On bounds on Dedekind sums and moments of $L$-functions over subgroups

Marc Munsch, Igor E. Shparlinski

Published 2023-05-07Version 1

We obtain new bounds, pointwisely and on average, for Dedekind sums $s(\lambda,p)$ modulo a prime $p$ with $\lambda$ of small multiplicative order $d$ modulo $p$. As an application, we use these bounds to extend an asymptotic formula of S.~Louboutin and M.~Munsch (2022) for the second moment of Dirichlet $L$-functions over subgroups of the group of multiplicative characters modulo $p$. Assuming the infinitude of Mersenne primes, the range of our results is optimal. Moreover, we estimate correlations between Dedekind sums and use this to obtain an asymptotic formula for the fourth moment of these $L$-functions, also in the optimal range. Finally, we relate higher moments to more complicated correlations of Dedekind sums, formulate a conjecture for their asymptotic behaviour and prove it in a slightly shorter than optimal range for all primes $p$ and in the full range for almost all $p$.