Related papers: On the prime zeta function and the Riemann hypothe…

We express the Riemann zeta function $\zeta\left(s\right)$ of argument $s=\sigma+i\tau$ with imaginary part $\tau$ in terms of three absolutely convergent series. The resulting simple algorithm allows to compute, to arbitrary precision,…

We prove that the Riemann zeta-function is not universal on the critical line by using the fact that the Hardy Z-function is real, and some elementary considerations. This is a related to a recent result of Garunkstis and Steuding. We also…

We investigate the values of the Riemann zeta function at odd integers and the Dirichlet beta function at even integers, by collecting several distinct analytic frameworks converging to these values, thus providing a unifying perspective.…

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 discusses a few main topics in Number Theory, such as the M\"{o}bius function and its generalization, leading up to the derivation of neat power series for the prime counting function, $\pi(x)$, and the prime-power counting…

As a generalization of the Dedekind zeta function, Weng defined the high rank zeta functions and proved that they have standard properties of zeta functions, namely, meromorphic continuation, functional equation, and having only two simple…

The purpose of the present paper is to reveal the relation between the behavior of the logarithm of the Riemann zeta-function $\log{\zeta(s)}$ and the distribution of zeros of the Riemann zeta-function. We already know some examples for the…

We show the estimates \inf_T \int_T^{T+\delta} |\zeta(1+it)|^{-1} dt =e^{-\gamma}/4 \delta^2+ O(\delta^4) and \inf_T \int_T^{T+\delta} |\zeta(1+it)| dt =e^{-\gamma} \pi^2/24 \delta^2+ O(\delta^4) as well as corresponding results for…

By the second mean-value theorem of calculus (Gauss-Bonnet theorem) we prove that the class of functions ${\mit \Xi}(z)$ with an integral representation of the form $\int_{0}^{+\infty}du\,{\mit \Omega}(u)\,{\rm ch}(uz)$ with a real-valued…

We make plausible the existence of counterexamples to the Riemann hypothesis located in the neighbourhood of unusually large peaks of $\vert \zeta \vert$. The main ingredient in our argument is an identity which links the zeros of a…

We prove that the Riemann hypothesis is equivalent to the condition $\int_{2}^x\left(\pi(t)-\text{li}(t)\right)\mathrm{d}t<0$ for all $x>2$. Here, $\pi(t)$ is the prime-counting function and $\text{li}(t)$ is the logarithmic integral. This…

The individual terms of the series representing the Riemann zeta function are examined geometrically from their accumulated plot in the complex plane. Symmetry is identified and determined mathematically for comparison with more traditional…

We have looked at the evaluation of the Riemann Zeta function at odd arguments and have provided a simple formula to approximate the value with exponential convergence. We have compared it with various other formulae present in literature.…

This is a reformulation and refutation of a proposed proof of the Riemann hypothesis published in 2013 (arXiv:1305.0323) and in 2014 (arXiv:1402.2822). Proceeding by contradiction, the author wants to prove that if zeta(s)=0 where 1/2<Re…

We study generalised prime systems $\mathcal{P}$ $(1<p_1\leq p_2\leq...,$ with $p_j\in\R$ tending to infinity) and the associated Beurling zeta function $\zeta_{\mathcal{P}}(s) =\prod_{j=1}^{\infty} (1-p_j^{-s})^{-1}$. Under appropriate…

We unconditionally prove a central limit theorem for linear statistics of the zeros of the Riemann zeta function with diverging variance. Previously, theorems of this sort have been proved under the assumption of the Riemann hypothesis. The…

We investigate the distribution of the Riemann zeta-function on the line $\Re(s)=\sigma$. For $\tfrac 12 < \sigma \le 1$ we obtain an upper bound on the discrepancy between the distribution of $\zeta(s)$ and that of its random model,…

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…

Results of extensive computations of moments of the Riemann zeta function on the critical line are presented. Calculated values are compared with predictions motivated by random matrix theory. The results can help in deciding between those…

An explicit identity of sums of powers of complex functions presented via this a closed-form formula of Riemann zeta function produced at any given non-zero complex numbers. The closed-form formula showed us Riemann zeta function has no…