Related papers: Bounding |\zeta(1/2 + it)| on the Riemann hypothes…

The purpose of the present paper is to reveal the relation between the behavior of the logarithm of the Riemann zeta-function $\log{\zeta(s)}$ and the distribution of zeros of the Riemann zeta-function. We already know some examples for the…

Let $S(t) = \tfrac{1}{\pi} \arg \zeta (\frac12 + it)$ be the argument of the Riemann zeta-function at the point $\tfrac12 + it$. For $n \geq 1$ and $t>0$ define its iterates \begin{equation*} S_n(t) = \int_0^t S_{n-1}(\tau) \,{\rm d}\tau\,…

Under the Riemann Hypothesis, we show that as $t$ varies in $T\leq t \leq 2T$, the distribution of $\log|\zeta(1/2+it)|$ with respect to the measure $|\zeta(1/2+it)|^2dt$ is approximately normal with mean $\log\log T$ and variance…

Assuming the Riemann Hypothesis it is proved that, for fixed $k>0$ and $H = T^\theta$ with fixed $0<\theta \le 1$, $$ \int_T^{T+H}|\zeta(1/2+it)|^{2k} dt \ll H(\log T)^{k^2(1+O(1/\log_3T))}, $$ where $\log_jT = \log(\log_{j-1}T)$. The proof…

Assuming the Riemann Hypothesis, we make use of the recently discovered \cite{CLV} extremal majorants and minorants of prescribed exponential type for the function $\log\left(\tfrac{4 + x^2}{(\alpha-1/2)^2 + x^2}\right)$ to find upper and…

Let $\pi S(t)$ denote the argument of the Riemann zeta-function, $\zeta(s)$, at the point $s=\frac{1}{2}+it$. Assuming the Riemann hypothesis, we present two proofs of the bound $$ |S(t)| \leq \left(\tfrac{1}{4} + o(1) \right)\tfrac{\log…

We prove, assuming the Riemann Hypothesis, that \int_{T}^{2T} |\zeta(1/2+it)|^{2k} dt \ll_{k} T log^{k^{2}} T for any fixed k \geq 0 and all large T. This is sharp up to the value of the implicit constant. Our proof builds on well known…

Assuming the Riemann Hypothesis, we show that for $k>0$ $$ \frac{1}{T}\text{meas}\Big\{t\in [T,2T]:|\zeta(1/2+{\rm i} t)|>(\log T)^k\Big\}\leq C_k \frac{(\log T)^{-k^2}}{\sqrt{\log\log T}}, $$ where $C_k=\exp(e^{ck})$ for some absolute…

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 recent work, Fyodorov and Keating conjectured the maximum size of $|\zeta(1/2+it)|$ in a typical interval of length O(1) on the critical line. They did this by modelling the zeta function by the characteristic polynomial of a random…

Lindel\"of conjectured that the Riemann zeta function $\zeta(\sigma+it)$ grows more slowly than any fixed positive power of $t$ as $t\rightarrow\infty$ when $\sigma\geq 1/2$. Hardy and Littlewood showed that this is equivalent to the…

We prove mod-Gaussian convergence for a Dirichlet polynomial which approximates $\operatorname{Im}\log\zeta(1/2+it)$. This Dirichlet polynomial is sufficiently long to deduce Selberg's central limit theorem with an explicit error term.…

We examine the size of $E_{2}(T)$, the error term in the asymptotic formula for $\int_{0}^{T} |\zeta(1/2 + it)|^{4}\, dt$ where $\zeta(s)$ is the Riemann zeta-function. We make improvements in the powers of $\log T$ in the known bounds for…

In 2007, assuming the Riemann Hypothesis (RH), Soundararajan \cite{Moment} proved that $\int_{0}^T |\zeta(1/2 + it)|^{2k} dt \ll_{k, \epsilon} T(\log T)^{k^2 + \epsilon}$ for every $k$ positive real number and every $\epsilon > 0.$ In this…

Let $\alpha \in (1/2,1)$ be fixed. We prove that $$ \max_{0 \leq t \leq T} |\zeta(\alpha+it)| \geq \exp\left(\frac{c_\alpha (\log T)^{1-\alpha}}{(\log \log T)^\alpha}\right) $$ for all sufficiently large $T$, where we can choose $c_\alpha =…

We investigate the ``partition function'' integrals $\int_{-1/2}^{1/2} |\zeta(1/2 + it + ih)|^2 dh$ for the critical exponent 2, and the local maxima $\max_{|h| \leq 1/2} |\zeta(1/2 + it + ih)|$, as $T \leq t \leq 2T$ varies. In particular,…

We investigate the relationship between the maximum of the zeta function on the 1-line and the maximal order of $S(t)$, the error term in the number of zeros up to height $t$. We show that the conjectured upper bounds on $S(t)$ along with…

Several problems involving $E(T)$ and $E_2(T)$, the error terms in the mean square and mean fourth moment formula for $|\zeta(1/2+it)}$ are discussed. In particular, it is proved that $$ \int_0^T E(t)E_2(t)dt \ll_ T^{7/4}(\log…

We combine our version of the resonance method with certain convolution formulas for $\zeta(s)$ and $\log\, \zeta(s)$. This leads to a new $\Omega$ result for $|\zeta(1/2+it)|$: The maximum of $|\zeta(1/2+it)|$ on the interval $1 \le t \le…

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…