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Automorphic line measures in the half-plane and the Grand Riemann Hypothesis

Andre Unterberger

TL;DR

The paper constructs a one-parameter family of automorphic measures supported on unions of congruent hyperbolic lines in the half-plane and analyzes their spectral decompositions, linking the presence or absence of Eisenstein components to zeros of $\zeta$ and related $L$-functions. It then develops a pseudodifferential-arithmetic framework, associating a symbol to an $L$-function and building distributions ${\mathfrak T}_N$, ${\mathfrak T}_{\infty}$, and ${\mathfrak T}_{\infty/2}$ to encode $L$-function data within Weyl calculus. By studying the resulting Hermitian forms and leveraging an extension of prior spectral-analytic methods, the work argues that the Grand Riemann Hypothesis (and a special case thereof) is disproved in this setting. The construction relies on a palindromic symmetry in local factors and a careful analysis of the continuous and discrete spectra, with implications for the spectral interpretation of zeta zeros and automorphic $L$-functions. The results showcase a novel bridge between automorphic line measures, spectral theory, and $L$-function zeros, providing a framework that challenges GRH within the proposed automorphic/arithmetic scheme.

Abstract

Poincare-type series, such as Selberg's, are known to produce automorphic functions, in the hyperbolic half-plane, the decompositions of which into eigenfunctions (genuine or generalized) of the automorphic Laplacian contain all modular forms of nonholomorphic type. We introduce a one-parameter family of explicit automorphic measures supported by discrete unions of congruent lines with the same property, except for one value of the real parameter, for which they miss exactly the Eisenstein series associated to non-trivial zeros of zeta, and the Hecke eigenforms the $L$-functions associated to which vanish as $\frac{1}{2}$. The Grand Riemann Hypothesis, a special case of which needs being analyzed, is disproved

Automorphic line measures in the half-plane and the Grand Riemann Hypothesis

TL;DR

The paper constructs a one-parameter family of automorphic measures supported on unions of congruent hyperbolic lines in the half-plane and analyzes their spectral decompositions, linking the presence or absence of Eisenstein components to zeros of and related -functions. It then develops a pseudodifferential-arithmetic framework, associating a symbol to an -function and building distributions , , and to encode -function data within Weyl calculus. By studying the resulting Hermitian forms and leveraging an extension of prior spectral-analytic methods, the work argues that the Grand Riemann Hypothesis (and a special case thereof) is disproved in this setting. The construction relies on a palindromic symmetry in local factors and a careful analysis of the continuous and discrete spectra, with implications for the spectral interpretation of zeta zeros and automorphic -functions. The results showcase a novel bridge between automorphic line measures, spectral theory, and -function zeros, providing a framework that challenges GRH within the proposed automorphic/arithmetic scheme.

Abstract

Poincare-type series, such as Selberg's, are known to produce automorphic functions, in the hyperbolic half-plane, the decompositions of which into eigenfunctions (genuine or generalized) of the automorphic Laplacian contain all modular forms of nonholomorphic type. We introduce a one-parameter family of explicit automorphic measures supported by discrete unions of congruent lines with the same property, except for one value of the real parameter, for which they miss exactly the Eisenstein series associated to non-trivial zeros of zeta, and the Hecke eigenforms the -functions associated to which vanish as . The Grand Riemann Hypothesis, a special case of which needs being analyzed, is disproved
Paper Structure (5 sections, 9 theorems, 69 equations)

This paper contains 5 sections, 9 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. An unusual class of automorphic functions
  3. Introducing $L$-functions
  4. Associating a symbol to an $L$-function
  5. The pseudodifferential arithmetic of ${\mathfrak T}_N$, $L$-function case

Key Result

Proposition 2.1

Assuming that $\nu\notin \mathbb Z$ and $\rho\pm \nu\notin 2\mathbb Z$, set for $t\neq 0$ where fractional powers $z_+^{\alpha}$ are characterized for $z\notin ]-\infty,0]$ by choosing the argument of $z$ in $]-\pi,\pi[$. The function $\chi_{\rho,\,\nu}(t)$ is analytic in $\mathbb R\backslash \{0\}$ and one has for some constant $C>0$ the inequality It extends as a $C^{\infty}$ function to each o

Theorems & Definitions (13)

  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • Theorem 2.5
  • Lemma 4.1
  • proof
  • Proposition 5.1
  • Lemma 5.2
  • ...and 3 more