On the mean square of the divisor function in short intervals

Ivić, Aleksandar

doi:10.5802/jtnb.669

Ivić, Aleksandar ¹

¹ Katedra Matematike RGF-a Universitet u Beogradu, Đušina 7 11000 Beograd, Serbia

Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 2, pp. 251-261.

Résumé
Abstract

On donne des estimations pour la moyenne quadratique de

\int_{X}^{2 X} {(𝔻_{k} (x + h) - 𝔻_{k} (x))}^{2} d x,

où $h = h (X) ≫ 1, h = o (x) quand X \to \infty$ et $h$ se trouve dans un intervalle convenable. Pour $k \geq 2$ un entier fixé, $𝔻_{k} (x)$ et le terme d’erreur pour la fonction sommatoire de la fonction des diviseurs $d_{k} (n)$ , generée par $ζ^{k} (s)$ .

We provide upper bounds for the mean square integral

\int_{X}^{2 X} {(𝔻_{k} (x + h) - 𝔻_{k} (x))}^{2} d x,

where $h = h (X) ≫ 1, h = o (x) as X \to \infty$ and $h$ lies in a suitable range. For $k \geq 2$ a fixed integer, $𝔻_{k} (x)$ is the error term in the asymptotic formula for the summatory function of the divisor function $d_{k} (n)$ , generated by $ζ^{k} (s)$ .

DOI : 10.5802/jtnb.669

Affiliations des auteurs :

Ivić, Aleksandar ¹

¹ Katedra Matematike RGF-a Universitet u Beogradu, Đušina 7 11000 Beograd, Serbia

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     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},
     mrnumber = {2541424},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/jtnb.669/}
}

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Ivić, Aleksandar. 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://www.numdam.org/articles/10.5802/jtnb.669/

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