相关论文: A note on the mean square of the Riemann zeta-func…

Assume the Riemann hypothesis. On the right-hand side of the critical strip, we obtain an asymptotic formula for the discrete mean square of the Riemann zeta-function over imaginary parts of its zeros.

We obtain an asymptotic formula for the second discrete moment of the Riemann zeta function over the arithmetic progression $\frac{1}{2} + in$. It shows that the first main term is equal to that of the continuous mean value.

This review article brings forth some recent results in the theory of the Riemann zeta-function $qzeta(s)$.

This note contains a short proof of the functional equation for the zeta function.

Improving earlier work of Balasubramanian, Conrey and Heath-Brown, we obtain an asymptotic formula for the mean-square of the Riemann zeta-function times an arbitrary Dirichlet polynomial of length $T^{1/2 + \delta}$, with $\delta =…

The purpose of this paper is to prove that the so-called Quasi-Riemann Hypothesis for the Zeta-function implies the Riemann Hypothesis

This is a review of some of the interesting properties of the Riemann Zeta Function.

A new method for continuing the usual Dirichlet series that defines the Riemann zeta function ${\zeta}(s)$ is presented. Numerical experiments demonstrating the computational efficacy of the resulting continuation are discussed.

In this paper, we find a new recurrence formula fo the Euler zeta functions.

This paper is closely related to the recent work [BW17] of the same authors and our purpose is to elaborate more on some of the results and methods from [BW17]. More specifically our goal is two-fold. Firstly, we will indicate how a simple…

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…

A quite fast proof of the functional equation of the Riemann zeta function. It is a modification of a proof usually overlooked in Titchmarsh's monograph.

We refine a previous work of K. Matsumoto and H. Ishikawa, obtaining an asymptotic formula for the mean square of the product of the Riemann zeta-function and a Dirichlet polynomial in the critical strip (1/4<$\sigma$<1/2), by obtaining an…

It is proved that, for $T^\epsilon\le G = G(T) \le {1\over2}\sqrt{T}$, $$ \int_T^{2T}\Bigl(I_1(t+G)-I_1(t)\Bigr)^2 dt = TG\sum_{j=0}^3a_j\log^j \Bigl({\sqrt{T}\over G}\Bigr) + O_\epsilon(T^{1+\epsilon}+ T^{1/2+\epsilon}G^2) $$ with some…

In this paper we give criteria about estimation of derivatives of the Riemann Zeta Function on the line $\sigma=1$.

We show that there is a contradiction between the Riemann's Hypothesis and some form of the theorem on the universality of the zeta function.

This paper considers some infinite series involving the Riemann zeta function.

We deal 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 show some evidence to indicate the hypothesis in this note.

We obtain another proof of Hermite's integral for the Hurwitz zeta function.

A generalization of a well-known relation between the Riemann zeta function $\zeta(s)$ and Bernoulli numbers $B_n$ is obtained. The formula is a new representation of the Riemann zeta function in terms of a nested series of Bernoulli…