Higher moments of the error term in the divisor problem

Details Higher moments of the error term in the divisor problem Higher moments of the error term in the divisor problem

2009-04-15

arxiv.org/abs/0904.2271v3

Mat. Zametki 88(2010), 374-383

Mathematics

Number Theory

12 pages

Scientific paper

It is proved that, if $k\ge2$ is a fixed integer and $1 \ll H \le X/2$, then $$ \int_{X-H}^{X+H}\Delta^4_k(x)\d x \ll_\epsilon X^\epsilon\Bigl(HX^{(2k-2)/k} + H^{(2k-3)/(2k+1)}X^{(8k-8)/(2k+1)}\Bigr), $$ where $\Delta_k(x)$ is the error term in the general Dirichlet divisor problem. The proof uses the Vorono{\"\i}--type formula for $\Delta_k(x)$, and the bound of Robert--Sargos for the number of integers when the difference of four $k$--th roots is small. We also investigate the size of the error term in the asymptotic formula for the $m$-th moment of $\Delta_2(x)$.

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