On the mean square of the divisor function in short intervals

Aleksandar Ivić

doi:10.5802/jtnb.669

Aleksandar Ivić

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 $ζ^{k} (s)$ .

Publié le : 2009-08-24

MR 2541424 | Zbl pre05620649

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/

Bibliographie

[1] G. Coppola, S. Salerno, On the symmetry of the divisor function in almost all short intervals. Acta Arith. 113(2004), 189–201. | EuDML 278625 | MR 2049565 | Zbl 1122.11062

[2] A. Ivić, The Riemann zeta-function. John Wiley & Sons, New York, 1985 (2nd ed., Dover, Mineola, N.Y., 2003). | MR 792089 | Zbl 0556.10026

[3] A. Ivić, On the divisor function and the Riemann zeta-function in short intervals. To appear in the Ramanujan Journal, see arXiv:0707.1756. | Zbl 1226.11086

[4] M. Jutila, On the divisor problem for short intervals. Ann. Univer. Turkuensis Ser. A I 186 (1984), 23–30. | MR 748516 | Zbl 0536.10032

[5] P. Shiu, A Brun-Titchmarsh theorem for multiplicative functions. J. Reine Angew. Math. 313 (1980), 161–170. | EuDML 152201 | MR 552470 | Zbl 0412.10030

[6] E.C. Titchmarsh, The theory of the Riemann zeta-function (2nd ed.). University Press, Oxford, 1986. | MR 882550 | Zbl 0601.10026

[7] W. Zhang, On the divisor problem. Kexue Tongbao 33 (1988), 1484–1485. | MR 969977