Aymone, Marco Real zeros of random Dirichlet series. (English) Zbl 1422.60072 Electron. Commun. Probab. 24, Paper No. 54, 8 p. (2019). 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. Cited in 6 Documents 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) Keywords:random series; zeros of random analytic functions; Dirichlet series PDFBibTeX XMLCite \textit{M. Aymone}, Electron. Commun. Probab. 24, Paper No. 54, 8 p. (2019; Zbl 1422.60072) Full Text: DOI arXiv Euclid References: [1] A. Bovier and P. Picco, A law of the iterated logarithm for random geometric series, Ann. Probab., 21 (1993), pp. 168-184. · Zbl 0770.60029 [2] J.-P. Kahane, Some random series of functions, vol. 5 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, second ed., 1985. · Zbl 0571.60002 [3] M. R. Krishnapur, Zeros of random analytic functions, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)-University of California, Berkeley. · Zbl 1120.82007 · doi:10.1007/s10955-006-9159-y [4] 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 [5] H. Nguyen, O. Nguyen, and V. Vu, On the number of real roots of random polynomials, Commun. Contemp. Math., 18 (2016), 1550052, 17 pp. · Zbl 1385.60019 [6] A. N. Shiryaev, Probability, vol. 95 of Graduate Texts in Mathematics, Springer-Verlag, New York, second ed., 1996. Translated from the first (1980) Russian edition by R. P. Boas. 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.