On the Mean Square of the Riemann Zeta Function and the Divisor Problem
نویسندگان
چکیده
Let ∆(T ) and E(T ) be the error terms in the classical Dirichlet divisor problem and in the asymptotic formula for the mean square of the Riemann zeta function in the critical strip, respectively. We show that ∆(T ) and E(T ) are asymptotic integral transforms of each other. We then use this integral representation of ∆(T ) to give a new proof of a result of M. Jutila.
منابع مشابه
Recent Progress on the Dirichlet Divisor Problem and the Mean Square of the Riemann Zeta-function
Let ∆(x) and E(t) denote respectively the remainder terms in the Dirichlet divisor problem and the mean square formula for the Riemann zeta-function on the critical line. This article is a survey of recent developments on the research of these famous error terms in number theory. These include upper bounds, Ω-results, sign changes, moments and distribution, etc. A few open problems will also be...
متن کاملSubconvexity for the Riemann Zeta-function and the Divisor Problem
A simple proof of the classical subconvexity bound ζ( 1 2 + it) ≪ε t1/6+ε for the Riemann zeta-function is given, and estimation by more refined techniques is discussed. The connections between the Dirichlet divisor problem and the mean square of |ζ( 1 2 + it)| are analysed. 1. Convexity for the Riemann zeta-function Let as usual (1.1) ζ(s) = ∞
متن کاملOn the Riemann Zeta-function and the Divisor Problem Iv
Let ∆(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 |ζ( 1 2 + it)|. If E∗(t) = E(t)− 2π∆∗(t/2π) with ∆∗(x) = −∆(x) + 2∆(2x)− 1 2 ∆(4x), then it is proved that
متن کاملOn the Riemann Zeta-function and the Divisor Problem Ii
Let ∆(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 |ζ( 1 2 + it)|. If E∗(t) = E(t) − 2π∆∗(t/2π) with ∆∗(x) = −∆(x) + 2∆(2x) − 1 2 ∆(4x), then we obtain ∫ T 0 |E(t)| dt ≪ε T 2+ε and ∫ T 0 |E∗(t)| 544 75 dt ≪ε T 601 225 . It is also shown how bounds for moments of |E∗(t)| lead to bounds for moments of |ζ( 1 2 ...
متن کاملOn the Riemann Zeta-function and the Divisor Problem
Let ∆(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 |ζ( 1 2 + it)|. If E∗(t) = E(t) − 2π∆∗(t/2π) with ∆∗(x) = −∆(x) + 2∆(2x) − 1 2 ∆(4x), then we obtain ∫ T 0 (E(t)) dt ≪ε T . We also show how our method of proof yields the bound R
متن کامل