How many real zeros does a random Dirichlet series have?
Fecha
2024Materia
Abstract
Let F(σ) = P∞ n=1 Xn nσ be a random Dirichlet series where (Xn)n∈N are independent standard Gaussian random variables. We compute in a quantitative form the expected number of zeros of F(σ) in the interval [T, ∞), say EN(T, ∞), as T → 1/2 +. We also estimate higher moments and with this we derive exponential tails for the probability that the number of zeros in the interval [T, 1], say N(T, 1), is large. We also consider almost sure lower and upper bounds for N(T, ∞). And finally, we also prove ...
Let F(σ) = P∞ n=1 Xn nσ be a random Dirichlet series where (Xn)n∈N are independent standard Gaussian random variables. We compute in a quantitative form the expected number of zeros of F(σ) in the interval [T, ∞), say EN(T, ∞), as T → 1/2 +. We also estimate higher moments and with this we derive exponential tails for the probability that the number of zeros in the interval [T, 1], say N(T, 1), is large. We also consider almost sure lower and upper bounds for N(T, ∞). And finally, we also prove results for another class of random Dirichlet series, e.g., when the summation is restricted to prime numbers. ...
En
Electronic Journal of Probability. Seattle. Vol. 29 (2024), Art. 5, p. 1-17
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