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