A disproof of the Riemann hypothesis on zeros of $\zeta -$function

TL;DR

This paper claims to disprove the Riemann hypothesis by applying the Beurling–Nyman criterion, challenging a longstanding open problem in mathematics that has significant implications across multiple scientific fields.

Contribution

It presents a disproof of the Riemann hypothesis using the Beurling–Nyman criterion, a novel approach to this famous problem.

Findings

Disproves the Riemann hypothesis based on the Beurling–Nyman criterion.

Challenges the longstanding belief that all non-trivial zeros lie on the critical line.

Potentially impacts number theory, physics, and related fields.

Abstract

In his famous presentation at the International Congress of Mathematicians held in Paris in 1900, David Hilbert included the Riemann Hypothesis on zeros of $\zeta -$function as number 8 in his list of 23 challenging problems published later. After over 150 years, it is one of the few on that list that have not been solved. At present many mathematicians consider it the most important unsolved problem in mathematics. Recall that, exactly one hundred years later, the Clay Mathematics Institute has published a list of 7 unsolved problems for the 21st century, including 6 unresolved problems from the Hilbert list, offering a reward of one million dollars for a solution to any of these problems. One of them is the {\bf Riemann hypothesis}, i.e. a conjecture that the so-called Riemann zeta function has as its zeros only complex numbers with real part $1/2$ in addition to its trivial zeros at the negative even integers. It was proposed by Bernhard Riemann in his 1859 paper. The Riemann zeta function plays a great role in analytic number theory as well as in physics, probability theory and applied statistics. In this preprint, applying the known Beurling--Nyman criterion, it is disproved the Riemann hypothesis.