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Real zeros of random Dirichlet series. (English) Zbl 1422.60072
Summary: Let \(F(\sigma )\) be the random Dirichlet series \(F(\sigma )=\sum _{p\in \mathcal{P} } \frac{X_p} {p^{\sigma }}\), where \(\mathcal{P} \) is an increasing sequence of positive real numbers and \((X_p)_{p\in \mathcal{P} }\) is a sequence of i.i.d. random variables with \(\mathbb{P} (X_1=1)=\mathbb{P} (X_1=-1)=1/2\). We prove that, for certain conditions on \(\mathcal{P} \), if \(\sum _{p\in \mathcal{P} }\frac{1} {p}<\infty \) then with positive probability \(F(\sigma )\) has no real zeros while if \(\sum _{p\in \mathcal{P} }\frac{1} {p}=\infty \), almost surely \(F(\sigma )\) has an infinite number of real zeros.

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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