Moments over short intervals
As usual, let \(\Delta(x) = \sum_{n\leq x}d(n) - x(\log x + 2\gamma - 1)\) denote the error term in the Dirichlet divisor problem (\(d(n)\) is the number of divisors of \(n\) and \(\gamma\) is Euler's constant). \textit{W. G. Nowak} [Acta Arith. 109, 329--341 (2003; Zbl 1065.11076)] proved a result on the power moments \(\Delta^k(x)\) in short intervals (\(k = 2,3,4\)), which the authors generalize and improve. Their main result states the following: suppose that the estimate \(\Delta(x) \ll_\varepsilon x^{\beta+\varepsilon}\) holds for a certain \(\beta \geq 1/4\). Let \(k < \min\{11,8\beta/(4\beta-1)\}\) and let \(\Lambda = \Lambda(X)\) be an increasing function of \(X\) such that \(X \gg \Lambda(X) \gg X^\lambda\) for some constant \(\lambda\), where \[ \lambda > {1\over2} + \max\left\{0,\,(k-2)\left(\beta - {1\over4}\right) \right\}.\tag{1} \] Then the limit \[ \lim_{X\to\infty}{1\over\Lambda X^{1/4}}\int_X^{X+\Lambda}| \Delta(x)| ^k \,dx\tag{2} \] exists. Furthermore, if \(k\,\left(<\min\left\{11, 8\beta/(4\beta-1)\right\}\right)\) is an odd integer and (1) holds, then \[ \lim_{X\to\infty}{1\over\Lambda X^{1/4}} \int_X^{X+\Lambda}\Delta(x)^k \,dx\tag{3} \] also exists. By using the sharpest currently known bound \(\beta \leq 131/416\) [due to \textit{M. N. Huxley}, Proc. Lond. Math. Soc. 87, 591--609 (2003; Zbl 1065.11079)], it follows that (2) exists for \(0 \leq k \leq 9.7\dots\) and (3) for \(k = 1,3,5,7,9\) if \(\Lambda \gg X^\lambda\) with \[ \lambda > {1\over2} + \max\left\{0,\,(k-2){27\over416}\right\}. \] In particular, we need only \(\lambda > 0.5649\ldots\) for \(k=3\) and \(\lambda > 0.6298\ldots\) for \(k=4\). The method of proof uses results on power moments of \(\Delta(x)\) due to the reviewer [Invent. Math. 71, 513--520 (1983; Zbl 0489.10045)], and the ideas of \textit{D. R. Heath-Brown} [Acta Arith. 60, 389--414 (1992; Zbl 0725.11045)] who both investigated the analogous problem of moments of \(\Delta(x)\) over ``long'' intervals.
scientific article; zbMATH DE number 2162319