Related papers: On the zeta function on the line Re(s) = 1

We discuss modifications in the integral representation of the Riemann zeta-function that lead to generalizations of the Riemann functional equation that preserves the symmetry $s\to (1-s)$ in the critical strip. By modifying one integral…

We provide explicit ranges for $\sigma$ for which the asymptotic formula \begin{equation*} \int_0^T|\zeta(1/2+it)|^4|\zeta(\sigma+it)|^{2j}dt \;\sim\; T\sum_{k=0}^4a_{k,j}(\sigma)\log^k T \quad(j\in\mathbb N) \end{equation*} holds as…

We supplement a very recent paper of R. Crandall concerned with the multiprecision computation of several important special functions and numbers. We show an alternative series representation for the Riemann and Hurwitz zeta functions…

Starting from the symmetrical reflection functional equation of the zeta function, we have found that the sigma values satisfying zeta(s) = 0 must also satisfy both |zeta(s)| = |zeta(1 - s)| and |gamma(s/2)zeta(s)| = |gamma((1 - s)/2)zeta(1…

Levinson and Montgomery proved that the Riemann zeta-function $\zeta(s)$ and its derivative have approximately the same number of non-real zeros left of the critical line. R. Spira showed that $\zeta'(1/2+it)=0$ implies $\zeta(1/2+it)=0$.…

In this paper we perform a detailed analysis of Riemann's hypothesis, dealing with the zeros of the analytically-extended zeta function. We use the functional equation $\zeta(s) = 2^{s}\pi^{s-1}\sin{(\displaystyle \pi…

The Riemann zeta function can be written as the Mellin transform of the unit interval map w(x) = floor(1/x)*(-1+x*floor(1/x)+x) multiplied by s((s+1)/(s-1)). A finite-sum approximation to \zeta (s) denoted by \zeta_w(N;s) which has real…

It is proved that Epstein's zeta-function $\zeta_{Q}(s)$, related to a positive definite integral binary quadratic form, has a zero $1/2 + i\gamma$ with $ T \leq \gamma \leq T + T^{{3/7} +\varepsilon} $ for sufficiently large positive…

Assuming the Generalized Riemann Hypothesis, we provide uniform upper and lower bounds with explicit main terms for $\log{\left|\cL(s)\right|}$ for $\sigma \in (1/2,1)$ and for functions in the Selberg class. In particular, we focus on the…

This paper contains a small improvement to the explicit bounds on the growth of the function $S(T)$. It is shown how more substantial improvements are possible if one has better explicit bounds on the growth of $|\zeta(\frac{1}{2}+it)|$.

This paper studies the non-holomorphic Eisenstein series E(z,s) for the modular surface, and shows that integration with respect to certain non-negative measures gives meromorphic functions of s that have all their zeros on the critical…

Let $S(t) \;:=\; \frac{\displaystyle 1}{\displaystyle \pi}\arg \zeta(\frac{1}{2} + it)$. We prove that, for $T^{\,27/82+\varepsilon} \le H \le T$, we have $$ {\rm mes}\Bigl\{t\in [T, T+H]\;:\; S(t)>0\Bigr\} = \frac{H}{2} +…

In the present paper, we show that under the Riemann hypothesis, and for fixed $h, \epsilon > 0$, the supremum of the real and the imaginary parts of $\log \zeta (1/2 + it)$ for $t \in [UT -h, UT + h]$ are in the interval $[(1-\epsilon)…

We study lower bounds for the Riemann zeta function $\zeta(s)$ along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the…

A scaling and renormalization approach to the Riemann zeta function, $\zeta$, evaluated at $-1$ is presented in two ways. In the first, one takes the difference between $U_{n}:=\sum_{q=1}^{n}q$ and $4U_{\left\lfloor \frac{n}{2}\right\rfloor…

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…

We give a representation of the classical Riemann $\zeta$-function in the half plane $\Re s>0$ in terms of a Mellin transform involving the real part of the dilogarithm function with an argument on the unit circle (associated Clausen…

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)…

Various properties of the Mellin transform function $$ {\cal M}_k(s) := \int_1^\infty Z^k(x)x^{-s}dx $$ are investigated, where $$ Z(t) := \zeta(1/2+it){\bigl(\chi(1/2+it)\bigr)}^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s) $$ is Hardy's…

A discussion involving the evaluation of the sum $\sum_{0<\gamma\le T} |\zeta(1/2+i\gamma)|^2$ is presented, where $\gamma$ denotes imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. Three theorems involving certain…