Abstract
We study the influence of the multipliers ξ(n) on the angular distribution of zeroes of the Taylor series \({F_\xi }\left( z \right) = \sum\limits_{n \geqslant 0} {\xi \left( n \right)} \frac{{{z^n}}}{{n!}}\).
We show that the distribution of zeroes of Fξ is governed by certain autocorrelations of the sequence ξ. Using this guiding principle, we consider several examples of random and pseudo-random sequences ξ and, in particular, answer some questions posed by Chen and Littlewood in 1967.
As a by-product, we show that if ξ is a stationary random integer-valued sequence, then either it is periodic, or its spectral measure has no gaps in its support. The same conclusion is true if ξ is a complex-valued stationary ergodic sequence that takes values in a uniformly discrete set.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. S. Azarin, Asymptotic behavior of subharmonic functions of finite order, Mat. Sb., 108 (1979) 147–167; English translation: Math. USSR-Sb. 36 (1980), 135–154.
L. Bieberbach, Analytische Fortsetzung, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1955.
S. Bochner and F. Bohnenblust, Analytic functions with almost periodic coefficients, Ann. of Math. (2), 35 (1934), 152–161.
J. Breuer and B. Simon, Natural boundaries and spectral theory, Adv. Math., 226 (2011), 4902–4920.
Y. W. Chen and J. E. Littlewood, Some new properties of power series, Indian J. Math., 9 (1967), 289–324.
J. Dufresnoy and Ch. Pisot, Prolongement analytique de la série de Taylor, Ann. Sci. École Norm. Sup. (3), 68 (1951), 105–124.
A. Eremenko and I. Ostrovskii, On the “pits effect” of Littlewood and Offord, Bull. Lond. Math. Soc., 39 (2007), 929–939.
L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, second edition. Springer–Verlag, Berlin, 1990.
L. Hörmander, Notions of Convexity, Birkhäuser Boston, Boston, 1994.
I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing, Groningen, 1971.
Z. Kabluchko and D. Zaporozhets, Asymptotic distribution of complex zeros of random analytic functions, Ann. Probab., 42 (2014), 1374–1395.
V. Katsnelson, Stieltjes functions and Hurwitz stable entire functions, Complex Anal. Oper. Theory, 5 (2011), 611–630.
Y. Katznelson, An Introduction to Harmonic Analysis, third edition, Cambridge University Press, Cambridge, 2004.
B. Ya. Levin, Distribution of Zeros of Entire Functions, revised edition, American Mathematical Society, Providence, R.I., 1980.
J. E. Littlewood, A “pits effect” for all smooth enough integral functions with a coefficient factor exp(n 2aπi), \(\alpha = \frac{4}{2}\left( {\sqrt 5 - 1} \right)\), J. London Math. Soc. 43 (1968), 79–92.
J. E. Littlewood and A. C. Offord, On the distribution of zeros and a-values of a random integral function. II, Ann. of Math. (2) 49 (1948), 885–952; errata, 50 (1949), 990–991.
H. L. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, American Mathematical Society, Providence, RI, 1994.
M. Nassif, On the behaviour of the function \(f\left( z \right) = \sum\nolimits_{n = 0}^\infty {{e^{\sqrt 2 \pi i{n^2}}}} \frac{{{}^z2n}}{{n!}}\), Proc. London Math. Soc. (2) 54 (1952), 201–214.
E. C. Titchmarsh, The Theory of the Riemann Zeta-function, second edition, The Clarendon Press, Oxford University Press, New York, 1986.
H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313–352.
Author information
Authors and Affiliations
Corresponding author
Additional information
To Alex Eremenko on the occasion of his birthday
A. Nishry was supported by U.S. National Science Foundation Grant DMS-1128155.
A. Nishry and M. Sodin were supported by Grant No. 166/11 of the Israel Science Foundation of the Israel Academy of Sciences and Humanities.
M. Sodin was supported by Grant No. 2012037 of the United States–Israel Binational Science Foundation.
Rights and permissions
About this article
Cite this article
Borichev, A., Nishry, A. & Sodin, M. Entire functions of exponential type represented by pseudo-random and random Taylor series. JAMA 133, 361–396 (2017). https://doi.org/10.1007/s11854-017-0037-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-017-0037-0