On some reasons for doubting the Riemann hypothesis

TL;DR

This paper discusses various classical and research-related arguments questioning the validity of the Riemann hypothesis, including phenomena, special functions, and zero distributions.

Contribution

It provides an extensive discussion of classical and recent arguments against the Riemann hypothesis, connecting them with the author's research.

Findings

Highlights the Lehmer phenomenon and Davenport-Heilbronn zeta-function as counterarguments.

Analyzes large and mean values of |z(1/2+it)| on the critical line.

Examines zeros of convolution functions related to the hypothesis.

Abstract

Several arguments against the truth of the Riemann hypothesis are extensively discussed. These include the Lehmer phenomenon, the Davenport-Heilbronn zeta-function, large and mean values of $|\zeta(1/2+it)|$ on the critical line, and zeros of a class of convolution functions. The first two topics are classical, and the remaining ones are connected with the author's research.