相关论文: Some identities for the Riemann zeta-function

The following theorem is proven: Both real and imaginary parts of the function F(s) defined as F(s)=zeta(s)*Gamma(s/2)*pi**(-s/2)=xi(s)/(s*(s-1)), and whose zeroes exactly coincide with the non-trivial zeroes of the Riemann zeta-function,…

The modified Mellin transform ${\cal Z}_k(s) = \int_1^\infty |\zeta({1\over2}+ix|^{2k}x^{-s}{\rm d} x$ ($k\ge1$ is a fixed integer, $s = \sigma + it$) is used to obtain estimates for $$…

We found, by Hurwitz's Zeta Function, a new functional equation for Riemann Zeta Function. Considering this equation for $s=2$ and $s=1$, we determine a relation between the values of Riemann zeta Function on positive integers. The Matrix…

An incomplete Riemann zeta function can be expressed as a lower-bounded, improper Riemann-Liouville fractional integral, which, when evaluated at $0$, is equivalent to the complete Riemann zeta function. Solutions to Landau's problem with…

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

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.

This paper studies the connections between the zeros and their distribution functions for two particular Dirichlet $L$ functions: the Riemann zeta function, and the Catalan beta function, also known as the Dirichlet beta function. It is…

We, by making use of elementary arguments, deduce integral representations of the Legendre chi function $\chi_{s}(x)$ valid for $|z|<1$ and $\Re(s)>1$. Our earlier established results on the integral representations for the Riemann zeta…

The leading asymptotic behaviour as $t\to \infty$ of the celebrated Riemann zeta function $\zeta(s), \ s = \sigma + it, \quad 0<\sigma<1, \quad t>0 , \ t\to\infty,$ can be expressed in terms of a transcendental sum. The sharp estimation of…

The Dirichlet eta function can be divided into $n$-th partial sum $\eta_{n}(s)$ and remainder term $R_{n}(s)$. We focus on the remainder term which can be approximated by the expression for $n$. And then, to increase reliability, we make…

For the Tornheim double zeta function T(s1,s2,s3) of complex variables,we obtain its functional equations,which are new.Using the calculus of r-th order derivative of zeta(s,alpha) as a function of alpha(developed in author[7])as the…

We investigate the values of the Riemann zeta function at odd integers and the Dirichlet beta function at even integers, by collecting several distinct analytic frameworks converging to these values, thus providing a unifying perspective.…

We study the distribution of large (and small) values of several families of $L$-functions on a line $\text{Re(s)}=\sigma$ where $1/2<\sigma<1$. We consider the Riemann zeta function $\zeta(s)$ in the $t$-aspect, Dirichlet $L$-functions in…

We improve the estimation of the distribution of the nontrivial zeros of Riemann zeta function $\zeta(\sigma+it)$ for sufficiently large $t$, which is based on an exact calculation of some special logarithmic integrals of nonvanishing…

In this short note, we give a proof of the Riemann hypothesis for Goss $v$-adic zeta function $\zeta_{v}(s)$, when $v$ is a prime of $\mathbb{F}_{q}[t]$ of degree one.

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…

In one of his posthumous papers, conserved in G\"ottingen, Riemann considers the derivatives of $\log\zeta(s)$ at the point $1/2$, giving explicit values for them. Around 2010 we shared Riemann's value of the second derivative with some…

We prove explicit upper bounds of the function $S_m(T)$, defined by the repeated integration of the argument of the Riemann zeta-function. The explicit upper bound of $S(T)$ and $S_1(T)$ have already been obtained by A. Fujii. Our result is…

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…

The functional equations of the Riemann zeta function, the Hurwitz zeta function, and the Lerch zeta function have been well known for a long time, and there is great importance in studying these zeta functions. For example, fundamental…