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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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