Related papers: Explicit formula in a imaginary quadratic number f…

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…

In this paper, we obtain explicit bounds for the real part of the logarithmic derivative of the Riemann zeta-function on the line $\re s=1$, assuming the Riemann hypothesis. The proof combines the Guinand--Weil explicit formula with…

We give explicit formulae for all of the terms in the asymptotic expansion of the mean fourth power of the Riemann zeta-function on the critical line.

It is well known that short-time expansions of heat kernels correlate to formal high-frequency expansions of spectral densities. It is also well known that the latter expansions are generally not literally true beyond the first term.…

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…

We survey some of the universality properties of the Riemann zeta function $\zeta(s)$ and then explain how to obtain a natural quantization of Voronin's universality theorem (and of its various extensions). Our work builds on the theory of…

This article describes a sequence of rational functions which converges locally uniformly to the zeta function. The numerators (and denominators) of these rational functions can be expressed as characteristic polynomials of matrices that…

For Hurwitz Zeta function,we consider its Taylor series expansion about various points as an analytic function of second variable in appropriate discs.We show that these Taylor are all polynomials in second variable for a non positive…

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

We compute the second moment of the Riemann zeta function for shifted arguments over a domain that extends the ones in the literature. We use the Riemann-Siegel formula for the error term in the approximate functional equation and take the…

As a generalization of [KMW], we introduce a higher Riemann zeta function for an abstract sequence. Then we explicitly determine its regularized product expression.

The Riemann zeta function and the distribution of its nontrivial zeros on the critical line remain central topics in analytic number theory and large-scale computation. This work develops a numerical framework that replaces classical…

We extend formulae of Mertens and Mirsky on the asymptotic behaviour of the standard Euler function to the Euler functions of principal rings of integers of imaginary quadratic number fields, giving versions in angular sector and with…

We revisit a representation for the Riemann zeta function $\zeta(s)$ expressed in terms of normalised incomplete gamma functions given by the author and S. Cang in Methods Appl. Anal. {\bf 4} (1997) 449--470. Use of the uniform asymptotics…

This article proves the Riemann hypothesis, which states that all non-trivial zeros of the zeta function have a real part equal to 1/2. We inspect in detail the integral form of the (symmetrized) completed zeta function, which is a product…

We build on a recent paper on Fourier expansions for the Riemann zeta function. We establish Fourier expansions for certain $L$-functions, and offer series representations involving the Whittaker function $W_{\gamma,\mu}(z)$ for the…

We give new integral and series representations of the Hurwitz zeta function. We also provide a closed-form expression of the coefficients of the Laurent expansion of the Hurwitz-zeta function about any point in the complex plane.

The modified zeta functions $\sum_{n \in K} n^{-s}$, where $K \subset \N$, converge absolutely for $\Re s > 1/2$. These generalise the Riemann zeta function which is known to have a meromorphic continuation to all of $\C$ with a single pole…

This is an anthology of series involving rational, factorial, and power functions expressed in terms of special functions. New finite expansions involving quotient functions expressed in terms of the Hurwitz-Lerch zeta function are given.…

The secondary zeta function is defined as a generalized zeta series over the imaginary parts of non-trivial zeros assuming (RH). This function admits Laurent series expansion at the double pole at $s=1$. In this article, we derive a new…