[Paper Review] Joint universality for dependent $L$-functions

Łukasz Pańkowski|arXiv (Cornell University)|Apr 15, 2016

Analytic Number Theory Research9 references20 citations

TL;DR

This paper establishes joint universality for Dirichlet $L$-functions under non-linear, non-uniform shifts of the form $L(s + ieta t^{a}"log^b t; \ chi)$, proving that any finite collection of non-vanishing analytic functions on a compact subset of the critical strip can be simultaneously approximated by such shifts, provided the exponent pairs $(a_j, b_j)$ are distinct and satisfy certain arithmetic conditions. The result extends classical universality beyond linear shifts and includes both continuous and discrete settings.

ABSTRACT

We prove that, for arbitrary Dirichlet $L$-functions $L(s;χ_1),\ldots,L(s;χ_n)$ (including the case when $χ_j$ is equivalent to $χ_l$ for $j e k$), suitable shifts of type $L(s+iα_jt^{a_j}\log^{b_j}t;χ_j)$ can simultaneously approximate any given analytic functions on a simply connected compact subset of the right open half of the critical strip, provided the pairs $(a_j,b_j)$ are distinct and satisfy certain conditions. Moreover, we consider a discrete analogue of this problem where $t$ runs over the set of positive integers.

Motivation & Objective

• To extend the classical joint universality theorem for $L$-functions beyond linear shifts to more general non-linear, non-uniform shifts of the form $t^{a_j} \log^{b_j} t$.
• To address the case of dependent or equivalent Dirichlet characters, where standard joint universality fails due to functional dependencies.
• To establish universality in both continuous and discrete settings, including when $t$ runs over positive integers.
• To provide a general framework based on uniform distribution modulo 1 and analytic approximation techniques applicable to $L$-functions with Euler products.
• To overcome arithmetic obstructions in the case of integer exponents by introducing auxiliary prime sets and modified shift sequences.

Proposed method

• Utilizes a general framework based on Lemma 2.1 and Lemma 3.1, which link uniform distribution modulo 1 to approximation of analytic functions.
• Applies the Shifts Universality Principle to non-linear functions $\gamma_j(t) = \alpha_j t^{a_j} \log^{b_j} t$, ensuring distinctness of exponent pairs $(a_j, b_j)$ to avoid degeneracy.
• Employs uniform distribution theory for sequences $\gamma_j(k)$ modulo 1, particularly when $a_j \in \mathbb{Z}$ and $b_j = 0$, by introducing auxiliary prime sets $\mathcal{A}_j$ and $\mathcal{P}_j$ to ensure irrationality conditions.
• Constructs modified $L$-functions $f_j^*(s)$ by removing Euler factors over selected primes to ensure the shift sequence remains uniformly distributed modulo 1.
• Uses Cauchy's integral formula and Carlson’s theorem to control error terms and ensure uniform approximation on compact sets.
• Adapts the continuous case to the discrete case via a scaling factor $q^*$, ensuring that the number of good integers $k \in [2,N]$ satisfying the approximation condition is at least $cN/q^*$.

Experimental results

Research questions

• RQ1 Can joint universality be established for Dirichlet $L$-functions under non-linear shifts $t^{a} \log^b t$ rather than just linear shifts?
• RQ2 What conditions on the exponent pairs $(a_j, b_j)$ ensure that the shifts $\gamma_j(t) = \alpha_j t^{a_j} \log^{b_j} t$ yield jointly universal behavior?
• RQ3 How can one handle the case when $L$-functions are dependent (e.g., associated with equivalent characters) or when $a_j \in \mathbb{Z}$ and $b_j = 0$?
• RQ4 Is it possible to achieve joint universality in the discrete setting where $t$ runs over positive integers?
• RQ5 Can the approximation be made effective in terms of density of good shifts, and what is the lower density of such shifts?

Key findings

• For any finite collection of non-vanishing analytic functions on a compact set $K$ with connected complement in $\{s \in \mathbb{C} : 1/2 < \operatorname{Re}(s) < 1\}$, there exist shifts $L(s + i\alpha_j t^{a_j} \log^{b_j} t; \chi_j)$ that simultaneously approximate them uniformly.
• The joint universality holds provided the pairs $(a_j, b_j)$ are distinct and satisfy the condition that $a_j \notin \mathbb{Z}$ or $b_j \neq 0$, or if $a_j \in \mathbb{Z}$ and $b_j = 0$, then the shift parameter $\alpha_j$ is chosen such that the associated exponential sum is irrational modulo 1.
• When $a_j \in \mathbb{Z}$ and $b_j = 0$, the method introduces a minimal prime set $\mathcal{P}_j$ and a scaling factor $q^*$ to ensure uniform distribution of the shift sequence modulo 1.
• The number of integers $k \in [2,N]$ for which the approximation holds is at least $cN/q^*$, where $c > 0$ is an absolute constant and $q^*$ depends on the arithmetic structure of the exponents.
• The result extends to the discrete setting: when $t$ runs over $\mathbb{N}$, the same approximation property holds with a positive lower density of good shifts.
• The proof relies on analytic techniques including Cauchy’s integral formula and Carlson’s theorem to control error terms and ensure convergence of the approximation.