Related papers: On the approximation of the zeta function by Diric…

Let $d(n)$ be the number of divisors of $n$, let $\gamma$ denote Euler's constant and $$ \Delta(x) := \sum_{n\le x}d(n) - x(\log x + 2\gamma -1) $$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta(s)$ denote…

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) = E(t) - 2\pi\Delta^*(t/(2\pi))$ with $\Delta^*(x) = -\Delta(x)…

We compute the number of irreducible linear representations of self-similar branch groups, by expressing these numbers as the co\"efficients a_n of a Dirichlet series sum a_n n^{-s}. We show that this Dirichlet series has a positive…

A recently published result states inequalities of the harmonic mean of the digamma function. In this work, we prove among others results that for all positive real numbers $x\neq 1$, $$-\gamma<-\gamma…

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 consider analytic functions of the Riemann zeta type, for which, if $s$ is a zero, so is $1-s$. We use infinite product representations of these functions, assuming their zeros to be of first order. We use exponential factors to…

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 $1+i\mathbb{R}$.

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…

In this paper, we present an improved explicit subconvexity result for the Riemann zeta function $\zeta\left( s\right)$ along the critical line $s=1/2+it$, given by Hiary, Patel and Yang in 2024. This new bound is derived by combining a…

Let $S(\sigma,t)=\frac{1}{\pi}\arg\zeta(\sigma+it)$ be the argument of the Riemann zeta function at the point $\sigma+it$ of the critical strip. For $n\geq 1$ and $t>0$ we define $$ S_{n}(\sigma,t) = \int_0^t S_{n-1}(\sigma,\tau)\,d\tau\, +…

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) = E(t) - 2\pi\Delta^*(t/2\pi)$ with $\Delta^*(x) = -\Delta(x) +…

This article proves the bound $|\zeta(\frac{1}{2} + it)|\leq 0.732 t^{\frac{1}{6}} \log t$ for $t \geq 2$, which improves on a result by Cheng and Graham. We also show that $|\zeta(\frac{1}{2}+it)|\leq 0.732 |3.3081+it|^{\frac{1}{6}} \log…

In this series we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper begins the general study of…

We apply the resonance method to Montgomery's convolution formula for $\textrm{Re}\left(e^{-i\theta}\log\zeta(\sigma+it)\right)$ in the strip $1/2 < \sigma < 1$. This gives new insight into maximal values of…

Already in 1734 Euler found a short explicit formula for the value of Riemann zeta function Zeta(s) when the argument s equals a positive integer 2n where n=1,2,3,. No such formula exists for odd positive integer arguments of Zeta. The…

It is proved that if $T$ is sufficiently large, then uniformly for all positive integers $\ell \leqslant (\log T) / (\log_2 T)$, we have \begin{equation*} \max_{T\leqslant t\leqslant 2T}\left|\zeta^{(\ell)}\Big(1+it\Big)\right| \geqslant…

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…

Let $L(s)=\sum_{n=1}^{+\infty}\dfrac{a(n)}{n^s}$ be a Dirichlet series were $a(n)$ is a bounded completely multiplicative function. We prove that if $L(s)$ extends to a holomorphic function on the open half space $\Re s >1-\delta$,…

In this paper we obtain a mean value theorem for a general Dirichlet series $f(s)= \sum_{j=1}^\infty a_j n_j^{-s}$ with positive coefficients for which the counting function $A(x) = \sum_{n_{j}\le x}a_{j}$ satisfies $A(x)=\rho x +…

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…