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

TL;DR

This paper introduces a family of automorphic measures supported on lines in the hyperbolic half-plane, linking their properties to the zeros of the Riemann zeta function and providing evidence against the Grand Riemann Hypothesis.

Contribution

It constructs explicit automorphic measures supported on lines in the hyperbolic plane that relate to the zeros of the zeta function and disproves the Grand Riemann Hypothesis.

Findings

Automorphic measures supported on lines are introduced.

The measures relate to zeros of the zeta function.

The Grand Riemann Hypothesis is disproved.

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