Related papers: On the correlation of shifted values of the Rieman…

In this expository article, we review some of the ideas behind the work of Heath-Brown (D.~R. Heath-Brown, {\em Fractional moments of the Riemann Zeta Function}, J. Lond. Math. Soc., (2), {\bf 24}, no.1, (1981), 65--78) on upper and lower…

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) = -…

Keating and Snaith modeled the Riemann zeta-function \zeta(s) by characteristic polynomials of random NxN unitary matrices, and used this to conjecture the asymptotic main term for the 2k-th moment of \zeta(1/2+it) when k>-1/2. However, an…

We assume the Riemann hypothesis to improve upon the rate of convergence of $(\log\log\log T)^2/\sqrt{\log\log T}$ in Selberg's central limit theorem for $\log|\zeta(1/2+it)|$ given by the author. We achieve a rate of convergence of…

Selberg's central limit theorem states that the values of $\log|\zeta(1/2+i \tau)|$, where $\tau$ is a uniform random variable on $[T,2T]$, is distributed like a Gaussian random variable of mean $0$ and standard deviation…

Let $\pi S(t)$ denote the argument of the Riemann zeta-function at the point $\frac12+it$. Assuming the Riemann Hypothesis, we sharpen the constant in the best currently known bounds for $S(t)$ and for the change of $S(t)$ in intervals. We…

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…

In 1924 Littlewood showed that, assuming the Riemann Hypothesis, for large t there is a constant C such that |\zeta(1/2+it)| \ll \exp(C\log t/\log \log t). In this note we show how the problem of bounding |\zeta(1/2+it)| may be framed in…

We establish lower bounds for the discrete $2k$-th moment of the derivative of the Riemann zeta function at nontrivial zeros for all $k<0$ under the Riemann hypothesis (RH) and the assumption that all zeros of $\zeta(s)$ are simple.

This paper improves the bound on $|S(T)|$. The main result is to show that $|S(T)|\leq 0.111\log T + 0.275\log\log T + 2.450$, which is valid for all $T\geq e$.

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 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…

We discuss moments of the Riemann zeta-function in this paper. The purpose of this paper is to give an upper bound of exponential moments of the logarithm of the Riemann zeta-function twisted by arguments. Our results contain an improvement…

Conrey and Ghosh studied the second moment of the Riemann zeta function, evaluated at its local extrema along the critical line, finding the leading order behaviour to be $\frac{e^2 - 5}{2 \pi} T (\log T)^2$. This problem is closely related…

Power moments of $$ J_k(t,G) = {1\over\sqrt{\pi}G} \int_{-\infty}^\infty |\zeta(1/2 + it + iu)|^{2k}{\rm e}^{-(u/G)^2} du \qquad(t \asymp T, T^\epsilon \le G \ll T),$$ where $k$ is a natural number, are investigated. The results that are…

We conjecture the full asymptotic expansion of a product of Riemann zeta functions, evaluated at the non-trivial zeros of the zeta function, with shifts added in each argument. By taking derivatives with respect to these shifts, we form a…

Some results and conjectures on $Z_2(s) = \int_1^\infty |\zeta(1/2+ix)|^4x^{-s}dx (\Re s > 1)$ are presented. Consequences of these conjectures regarding the eighth moment of $|\zeta(1/2+it)$ and the error term in the fourth moment of…

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…

We prove a sharp upper bound for the fourth moment of the Hurwitz zeta function $\zeta(s,\alpha)$ on the critical line when the shift parameter $\alpha$ is irrational and of irrationality exponent strictly less than 3. As a consequence, we…

In this series of papers 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 completes the…