Computing degree-1 L-functions rigorously

We describe a new, rigorous algorithm for efficiently and simultaneously computing many values of the Riemann zeta function on the critical line by exploiting the fast Fourier transform (FFT). We apply this to locating non-trivial zeros of zeta to high precision which are in turn used as input to our own implementation of the Lagarias and Odlyzko analytic algorithm to compute π(x), the prime counting function. We confirm the value of π(x) for x = 10, matching the largest unconditional result to date. We then turn to Dirichlet L-functions and detail a version of Booker’s rigorous algorithm for generic L-functions, tailored to this application. We employ this for computations with characters of relatively small modulus. For larger modulus, we describe a new algorithm and its implementation. Both again rely on the FFT to compute many values simultaneously and hence achieve efficiency. We use a combination of these two algorithms to extend the work of Rumely and verify the generalised Riemann hypothesis (the GRH) for all characters modulus q ≤ 100 000 to height T such that qT is at least 100 000 000. We then confirm rigorously the non-vanishing of Lχ(1/2) for all characters of modulus q ≤ 2 000 000 before finishing with some comparisons of computed data to predictions from random matrix theory.