[Paper Review] Perturbations of L-functions with or without non-trivial zeros off the critical line

P. M. Gauthier, Xavier Xarles|ArXiv.org|Nov 26, 2009

Analytic Number Theory Research16 references3 citations

TL;DR

This paper demonstrates that small perturbations of L-functions satisfying standard functional equations can either fail or satisfy the Grand Riemann Hypothesis, depending on the perturbation. It proves that functions violating the hypothesis are generic in the space of such L-functions, while also showing that functions satisfying the hypothesis can be approximated via perturbations, establishing both failure and persistence of the GRH under small deformations.

ABSTRACT

There exist small perturbations of L-functions, satisfying the appropriate functional equation, for which the analogue of the Riemann hypothesis fails radically. Moreover, this phenomenon is generic. However, there also exist small perturbations, for which the analogue of the Riemann hypothesis holds.

Motivation & Objective

• To investigate whether small perturbations of L-functions satisfying standard functional equations can fail to satisfy the Grand Riemann Hypothesis.
• To determine whether such perturbations that preserve the Grand Riemann Hypothesis also exist.
• To establish the genericity of perturbations that violate the Grand Riemann Hypothesis in the space of meromorphic and holomorphic functions satisfying the functional equation.
• To extend results known for the Riemann zeta-function to broader classes of L-functions, including those with functional equations involving gamma factors and conductor terms.
• To show that approximation by functions satisfying the Grand Riemann Hypothesis is possible even when such functions are not dense in the full space of perturbed L-functions.

Proposed method

• The authors consider perturbations of L-functions via multiplication by a meromorphic function ν(s) symmetric about s = 1/2, preserving the functional equation structure.
• They define the space MQ of meromorphic functions f satisfying the functional equation ΛQ(f,s) = ΛQ(f,1−s), and the subspace HQ of entire functions within MQ.
• Using the topology of uniform convergence on compacta, they analyze the density and openness of the subsets RMQ and RHQ (functions satisfying the Grand Riemann Hypothesis) within MQ and HQ.
• They apply a generalized version of the Walsh Lemma for simultaneous approximation and interpolation to construct perturbations that either preserve or break the GRH.
• They verify that the symmetry of the original L-function (e.g., real-valuedness or symmetry under complex conjugation) can be preserved in the approximating functions.
• They provide explicit examples of L-functions (e.g., associated to modular forms, elliptic curves, and zeta functions over function fields) for which the GRH holds, and show that perturbations can be constructed to either preserve or violate this property.

Experimental results

Research questions

• RQ1 Can small perturbations of L-functions satisfying the standard functional equation fail to satisfy the Grand Riemann Hypothesis?
• RQ2 Is the set of perturbations violating the Grand Riemann Hypothesis generic within the space of meromorphic and entire functions satisfying the functional equation?
• RQ3 Can perturbations be constructed such that the Grand Riemann Hypothesis still holds for the perturbed L-function?
• RQ4 Are there explicit classes of L-functions for which the Grand Riemann Hypothesis is known to hold, and can such functions be approximated by perturbations that preserve or violate the hypothesis?
• RQ5 Can the symmetry properties (e.g., real-valuedness or conjugate symmetry) of the original L-function be preserved in the approximating perturbations?

Key findings

• The set of meromorphic functions in MQ that violate the Grand Riemann Hypothesis is both open and dense, indicating that such failures are generic in the space of perturbations.
• Similarly, the set of entire functions in HQ that violate the Grand Riemann Hypothesis is also open and dense, confirming the genericity of GRH failure in the holomorphic case.
• There exist small perturbations of L-functions that preserve the functional equation and satisfy the Grand Riemann Hypothesis, demonstrating that GRH-preserving approximations are constructively possible.
• The results extend to L-functions associated to Dirichlet characters, Hecke characters, modular forms, and automorphic forms, provided the functional equation is of the standard form involving gamma factors and conductor terms.
• For L-functions with root number W(ρ) = 1 and real representations, or those arising from elliptic curves over Q with L(1/2) ≠ 0, the Grand Riemann Hypothesis holds, and such functions can be approximated by perturbations that preserve or violate the hypothesis.
• The construction of approximating functions respects symmetry: if the original L-function is symmetric under complex conjugation or real-valued, the approximating functions can be chosen to preserve these properties.