Related papers: Vinogradov's integral and bounds for the Riemann z…

In this article we prove an explicit sub-Weyl bound for the Riemann zeta function $\zeta(s)$ on the critical line $s = 1/2 + it$. In particular, we show that $|\zeta(1/2 + it)| \le 66.7\, t^{27/164}$ for $t \ge 3$. Combined, our results…

We investigate the extreme values of the Riemann zeta function $\zeta(s)$. On the 1-line, we obtain a lower bound evaluation $$\max_{t\in[1,T]}|\zeta(1+\i t)|\ge {\rm e}^\gamma(\log_2T+\log_3T+c),$$ with an effective constant $c$ which…

We show the estimates \inf_T \int_T^{T+\delta} |\zeta(1+it)|^{-1} dt =e^{-\gamma}/4 \delta^2+ O(\delta^4) and \inf_T \int_T^{T+\delta} |\zeta(1+it)| dt =e^{-\gamma} \pi^2/24 \delta^2+ O(\delta^4) as well as corresponding results for…

We prove that $|\zeta(\sigma+it)|\le 70.7 |t|^{4.438 (1-\sigma)^{3/2}}\log^{2/3}|t|$ for $1/2\le\sigma\le 1$ and $|t|\ge 3$. As a consequence, we improve the explicit zero-free region for $\zeta(s)$, showing that $\zeta(\sigma+it)$ has no…

In this article, we give explicit bounds of order $\log t$ for $\sigma$ close to $1$, for two quantities: $|\zeta'(\sigma +it)/\zeta(\sigma +it)|$ and $|1/\zeta(\sigma +it)|$. We correct an error in the literature, and especially in the…

It is proved that $$\int_{T}^{2T} \left|\frac{\zeta\left(\frac{1}{2}+{\rm i} t\right)}{\zeta\left(1+2{\rm i} t\right)}\right|^2 {\rm d} t = \frac{1}{\zeta(2)} T \log T + \left( \frac{\log \frac{2}{\pi} + 2\gamma -1 }{\zeta(2)} -4…

The research shows that Riemann proved that all of zeros of Riemann's zeta function are on $\sigma=1/2$ based on the functional equation \begin{align*} \pi^{-\frac{s}{2}}\Gamma \left( \frac{s}{2} \right) \zeta(s)&={\frac{1}{s(s-1)} +…

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

Let $\zeta(s)$ and $Z(t)$ be the Riemann zeta function and Hardy's function respectively. We show asymptotic formulas for $\int_0^T Z(t)\zeta(1/2+it)dt$ and $\int_0^T Z^2(t) \zeta(1/2+it)dt$. Furthermore we derive an upper bound for…

A simple proof of the classical subconvexity bound $\zeta(1/2+it) \ll_\epsilon t^{1/6+\epsilon}$ for the Riemann zeta-function is given, and estimation by more refined techniques is discussed. The connections between the Dirichlet divisor…

In this work, we show that for all $t\geq e$, \[|\zeta(1+it)|\leq 0.6443 \log t. \] The equality is achieved when $t=17.7477$. We also use the Riemann-Siegel formula and numerical computations to show that \[|\zeta(1+it)|\leq\frac{1}{2}\log…

We improve Montgomery's $\Omega$-results for $|\zeta(\sigma+it)|$ in the strip $1/2<\sigma<1$ and give in particular lower bounds for the maximum of $|\zeta(\sigma+it)|$ on $\sqrt{T}\le t \le T$ that are uniform in $\sigma$. We give similar…

An explicit subconvex bound for the Riemann zeta function $\zeta(s)$ on the critical line $s=1/2+it$ is proved. Previous subconvex bounds relied on an incorrect version of the Kusmin-Landau lemma. After accounting for the needed correction…

Assuming the Riemann Hypothesis, we provide explicit upper bounds for moduli of $S(t)$, $S_1(t)$, and $\zeta\left(1/2+\mathrm{i}t\right)$ while comparing them with recently proven unconditional ones. As a corollary we obtain a conditional…

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 give a slight refinement to the process by which estimates for exponential sums are extracted from bounds for Vinogradov's mean value. Coupling this with the recent works of Wooley, and of Bourgain, Demeter and Guth, providing optimal…

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…

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) = -…

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…

We consider the real part $\Re(\zeta(s))$ of the Riemann zeta-function $\zeta(s)$ in the half-plane $\Re(s) \ge 1$. We show how to compute accurately the constant $\sigma_0 = 1.19\ldots$ which is defined to be the supremum of $\sigma$ such…