Related papers: A note on Hardy's theorem

We review generalized zeta functions built over the Riemann zeros (in short: "superzeta" functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz…

Riemann numerically approximated at least three zeta zeros. According to Edwards, Riemann even took steps to verify that the lowest zero he computed was indeed the first zeta zero. This approach to verification is developed, improved, and…

Assuming GRH, we prove an explicit upper bound for the number of zeros of a Dedekind zeta function having imaginary part in $[T-a,T+a]$. We also prove a bound for the multiplicity of the zeros.

We show that at least 19/27 of the zeros of the Riemann zeta-function are simple, assuming the Riemann Hypothesis (RH). This was previously established by Conrey, Ghosh and Gonek [Proc. London Math. Soc. 76 (1998), 497--522] under the…

We prove some new bounds for the maximum of Riemann zeta-function on very short segments of the critical line. All the theorems are based on the Riemann hypothesis.

It is shown that a new series representation of Riemann s Zeta function obtained by Tyagi and Holm leads to an interesting new recursion for Bernoulli numbers of even index as well as new representations of, and infinite series involving,…

We prove that there exist infinitely many consecutive zeros of the Riemann zeta-function on the critical line whose gaps are greater than $3.18$ times the average spacing. Using a modification of our method, we also show that there are even…

We show that the analytic continuations of Helson zeta functions $ \zeta_\chi (s)= \sum_1^{\infty}\chi(n)n^{-s} $ can have essentially arbitrary poles and zeroes in the strip $ 21/40 < \Re s < 1 $ (unconditionally), and in the whole…

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…

In 1975 Voronin proved the universality theorem for the Riemann zeta-function $\zeta(s)$ which roughly says that any admissible function $f(s)$ is approximated by $\zeta(s)$. A few years later Reich proved a discrete analogue of this…

it is proved that at least 41.28% zeros of the Riemann zeta function are on the critical line

The finite Dirichlet series from the title are defined by the condition that they vanish at as many initial zeroes of the zeta function as possible. It turned out that such series can produce extremely good approximations to the values of…

A proof of the Riemann hypothesis is proposed by relying on the properties of the Mellin transform. The function $\mathfrak{G}_{\eta}\left(t\right)$ is defined on the set $\bar{\mathbb{R}}_+$ of the non-negative real numbers, in term of a…

We formulate a parametrized uniformly absolutely globally convergent series of $\zeta$(s) denoted by Z(s, x). When expressed in closed form, it is given by Z(s, x) = (s -- 1)$\zeta$(s) + 1 x Li s z z -- 1 dz, where Li s (x) is the…

This paper studies the non-holomorphic Eisenstein series E(z,s) for the modular surface, and shows that integration with respect to certain non-negative measures gives meromorphic functions of s that have all their zeros on the critical…

The meromorphic function $W(s)$ introduced in the Riemann-Zeta function $\zeta(s) = W(s) \zeta(1-s)$ maps the line of $s = 1/2 + it$ onto the unit circle in $W$-space. $|W(s)| = 0$ gives the trivial zeroes of the Riemann-Zeta function…

Let $\Xi(t)$ be a function relating to the Riemann zeta function $\zeta (s)$ with $s = \frac{1} {2} + it$. In this paper, we construct a function $v$ containing $t$ and $\Xi(t)$, and prove that $v$ satisfies a nonadjoint boundary value…

In this paper, some new results are reported for the study of Riemann zeta function $\zeta(s)$ in the critical strip $0<Re(s)<1$, such as $\zeta(s)$ expressed in a generalized Euler product only involving prime numbers. Particularly, some…

Let $y\ne 0$ and $C>0$. Under the Riemann Hypothesis, there is a number $T_*>0$ $($depending on $y$ and $C)$ such that for every $T\ge T_*$, both \[ \zeta(\tfrac12+i\gamma)=0 \quad\text{and}\quad\zeta(\tfrac12+i(\gamma+y))\ne 0 \] hold for…

We prove a density theorem for the auxiliar function $\mathop{\mathcal R}(s)$ found by Siegel in Riemann papers. Let $\alpha$ be a real number with $\frac12< \alpha\le 1$, and let $N(\alpha,T)$ be the number of zeros $\rho=\beta+i\gamma$ of…