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We study universal quadratic forms over totally real number fields using Dedekind zeta functions. In particular, we prove an explicit lower bound for the rank of universal quadratic forms over a given number field K, under the assumption that the codifferent of K is generated by a totally positive element. Motivated by a possible path to remove that assumption, we also investigate the smallest number of generators for the positive part of ideals in totally real numbers fields. [ABSTRACT FROM AUTHOR]

Two algebraic number fields are called arithmetically equivalent if the Dedekind zeta functions of the fields coincide. We show that if a G-extension contains non-conjugate arithmetically equivalent fields and there is an injection from G to another group H inducing an isoclinism between G and H, then there are non-conjugate arithmetically equivalent fields inside an H-extension. [ABSTRACT FROM AUTHOR]

Let K3 be a non-normal cubic extension over Q. In this paper, we investigate the higher moments of coefficients aK3 (n) of the Dedekind zeta function of the following type Σ(n≤x) al K3 (n), where l ≥ 4 is any fixed positive integer. [ABSTRACT FROM AUTHOR]

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]