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

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

In the case of the Riemann zeta function, the difficult last section (Section 10) of arXiv:2208.12937 underwent repeated corrections, which led finally to the conclusion that RH does not hold. This propagates to the present preprint, whcih deals with more general $L$-functions