Related papers: Some notes about the Zeta function and the Riemann…
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…
Global mapping properties of the Riemann Zeta function are used to investigate its non trivial zeros.
A key theorem formulated in the context of functional Mellin transforms generalizes the important relationship $\exp\mathrm{tr} M=\det\exp M$. Along with the involution symmetry of the zeta function, the theorem suggests a strategy for…
On the critical line the conditional distribution of the zeta function's magnitude around zeta zeros exists and predicts the well-known pair correlation between nontrivial zeta zeros. However, this conditional distribution does not exist at…
The Riemann hypothesis, which states that the non-trivial zeros of the Riemann zeta function all lie on a certain line in the complex plane, is one of the most important unresolved problems in mathematics. Inspired by the P\'olya-Hilbert…
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…
A fresh approach to the long debated question is proposed, starting from the GRAM-BACKLUND analytical continuation of the Zeta function (G-B Zeta expression). Consideration is given to the symmetric (even-exponent) and anti-symmetric (odd…
Let $Z(t)$ be the classical Hardy function in the theory of the Riemann zeta-function. The main result in this paper is that if the Riemann hypothesis is true then for any positive integer $n$ there exists a $t_{n}>0$ such that for…
For $N \in \mathbb{N}$ consider the $N$-th section of the approximate functional equation $$ \zeta_N(s)= \sum_{n =1 }^N B_n(s),$$ where $$ B_n(s)= \frac{1}{2} \left [ n^{-s} + \chi(s) \cdot n^{s-1} \right ].$$ Our aim in this work is to…
The goal of this paper is to give a relatively simple proof of some known zero density estimates for Riemann zeta function which are sufficiently strong to break the density hypothesis in a nontrivial part of the critical strip. Apart from…
In this paper, we focus on the explicit expression of an extended version of Riemann zeta function. We use two different methods, Mellin inversion formula and Cauchy's residue theorem, to calculate a Mellin-Barnes type integral of the…
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…
In this paper we study the distribution of the non-trivial zeros of the Riemann zeta-function $\zeta(s)$ (and other L-functions) using Montgomery's pair correlation approach. We use semidefinite programming to improve upon numerous…
We introduce a differential topological proof and an analytical proof of Riemann hypothesis according to the saddle point method because Riemann calculated the integral representation of zeta function on the critical line by this method.…
The positivity of the sum from the title is the first condition in the well-known criterium for the validity of the Riemann Hypothesis suggested by X.-J. Li. In the paper this value is represented as an infinite sum with positive summands.
A strategy for proving Riemann hypothesis is suggested. The vanishing of the Rieman Zeta reduces to an orthogonality condition for the eigenfunctions of a non-Hermitian operator $D^+$ having the zeros of Riemann Zeta as its eigenvalues. The…
A previous exploration of the Riemann functional equation that focussed on the critical line, is extended over the complex plane. Significant results include a simpler derivation of the fundamental equation developed previously, and its…
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…
Results of a multipart work are outlined. Use is made therein of the conjunction of the Riemann hypothesis, RH, and hypotheses advanced by the author. Let z(n) be the nth nonreal zero of the Riemann zeta-function with positive imaginary…
We establish a uniform upper estimate for the values of zeta(s)/zeta(s+A), 0<= A, on the critical line (conditionally on the Riemann Hypothesis). We use this to give a variant, purely complex analytic, to Baez-Duarte's proof of a…