×

On the divisor problem for short intervals. (English) Zbl 0536.10032

Let \(D(x)=\sum_{n\leq x}d(n)=x \log x+(2\gamma -1) x+\Delta(x)\), \[ \int^{T}_{0}| \zeta(\tfrac{1}{2}+it)|^ 2\, dt=T\, \log(T/2\pi)+(2\gamma -1) T+E(T), \] where \(\gamma\) is Euler’s constant. The author has already successfully investigated the analogy between \(\Delta(x)\) and \(E(T)\) [Ark. Mat. 21, 75–96 (1983; Zbl 0513.10040); Colloq. Math. Soc. János Bolyai 34, 807–823 (1984; Zbl 0549.10029)]. The purpose of this paper is to study the differences \(\Delta(x+U)-\Delta(x)\) and \(E(t+U)-E(t)\) in the mean value sense. The main result is that, for \(X\geq 2\) and \(1\leq U\ll X^{1/2}\ll H\leq X\), \[ \begin{split} \int^{X+H}_{X}(\Delta(x+U)-\Delta(x))^ 2 \,dx = \\ (4\pi^ 2)^{- 1}\sum_{n\leq X/(2U)}d^ 2(n) n^{- 3/2}\int^{X+H}_{X}x^{1/2}\left| \exp(2\pi i (n/x)^{1/2} U)- 1\right|^ 2 \,dx + O(X^{1+\varepsilon})+O(H U^{1/2} X^{\varepsilon}).\end{split} \] It is remarkable that an analogous formula holds if \(\Delta(x)\) is replaced by \(E(x)\).

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11N37 Asymptotic results on arithmetic functions
PDFBibTeX XMLCite