One level density of low-lying zeros of quadratic Hecke L-functions to prime moduli

Abstract

International audience In this paper, we study the one level density of low-lying zeros of a family of quadratic Hecke L-functions to prime moduli over the Gaussian field under the generalized Riemann hypothesis (GRH) and the ratios conjecture. As a corollary, we deduce that at least 75% of the members of this family do not vanish at the central point under GRH.

Lower-Order Terms of the One-Level Density of a Family of Quadratic Hecke -Functions

“…We deduce first from Lemma 2.3 and (2-10), after partial summation, that Next note that, similarly to [16, formula (4.33)], we have …”

“…Similarly to the treatment in [16, Section 4.1], we use the approximation where denotes a sum over nonzero integral ideals in and …”

“…We now proceed as in [16, Section 4.4]. Assuming the truth of the GRH, it follows from Lemma 5.1 that where is defined in (1-1).…”

“…It follows from (2-10), (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) that…”

“…We can therefore use Φ(X) defined in (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) and partial summation to extend the sum over l to all odd elements in O K by introducing an extra error term of size O(X ε Z −3/2 ). We now set Z = X 2/3−σ/3 , change the order of summation in (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) and apply (2-10) to derive the desired result.…”