Related papers: Non universality on the critical line

We estimate large and small values of $|\zeta(\rho')|$, where $\rho'$ runs over critical points of the zeta function in the right half of the critical strip, that is, the points where $\zeta'(\rho')=0$ and $1/2<\Re \rho'<1$.

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…

Several identities for the Riemann zeta-function $\zeta(s)$ are proved. For example, if $s = \sigma + it$ and $\sigma > 0$, then $$ \int_{-\infty}^\infty |{(1-2^{1-s})\zeta(s)\over s}|^2dt = {\pi\over\sigma}(1 -…

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…

Assuming the Riemann Hypothesis, we provide effective upper and lower estimates for $\left|\zeta(s)\right|$ right to the critical line. As an application we make explicit Titchmarsh's conditional bound for the Mertens function and…

Two identities extracted from the literature are coupled to obtain an integral equation for Riemann's $\xi(s)$ function, and thus $\zeta(s)$ indirectly. The equation has a number of simple properties from which useful derivations flow, the…

In article, we explore the secondary zeta function $Z(s)$, which is defined as a generalized zeta type of series over imaginary parts of non-trivial zeros of the Riemann zeta function $\zeta(s)$. This function has been analytically…

We prove under RH the existence of a very large positive and negative values of the argument of the Riemann zeta function on a very short intervals.

We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function is well approximated by short truncations of its Euler product. Conversely, if the…

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…

For a fixed integer $k\ge 3$ and fixed $1/2 < \sigma > 1$ we consider $$ \int_1^T |\zeta(\sigma + it)|^{2k}dt = \sum_{n=1}^\infty d_k^2(n)n^{-2\sigma}T + R(k,\sigma;T), $$ where $R(k,\sigma;T) = o(T) (T\to\infty)$ is the error term in the…

In 1946, A. Selberg proved $N(\sigma,T) \ll T^{1-\frac{1}{4} \left(\sigma-\frac{1}{2}\right)} \log{T}$ where $N(\sigma,T)$ is the number of nontrivial zeros $\rho$ of the Riemann zeta-function with $\Re\{\rho\}>\sigma$ and…

We consider generalised root identities for zeta functions of curves over finite fields, \zeta_{k}, and compare with the corresponding analysis for the Riemann zeta function. We verify numerically that, as for \zeta, the \zeta_{k} do…

In this paper, we will give a new proof for a known result of the mean square of Riemann zeta-function.

We give a new proof that the Riemann zeta function is nonzero in the half-plane $\{s\in{\mathbb C}:\sigma>1\}$. A novel feature of this proof is that it makes no use of the Euler product for $\zeta(s)$.

In this paper we investigate the positivity property of the real part of logarithmic derivative of the Riemann $\xi$-function for $1/2<\sigma<1$ and sufficiently large $t$. We give an explicit upper and lower bounds for $\Re\sum_{\rho}…

Let $\zeta_K(s)$ denote the Dedekind zeta-function associated to a number field $K$. In this paper, we give an effective upper bound for the height of first non-trivial zero other than $1/2$ of $\zeta_K(s)$ under the generalized Riemann…

An explicit subconvex bound for the Riemann zeta function $\zeta(s)$ on the critical line $s=1/2+it$ is proved. Previous subconvex bounds relied on an incorrect version of the Kusmin-Landau lemma. After accounting for the needed correction…

We have dealt 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 showed some evidence to indicate the hypothesis. We briefly…

We give a short proof of Levinson's result that more than 1/3 of the zeros of the zeta function are on the critical line.