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Distribution of zeroes of Rademacher Taylor series
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 4, pp. 759-784.

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.

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.

Publié le : 2016-09-11
DOI : https://doi.org/10.5802/afst.1510
@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},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
     number = {4},
     year = {2016},
     pages = {759-784},
     doi = {10.5802/afst.1510},
     language = {en},
     url = {afst.centre-mersenne.org/item/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, Série 6, Tome 25 (2016) no. 4, pp. 759-784. doi : 10.5802/afst.1510. https://afst.centre-mersenne.org/item/AFST_2016_6_25_4_759_0/

[1] Borichev (A.), Nishry (A.), Sodin (M.).— Entire functions of exponential type represented by pseudo-random and random Taylor series. J. d’Analyse Math., to appear.

[2] Favorov (S.Yu).— Growth and distribution of the values of holomorphic mappings of a finite-dimensional space into a Banach space. Siberian Math. J. 31, p. 137-146 (1990).

[3] Favorov (S.Yu).— On the growth of holomorphic mappings from a finite-dimensional space into a Banach space. Mat. Fiz. Anal. Geom. 1, p. 240-251 (1994).

[4] Hayman (W. K.).— Subhamronic functions, vol. 2. Academic Press (1989).

[5] Kabluchko (Z.), Zaporozhets (D.).— Asymptotic distribution of complex zeros of random analytic functions, Ann. Probab. 42, p. 1374-1395 (2014).

[6] Littlewood (J. E.), Offord (A. C.).— On the distribution of zeros and a-values of a random integral function (II), Ann. of Math. (2) 49.— (1948), 885-952; errata 50, p. 990-991 (1949).

[7] Mahola (M. P.), Filevich (V. P.).— The angular distribution of zeros of random analytic functions, Ufa Math. J. 12:4, p. 122-135 (2012).

[8] Mahola (M. P.), Filevich (V. P.).— The angular distribution of the values of analytic and random analytic functions, Mat. Stud. 38:2, p. 147-153 (2012).

[9] Nazarov (F.), Nishry (A.), Sodin (M.).— Log-integrability of Rademacher Fourier series, with applications to random analytic functions, Algebra & Analysis 25:3, p. 147-184 (2013).

[10] Offord (A. C.).— The distribution of the values of an entire function whose coefficients are independent random variables. (I) Proc. London Math. Soc. (3) 14a, p. 199-238 (1965).

[11] Offord (A. C.).— The distribution of zeros of power series whose coefficients are independent random variables. Indian J. Math. 9, p. 175-196 (1967).

[12] Offord (A. C.).— The distribution of the values of an entire function whose coefficients are independent random variables. (II). Math. Proc. Cambridge Phil. Soc. 118, p. 527-542 (1995).

[13] Ullrich (D. C.).— An extension of the Kahane-Khinchine inequality in a Banach space. Israel J. Math. 62, p. 56-62 (1988).

[14] Ullrich (D. C.).— Khinchin’s inequality and the zeros of Bloch functions. Duke Math. J. 57, p. 519-535 (1988).