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

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 just proved the Riemann hypothesis.