no. 8
Linear statistics for zeros of Riemann's zeta function
[Statisiques linéaires pour les zéros de la fonction zêta de Riemann]
Hughes, Chris1 ; Rudnick, Zeév1
1 Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Comptes Rendus. Mathématique, Tome 335 (2002) no. 8, pp. 667-670.

Nous considérons une fonction de comptage lisse des zéros de la fonction zêta de Riemann, normalisés au voisinage de la hauteur T. Nous montrons que les premiers moments sont Gaussiens, le nombre exact de tels moments dépendant de la moyenne choisie et de la fonction de comptage des zéros.

We consider a smooth counting function of the scaled zeros of the Riemann zeta function, around height T. We show that the first few moments tend to the Gaussian moments, with the exact number depending on the statistic considered.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02541-4
Hughes, Chris 1 ; Rudnick, Zeév 1

1 Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel 
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Hughes, Chris; Rudnick, Zeév. Linear statistics for zeros of Riemann's zeta function. Comptes Rendus. Mathématique, Tome 335 (2002) no. 8, pp. 667-670. doi : 10.1016/S1631-073X(02)02541-4. http://www.numdam.org/articles/10.1016/S1631-073X(02)02541-4/

[1] Fujii, A. Explicit formulas and oscillations (Hejhal, D.A.; Friedman, J.; Gutzwiller, M.C.; Odlyzko, A.M., eds.), Emerging Applications of Number Theory, Springer, 1999, pp. 219-267

[2] Hughes, C.P.; Rudnick, Z. Linear statistics of low-lying zeros of L-functions, 2002 | arXiv

[3] Selberg, A. Contributions to the theory of the Riemann zeta-function, Arch. Math. Naturvid., Volume 48 (1946) no. 5, pp. 89-155

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