# zbMATH — the first resource for mathematics

Law of the iterated logarithm for a random Dirichlet series. (English) Zbl 07252776
Summary: Let $$(X_n)_{n\in \mathbb{N}}$$ be a sequence of i.i.d. random variables with distribution $$\mathbb{P}(X_1=1)=\mathbb{P}(X_1=-1)=1/2$$. Let $$F(\sigma )=\sum_{n=1}^{\infty }X_nn^{-\sigma }$$. We prove that the following holds almost surely $\limsup_{\sigma \to 1/2^+}\frac{F(\sigma)}{\sqrt{2\mathbb{E} F(\sigma )^2 \log \log \mathbb{E} F(\sigma )^2}}=1.$
##### MSC:
 60G50 Sums of independent random variables; random walks 11M41 Other Dirichlet series and zeta functions 11R42 Zeta functions and $$L$$-functions of number fields 11R52 Quaternion and other division algebras: arithmetic, zeta functions 11S40 Zeta functions and $$L$$-functions 11S45 Algebras and orders, and their zeta functions 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F72 Spectral theory; trace formulas (e.g., that of Selberg)
Full Text:
##### References:
 [1] M. Aymone, Real zeros of random Dirichlet series, Electron. Commun. Probab., 24 (2019), Paper No. 54, 8. · Zbl 1422.60072 [2] A. Bovier and P. Picco, Limit theorems for Bernoulli convolutions, in Disordered systems (Temuco, 1991/1992), vol. 53 of Travaux en Cours, Hermann, Paris, 1996, pp. 135-158. · Zbl 0879.60019 [3] A. Bovier and P. Picco, A law of the iterated logarithm for random geometric series, Ann. Probab., 21 (1993), pp. 168-184. · Zbl 0931.41017 [4] J.-P. Kahane, Some random series of functions, vol. 5 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, second ed., 1985. [5] I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, vol. 113 of Graduate Texts in Mathematics, Springer-Verlag, New York, second ed., 1991. · Zbl 0734.60060 [6] H. L. Montgomery and R. C. Vaughan, Multiplicative number theory. I. Classical theory, vol. 97 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2007. · Zbl 1142.11001 [7] G.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.