Related papers: On a generalisation of the Riemann $\xi$-function

Let $x$ be a complex number which has a positive real part, and $w_1,\ldots,w_N$ be positive rational numbers. We show that $w^s \zeta_N (s, x \ |\ w_1,\ldots, w_N)$ can be expressed as a finite linear combination of the Hurwitz zeta…

A GGC (Generalized Gamma Convolution) representation of Riemann's Xi-function is constructed.

As a generalization of the Dedekind zeta function, Weng defined the high rank zeta functions and proved that they have standard properties of zeta functions, namely, meromorphic continuation, functional equation, and having only two simple…

This is a survey of results on the following problem. Consider a simply connected Riemann surface spread over the Riemann sphere. How are the properties of the uniformizing function of this surface related to the geometric properties of the…

The secondary zeta function $Z(s)=\sum_{n=1}^\infty\alpha_n^{-s}$, where $\rho_n=\frac12+i\alpha_n$ are the zeros of zeta with $\Im(\rho)>0$, extends to a meromorphic function on the hole complex plane. If we assume the Riemann hypothesis…

Hardy's theorem for the Riemann zeta-function $\zeta(s)$ says that it admits infinitely many complex zeros on the line $\Re({s}) = \frac{1}{2}$. In this note, we give a simple proof of this statement which, to the best of our knowledge, is…

Consider a closed analytic curve $\gamma$ in the complex plane and denote by > $D_+$ and $D_-$ the interior and exterior domains with respect to the curve. The point $z=0$ is assumed to be in $D_+$. Then according to Riemann theorem there…

In this paper, on the basis of a specific question raised in [6], we further continue our investigations on the uniqueness of a meromorphic function with its higher derivatives sharing two sets and answer the question affirmatively.…

Using a different approach, we derive integral representations for the Riemann zeta function and its generalizations (the Hurwitz zeta, $\zeta(-k,b)$, the polylogarithm, $\mathrm{Li}_{-k}(e^m)$, and the Lerch transcendent,…

We study some "density function" related to the value-distribution of $L$-functions. The first example of such a density function was given by Bohr and Jessen in 1930s for the Riemann zeta-function. In this paper, we construct the density…

The $\psi(x)$-function, which solves the equation $x = \sinh(aw)e^w$ for $0<a<1$, has a natural connection to the renowned Lambert $W$ function and also physical relevance through its connection to the Lenz-Ising model of ferromagnetism. We…

Y\i ld\i r\i m obtained an asymptotic formula of the discrete moment of $|\zeta(\frac{1}{2}+it)|$ over the zero of the higher derivatives of Hardy's $Z$-function. We give a generalization of his result on Hardy's $Z$-function.

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…

Let $K$ be a quadratic field, and let $\zeta_K$ its Dedekind zeta function. In this paper we introduce a factorization of $\zeta_K$ into two functions, $L_1$ and $L_2$, defined as partial Euler products of $\zeta_K$, which lead to a…

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

The main aim of this paper is to give a new generalization of Hurwitz-Lerch Zeta function of two variables.Also, we investigate several interesting properties such as integral representations, summation formula and a connection with…

We prove a mixed joint discrete universality theorem for a Matsumoto zeta-function $\varphi(s)$ (belonging to the Steuding subclass) and a periodic Hurwitz zeta-function $\zeta(s,\alpha;{\mathfrak{B}})$. For this purpose, certain…

In 1914, Hardy proved that infinitely many non-trivial zeros of the Riemann zeta function lie on the critical line using the transformation formula of the Jacobi theta function. Recently the first author obtained an integral representation…

In the paper, we introduce $q$-deformations of the Riemann zeta function, extend them to the whole complex plane, and establish certain estimates of the number of roots. The construction is based on the recent difference generalization of…

Sections of the Hardy $Z$-function are given by $Z_N(t) := \sum_{k=1}^{N} \frac{cos(\theta(t)-ln(k) t) }{\sqrt{k}}$ for any $N \in \mathbb{N}$. Sections approximate the Hardy $Z$-function in two ways: (a) $2Z_{\widetilde{N}(t)}(t)$ is the…