How many real zeros does a random Dirichlet series have?

2023/02/01 by Aymone, Marco, Frómeta, Susana, Misturini, Ricardo
#FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2302.00616

Abstract

Let F(σ)=∑_n=1^∞ (X_n)/(n^σ) be a random Dirichlet series where (X_n)_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 𝔼 N(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.

Related

• On the expected number of roots of a random Dirichlet polynomial
• Expected number of real zeros of random Taylor Series
• Real Zeros of Random Sums with I.I.D. Coefficients
• Precise tail behavior of some Dirichlet series
• Limit theorems for random Dirichlet series