Ratios conjecture for quadratic Hecke $L$-functions in the Gaussian field

TL;DR

This work extends the L-functions ratios conjecture to the family of quadratic Hecke L-functions over the Gaussian field $K=\mathbb{Q}(i)$, proving a one-shift ratio formula under GRH and establishing a sharp smoothed first-moment asymptotic with error $O(X^{1/2+\varepsilon})$. The authors build a three-variable Dirichlet series $A(s,w,z)$ from $L^{(2)}(w,\chi_n)/L^{(2)}(z,\chi_n)$, develop its meromorphic continuation via functional equations and Poisson summation, and extract main terms from explicit residues at $s=1$ and $s=1-\alpha$. The method follows the double Dirichlet-series framework of Čech while adapting it to quadratic Hecke twists in the Gaussian setting, and yields unconditional second-moment control and explicit first-moment formulas. The results provide precise mean-value information for this number-field family and illustrate the robustness of the ratios conjecture in a nontrivial CM-field context, with GRH enabling tight bounds on the auxiliary Dirichlet-series components.

Abstract

We develope the $L$-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of quadratic Hecke $L$-functions in the Gaussian field using multiple Dirichlet series under the generalized Riemann hypothesis. We also obtain an asymptotical formula for the first moment of central values of the same family of $L$-functions, obtaining an error term of size $O(X^{1/2+\varepsilon})$.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Lemma 2.2
• Lemma 2.4
• Proposition 2.5
• Lemma 2.7
• Definition 2.9
• Theorem 2.10
• Proposition 2.11