Related papers: The screw line of the Riemann zeta-function and it…

We introduce a screw function corresponding to the Riemann zeta-function and study its properties from various aspects. Typical results are several equivalent conditions for the Riemann hypothesis in terms of the screw function. One of them…

In this article, it is proved that the non-trivial zeros of the Riemann zeta function must lie on the critical line, known as the Riemann hypothesis.

Four propositions are considered concerning the relationship between the zeros of two combinations of the Riemann zeta function and the function itself. The first is the Riemann hypothesis, while the second relates to the zeros of a…

For the Riemann zeta-function, we introduce a function such that it is a characteristic function of an infinitely divisible distribution on the real line if and only if the Riemann Hypothesis is true.

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence \{\lambda_k\}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we…

This paper compares the distribution of zeros of the Riemann zeta function $\zeta(s)$ with those of a symmetric combination of zeta functions, denoted ${\cal T}_+(s)$, known to have all its zeros located on the critical line $\Re(s)=1/2$.…

We prove Riemann hypothesis. Method is to show the convexity of function which has zeros on open critical strip the same as zeta function.

We study the $M$-functions, which describe the limit theorem for the value-distributions of the secondary main terms in the asymptotic formulas for the summatory functions of the Goldbach counting function. One of the new aspects is a…

Riemann zeta function is an important object of number theory. It was also used for description of disordered systems in statistical mechanics. We show that Riemann zeta function is also useful for the description of integrable model. We…

In this paper we give criteria about estimation of derivatives of the Riemann Zeta Function on the line $\sigma=1$.

In this paper, a positive answer to the Riemann hypothesis is given by using a new result that predict the exact location of zeros of the alternating zeta function on the critical strip.

We present a quantum mechanical model which establishes the veracity of the Riemann hypothesis that the non-trivial zeros of the Riemann zeta-function lie on the critical line of $\zeta(s)$.

We introduce screw functions for Dirichlet series in the extended Selberg class. Then we prove that the Grand Riemann Hypothesis for a member of the extended Selberg class is equivalent to the nonpositivity of the corresponding screw…

In this paper we provide a proof of the Riemann Hypothesis by relating the non-trivial zeros of the zeta function to a certain Sturm-Liouville eigenvalue problem on a finite interval.

We prove the Riemann Hypothesis via an analytically regulated surface integral over the critical strip of the Riemann zeta function. The key idea is that the convergence of this normalized integral is equivalent to the condition that all…

In this manuscript, we show that the Riemann zeta function satisfies $\big(\zeta(s),\zeta(1-\overline{s})\big)\neq(0,0)$ for any $s$ in the critical strip, except on the critical line. This still holds even when the fractional part function…

We deal with the Euler's alternating series of the Riemann zeta function to define a regularized ratio appeared in the functional equation even in the critical strip and show some evidence to indicate the hypothesis in this note.

We investigate a dynamical basis for the Riemann hypothesis (RH) that the non-trivial zeros of the Riemann zeta function lie on the critical line x = 1/2. In the process we graphically explore, in as rich a way as possible, the diversity of…

We provide explicit bounds in the theory of the Riemann zeta-function at the line $\Re{s}=1$, assuming that the Riemann hypothesis holds until the height $T$. In particular, we improve some bounds, in finite regions, for the logarithmic…

By using new power inequalities we give an elementary proof of an important relation for the Riemann zeta-function |\zeta(1-s)| <= |\zeta(s)| in the strip 0< Re s<1/2,\ |\Im s| >= 12. Moreover, we establish a sufficient condition of the…