相关论文: The Values of the Riemann Zeta-Function on Discret…

We study lower bounds for the Riemann zeta function $\zeta(s)$ along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the…

This paper continues a series of investigations on converging representations for the Riemann Zeta function. We generalize some identities which involve Riemann's zeta function, and moreover we give new series and integrals for the zeta…

In 2017, Garunk\v{s}tis, Laurin\v{c}ikas and Macaitien\.{e} proved the discrete universality theorem for the Riemann zeta-function sifted by the nontrivial zeros of the Riemann zeta-function. This discrete universality has been extended in…

A relationship between the Riemann zeta function and a density on integer sets is explored. Several properties of the examined density are derived.

We obtain asymptotic formulae for the second discrete moments of the Riemann zeta function over arithmetic progressions $\frac{1}{2} + i(a n + b)$. It reveals noticeable relation between the discrete moments and the continuous moment of the…

An elementary approach for computing the values at negative integers of the Riemann zeta function is presented. The approach is based on a new method for ordering the integers and a new method for summation of divergent series. We show that…

We show that the $n$th derivative of the Riemann zeta function, when summed over the non-trivial zeros of zeta, is real and positive/negative in the mean for $n$ odd/even, respectively. We show this by giving a full asymptotic expansion of…

We consider iterated integrals of $\log\zeta(s)$ on certain vertical and horizontal lines. Here, the function $\zeta(s)$ is the Riemann zeta-function. It is a well known open problem whether or not the values of the Riemann zeta-function on…

In this article we compute a discrete mean value of the derivative of the Riemann zeta function. This mean value will be important for several applications concerning the size of $\zeta'(\rho)$ where $\zeta(s)$ is the Riemann zeta function…

A simple and elementary derivation of values at integer points for the Riemann's zeta and related functions is reported.

In this work we show that the Riemann hypothesis for the Dedekind zeta--function $\zeta_{\mathrm{K}}(s)$ of an algebraic number field $\mathrm{K}$ is equivalent to a problem of the rate of convergence of certain discrete measures defined…

We employ mean value estimates of Weyl sums in order to obtain discrete second moments of the Riemann Zeta-function with respect to polynomials near the vertical line $1+i\mathbb{R}$.

To evaluate Riemann's zeta function is important for many investigations related to the area of number theory, and to have quickly converging series at hand in particular. We investigate a class of summation formulae and find, as a special…

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…

For an arbitrary complex number $a\neq 0$ we consider the distribution of values of the Riemann zeta-function $\zeta$ at the $a$-points of the function $\Delta$ which appears in the functional equation $\zeta(s)=\Delta(s)\zeta(1-s)$. These…

The Riemann Hypothesis has been of central interest to mathematicians for a long time and many unsuccessful attempts have been made to either prove or disprove it. Since the Riemann zeta function is defined as a sum of the infinite number…

A family of Zeta functions built as Dirichlet series over the Riemann zeros are shown to have meromorphic extensions in the whole complex plane, for which numerous analytical features (the polar structure, plus countably many special…

We show that the twisted second moments of the Riemann zeta function averaged over the arithmetic progression $1/2 + i(an + b)$ with $a > 0$, $b$ real, exhibits a remarkable correspondance with the analogous continuous average and derive…

We prove a general result on representing the Riemann zeta function as a convergent infinite series in a complex vertical strip containing the critical line. We use this result to re-derive known expansions as well as to discover new series…

We present an explicit formula for a weighted sum over the zeros of the Riemann zeta function. This weighted sum is evaluated in terms of a sum over the prime numbers, weighted with help of the Hermite polynomials. From the explicit formula…