In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form [Formula: see text] 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.

Let H = Q(ζn + ζn −1 ) and ℓ be an odd prime such that q ≡ 1 (mod ℓ) for some prime factor q of n. We get a bound on the ℓ-rank of the class group of H (under some conditions) in terms of the ℓ-rank of the class group of real quadratic subfield contained in H. This is an extension of a recent work of E. Agathocleous (with alternate hypothesis) where she handles ℓ = 3 case. As an application of our main result we relate the ℓ-rank of real quadratic subfields of H.

Let F (over Q) be a totally real number field of narrow class number 1. We generalize a result of Kohnen on the determination of half integral weight modular forms by their Fourier coefficients supported on squarefree (algebraic) integers. We also give a soft proof that infinitely many Fourier coefficients supported on squarefree integers are non-vanishing.

No abstract