the Pair Correlation Conjecture and GUE hypothesis.