The study of L-functions such as the Riemann zeta function is one of the most classical fields of analytic number theory. The distribution of zeros of L-functions encode arithmetic information such as the distribution of primes and coefficients of modular forms. For example, the Generalized Riemann Hypothesis predicts that all the nontrivial zeros of L-functions have real part equal to ½, and knowing this for the Riemann zeta function would give us a very strong approximation for the prime counting function. With Liu, Thorner, and Zaharescu, I have been studying statistical approximations to the Generalized Riemann Hypothesis. As an application, we've explored the distributions of the fractional parts of the zeros of a class of L-functions. We would be able to make several of our applications more precise if we could quantify the error terms in some of our estimates, but this involves a certain Diophantine approximation question that does not seem straightforward, and it is something I hope to make progress on in the future.

- obeckwith

# Modular forms and L-functions

Updated: Mar 11, 2021