Related papers: On the Riemann zeta-function and the divisor probl…

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)…

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) = -…

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) +…

We obtain, for $T^\epsilon \le U=U(T)\le T^{1/2-\epsilon}$, asymptotic formulas for $$ \int_T^{2T}(E(t+U) - E(t))^2 dt,\quad \int_T^{2T}(\Delta(t+U) - \Delta(t))^2 dt, $$ where $\Delta(x)$ is the error term in the classical divisor problem,…

Let $\Delta(x)$ denote the error term in the classical Dirichlet divisor problem, and let the modified error term in the divisor problem be $\Delta^*(x) = -\Delta(x) + 2\Delta(2x) - \frac{1}{2}\Delta(4x)$. We show that $$…

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) +…

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and let $E(T)$ denote 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)…

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) +…

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

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…

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…

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…

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 prove that $$ \int_1^X\Delta(x)\Delta_3(x)\,dx \ll X^{13/9}\log^{10/3}X, \quad \int_1^X\Delta(x)\Delta_4(x)\,dx \ll_\varepsilon X^{25/16+\varepsilon}, $$ where $\Delta_k(x)$ is the error term in the asymptotic formula for the summatory…

If $(k,\ell)$ is an exponent pair such that $k+\ell<1$, then we have $$ \int_1^T|\zeta(1/2+it)|^4|\zeta(\sigma+it)|^2dt \ll_\epsilon T^{1+\epsilon}\quad(\sigma > \min({5\over6},\max(\ell-k, {5k+\ell\over4k+1})), $$ while if $(k,\ell)$ is an…

We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length $T^{\frac14 - \varepsilon}$. Our work relies crucially on Watt's theorem on averages of Kloosterman fractions. In the…

Let $\Delta(x)$ be the error term of the Dirichlet divisor problem. An asymptotic formula with the error term $O(T^{53/28+\epsilon})$ is established for the integral $\int_1^T\Delta^4(x)dx.$ Similar results are also established for some…

The "hybrid" moments $$ \int_T^{2T}|\zeta(1/2+it)|^k{(\int_{t-G}^{t+G}|\zeta(1/2+ix)|^\ell dx)}^m dt $$ of the Riemann zeta-function $\zeta(s)$ on the critical line $\Re s = 1/2$ are studied. The expected upper bound for the above…

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem. Our main results are the asymptotic formulas $$ \int_1^X \Delta^3(x){\rm d}x = BX^{7/4} + O_\epsilon(X^{\beta+\epsilon}) \qquad(B > 0) $$ and $$ \int_1^X…

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…