Sharp Upper Bounds for Moments of Dedekind Zeta Functions

Assuming the Generalised Riemann Hypothesis, we establish conjecturally sharp upper bounds for shifted moments of products of Dedekind zeta functions of arbitrary number fields. This improves and extends earlier bounds obtained by Milinovich and Turnage–Butterbaugh, and Hagen. As applications, we obtain improved bounds for the variance of the coefficients and upper bounds for the large deviations of Dedekind zeta functions. Our results apply for both Galois and non-Galois extensions.