Riemann's zeta-function and the divisor problem. II
In the first paper of this series [Ark. Mat. 21, 75-96 (1983; Zbl 0513.10040)] the relationship between \[ E(T)= \int_ 0^ T |\zeta({\textstyle {1\over 2}}+it)|^ 2 dt- T(\log(T/2\pi)+2\gamma-1) \quad\text{and}\quad \Delta(x)= \sum_{n\leq x} d(n)-x(\log x+2\gamma-1) \] was explored. The present paper makes the connection more precise, by giving an approximate formula for a certain mean-value of \(E(T)\), in terms of \(\Delta(x)\). As a corollary it is shown that if \(\Delta(x)\ll_ \varepsilon x^{\beta+\varepsilon}\) for any \(\varepsilon>0\), then \(E(T)\ll_ \varepsilon T^{(1+2\beta)/5+\varepsilon}\). Thus the conjectured bound \(\Delta(x)\ll x^{1/4+\varepsilon}\) would imply that \(E(T)\ll T^{3/10+\varepsilon}\). This improves on the corresponding bound from the previous paper.
scientific article; zbMATH DE number 493365