We find the asymptotics of the counting function of zeroes of random entire functions represented by Rademacher Taylor series. We also give the asymptotics of the weighted counting function, which takes into account the arguments of zeroes. These results answer several questions left open after the pioneering work of Littlewood and Offord of 1948.
The proofs are based on our recent result on the logarithmic integrability of Rademacher Fourier series.
Nous trouvons l’asymptotique de la fonction de comptage de zéros pour les fonctions entières aléatoires représentées par des séries de Taylor du type de Rademacher. Nous donnons aussi l’asymptotique pour la fonction de comptage à poids, qui prend en compte les arguments des zéros. Ces résultats répondent à certaines questions laissées ouvertes après le travail novateur de Littlewood et Offord en 1948.
Les preuves sont basées sur notre résultat récent sur l’intégrabilité logarithmique de séries de Fourier du type de Rademacher.
DOI: 10.5802/afst.1510
Fedor Nazarov 1; Alon Nishry 2; Mikhail Sodin 3
@article{AFST_2016_6_25_4_759_0,
author = {Fedor Nazarov and Alon Nishry and Mikhail Sodin},
title = {Distribution of zeroes of {Rademacher} {Taylor} series},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {759--784},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 25},
number = {4},
year = {2016},
doi = {10.5802/afst.1510},
zbl = {1352.30001},
mrnumber = {3564126},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1510/}
}
TY - JOUR AU - Fedor Nazarov AU - Alon Nishry AU - Mikhail Sodin TI - Distribution of zeroes of Rademacher Taylor series JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2016 SP - 759 EP - 784 VL - 25 IS - 4 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1510/ DO - 10.5802/afst.1510 LA - en ID - AFST_2016_6_25_4_759_0 ER -
%0 Journal Article %A Fedor Nazarov %A Alon Nishry %A Mikhail Sodin %T Distribution of zeroes of Rademacher Taylor series %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2016 %P 759-784 %V 25 %N 4 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1510/ %R 10.5802/afst.1510 %G en %F AFST_2016_6_25_4_759_0
Fedor Nazarov; Alon Nishry; Mikhail Sodin. Distribution of zeroes of Rademacher Taylor series. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 4, pp. 759-784. doi : 10.5802/afst.1510. https://afst.centre-mersenne.org/articles/10.5802/afst.1510/
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