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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 \[ \limsup_{\sigma \to 1/2^+}\frac{F(\sigma)}{\sqrt{2\mathbb{E} F(\sigma )^2 \log \log \mathbb{E} F(\sigma )^2}}=1. \]
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: DOI Euclid
[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.
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