相关论文: Shifted second moment of the Riemann zeta function…

We point out an interesting occurrence of the sine kernel in connection with the shifted moments of the Riemann zeta function along the critical line. We discuss rigorous results in this direction for the shifted second moment and for the…

We prove an asymptotic formula for the second moment (up to height $T$) of the Riemann zeta function with two shifts. The case we deal with is where the real parts of the shifts are very close to zero and the imaginary parts can grow up to…

In this article, we prove that the Riemann hypothesis implies a conjecture of Chandee on shifted moments of the Riemann zeta function. The proof is based on ideas of Harper concerning sharp upper bounds for the $2k$-th moments of the…

We examine the calculation of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. Previously this approach has proved unsuccessful in…

Assuming the Riemann Hypothesis we obtain an upper bound for the moments of the Riemann zeta-function on the critical line. Our bound is nearly as sharp as the conjectured asymptotic formulae for these moments. The method extends to moments…

In this article, we establish an asymptotic formula for the eighth moment of the Riemann zeta function, assuming the Riemann hypothesis and a quaternary additive divisor conjecture. This builds on the work of the first author on the sixth…

We compute the average of a product of two shifted squares of the Riemann zeta on the critical line with shifts up to size $T^{3/2-\varepsilon}$. We give an explicit expression for such an average and derive an approximate spectral…

Several second moment and other integral evaluations for the Riemann zeta function $\zeta(s)$, Hurwitz zeta function $\zeta(s,a)$, and Lerch zeta function $\Phi(z,s,a)$ are presented. Additional corollaries that are obtained include…

We prove a reciprocity relation for the twisted second moment of the Riemann Zeta function. This provides an analogue to a formula of Conrey for Dirichlet L-functions

Taking $t$ at random, uniformly from $[0,T]$, we consider the $k$th moment, with respect to $t$, of the random variable corresponding to the $2\beta$th moment of $\zeta(1/2+ix)$ over the interval $x\in(t, t+1]$, where $\zeta(s)$ is the…

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…

The pseudomoments of the Riemann zeta function, denoted $\mathcal{M}_k(N)$, are defined as the $2k$th integral moments of the $N$th partial sum of $\zeta(s)$ on the critical line. We improve the upper and lower bounds for the constants in…

We obtain asymptotic formulae for the second discrete moments of the Riemann zeta function over arithmetic progressions $\frac{1}{2} + i(a n + b)$. It reveals noticeable relation between the discrete moments and the continuous moment of the…

We establish sharp upper bounds for the $2k$th moment of the Riemann zeta function on the critical line, for all real $0 \leqslant k \leqslant 2$. This improves on earlier work of Ramachandra, Heath-Brown and Bettin-Chandee-Radziwi\l\l

We assume the Riemann Hypothesis and an quantitative form of the Twin Prime Conjecture, and obtain an asymptotic formula for the second moment of $S(T)$ with better error term.

We prove exact formulas for weighted $2k$th moments of the Riemann zeta function for all integer $k\geq 1$ in terms of the analytic continuation of an auto-correlation function. This latter enjoys several functional equations. One of them,…

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…

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…

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 $1+i\mathbb{R}$.

In various papers the authors have derived asymptotics for moments of certain cotangent sums related to the Riemann Hypothesis. S. Bettin has given an upper bound for the error term in these asymptotic results. In the present paper the…