On the expected number of roots of a random Dirichlet polynomial
Volume 220 / 2025
Acta Arithmetica 220 (2025), 305-321
MSC: Primary 11C08; Secondary 11K99
DOI: 10.4064/aa240130-23-4
Published online: 26 September 2025
Abstract
Let $T \gt 0$ and consider the random Dirichlet polynomial $$S_T(t)=\mathrm{Re}\sum _{n\leq T} X_n n^{-1/2-it},$$ where $(X_n)_{n}$ are i.i.d. Gaussian random variables with mean $0$ and variance $1$. We prove that the expected number of roots of $S_T(t)$ in the dyadic interval $[T,2T]$, say $\mathbb E N(T)$, is approximately $2/\sqrt{3}$ times the number of zeros of the Riemann $\zeta $ function in the critical strip up to height $T$. Moreover, we also compute the expected number of zeros in the same dyadic interval of the $k$th derivative of $S_T(t)$. Our proof requires the best upper bounds for the Riemann $\zeta $ function known up to date, and also estimates for the $L^2$ averages of certain Dirichlet polynomials.
