Estimates for Weyl sums in the theory of discrete universality

Abstract

We prove joint and disjoint discrete universality theorems for Dirichlet L-functions $$L(s,\chi )$$ and Hurwitz zeta-functions $$\zeta (s;\beta )$$ with rational parameter $$\beta \in (0,1]$$ . Our approach does not utilize Gallagher’s lemma, which is usually employed to prove discrete universality theorems. In our case, however, this would lead to certain difficulties when it is about estimating discrete second moments of L-functions. Therefore, we introduce a novel approach which is based only on Euler product representations and zero-density estimates of L-functions, as well as mean-value estimates for Weyl sums.