On the mean square of the divisor function in short intervals
Journal de Théorie des Nombres de Bordeaux, Tome 21 (2009) no. 2, pp. 251-261.

On donne des estimations pour la moyenne quadratique de

${\int }_{X}^{2X}{\left({𝔻}_{k}\left(x+h\right)-{𝔻}_{k}\left(x\right)\right)}^{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x,$

$h=h\left(X\right)\gg 1,\phantom{\rule{0.277778em}{0ex}}h=o\left(x\right)\phantom{\rule{0.277778em}{0ex}}\mathrm{quand}\phantom{\rule{0.277778em}{0ex}}X\to \infty$ et $h$ se trouve dans un intervalle convenable. Pour $k\ge 2$ un entier fixé, ${𝔻}_{k}\left(x\right)$ et le terme d’erreur pour la fonction sommatoire de la fonction des diviseurs ${d}_{k}\left(n\right)$, generée par ${\zeta }^{k}\left(s\right)$.

We provide upper bounds for the mean square integral

${\int }_{X}^{2X}{\left({𝔻}_{k}\left(x+h\right)-{𝔻}_{k}\left(x\right)\right)}^{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x,$

where $h=h\left(X\right)\gg 1,\phantom{\rule{0.277778em}{0ex}}h=o\left(x\right)\phantom{\rule{0.277778em}{0ex}}\mathrm{as}\phantom{\rule{0.277778em}{0ex}}X\to \infty$ and $h$ lies in a suitable range. For $k\ge 2$ a fixed integer, ${𝔻}_{k}\left(x\right)$ is the error term in the asymptotic formula for the summatory function of the divisor function ${d}_{k}\left(n\right)$, generated by ${\zeta }^{k}\left(s\right)$.

Publié le : 2009-08-24
DOI : https://doi.org/10.5802/jtnb.669
@article{JTNB_2009__21_2_251_0,
author = {Aleksandar Ivi\'c},
title = {On the mean square of the divisor function in short intervals},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {251--261},
publisher = {Universit\'e Bordeaux 1},
volume = {21},
number = {2},
year = {2009},
doi = {10.5802/jtnb.669},
zbl = {pre05620649},
mrnumber = {2541424},
language = {en},
url = {jtnb.centre-mersenne.org/item/JTNB_2009__21_2_251_0/}
}
Aleksandar Ivić. On the mean square of the divisor function in short intervals. Journal de Théorie des Nombres de Bordeaux, Tome 21 (2009) no. 2, pp. 251-261. doi : 10.5802/jtnb.669. https://jtnb.centre-mersenne.org/item/JTNB_2009__21_2_251_0/

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