Geodesic
Higher Moments of the Error Term in the Divisor Problem
Matematičeskie zametki, Tome 88 (2010) no. 3, pp. 374-383

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It is proved that, if $k\ge 2$ is a fixed integer and $1\ll H\le(1/2)X$, then $$ \int_{X-H}^{X+H}\Delta^4_k(x)\,dx \ll_\varepsilon X^\varepsilon (HX^{(2k-2)/k}+H^{(2k-3)/(2k+1)}X^{(8k-8)/(2k+1)}), $$ where $\Delta_k(x)$ is the error term in the general Dirichlet divisor problem. The proof uses a Voronoï–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. The size of the error term in the asymptotic formula for the $m$th moment of $\Delta_2(x)$ is also investigated.
Keywords: Dirichlet divisor problem, higher moments, mean fourth power, residue theorem.
Mots-clés : Voronoï formula, Euler's constant $\gamma$ 
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     author = {A. Ivi\'c and W. Zhai},
     title = {Higher {Moments} of the {Error} {Term} in the {Divisor} {Problem}},
     journal = {Matemati\v{c}eskie zametki},
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     number = {3},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_88_3_a5/}
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A. Ivić; W. Zhai. Higher Moments of the Error Term in the Divisor Problem. Matematičeskie zametki, Tome 88 (2010) no. 3, pp. 374-383. http://geodesic.mathdoc.fr/item/MZM_2010_88_3_a5/