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In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form Δ should have an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function. This conjecture was proven by Hafner and Stopple. In 2017 and 2019, Chakraborty et al. established an asymptotic relation between Lambert series associated to any primitive cusp form (for full modular group, congruence subgroup and in Maass form case) and the nontrivial zeros of the Riemann zeta function. In this paper, we study Lambert series associated with primitive Hilbert modular form and establish similar kind of asymptotic expansion. [ABSTRACT FROM AUTHOR]

For a number field K , let ζ K (s) be the Dedekind zeta function associated to K. In this paper, we study non-vanishing and transcendence of ζ K as well as its derivative ζ K ′ at s = 1 / 2. En route, we strengthen a result proved by Ram Murty and Tanabe [On the nature of e γ and non-vanishing of L -series at s = 1 / 2 , J. Number Theory 161 (2016) 444–456]. [ABSTRACT FROM AUTHOR]

Building on Stechkin and Kadiri's ideas we derive an explicit zero-free region of the real axis for Dedekind zeta functions of number fields. We then explain how this new region enables us to improve upon the previously known explicit lower bounds for class numbers of number fields and relative class numbers of CM-fields. [ABSTRACT FROM AUTHOR]

We compute the second moment of the Dedekind zeta function of a quadratic field times an arbitrary Dirichlet polynomial of length T1/11-∊. Our result generalizes a formula of Hughes and Young concerning the fourth moment of the Riemann zeta function. [ABSTRACT FROM AUTHOR]