Effective hybrid joint universality for Dirichlet $L$-functions and its application

TL;DR

The paper proves an effective, quantitative lower-density bound for hybrid joint universality of Dirichlet $L$-functions with prime modulus, establishing a positive measure of shifts $\tau$ for which a family of $L(s+i\tau,\chi)$ simultaneously approximate prescribed non-vanishing analytic targets with controlled phase alignments. The authors develop an approach that combines approximation by trigonometric polynomials, finite Euler products, and a multi-dimensional Weyl criterion to obtain explicit density bounds. These results yield concrete lower-density statements for Hurwitz zeta-functions with rational parameters (prime denominators) and outline a path, via normal families, toward extending universality to algebraic irrational parameters. Overall, the work provides a constructive framework for effective density estimates in hybrid universality and demonstrates its applicability to Hurwitz zeta-functions through Dirichlet $L$-functions.

Abstract

In 2003, Garunkštis provided a lower bound for the lower density of the universality theorem for the Riemann zeta-function. In this paper, we generalize this result for the hybrid joint universality theorem for Dirichlet $L$-functions whose moduli are prime numbers. Furthermore, by its application, we estimate a lower bound of the lower density of the universality theorem for Hurwitz zeta-functions with rational parameters.

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.2: Gon
• Theorem 1.3: GLMS
• Theorem 1.4: Ga03
• Theorem 1.5: Gon, KK
• Theorem 1.6
• Corollary 1.7
• Proposition 2.1
• Lemma 2.2
• proof