On the expected number of roots of a random Dirichlet polynomial

TL;DR

This work analyzes zeros of the real part of a random Dirichlet polynomial with Gaussian coefficients, establishing a precise link between the expected zero count in [T,2T] and the distribution of zeros of the Riemann zeta function. Using the Edelman–Kostlan framework, the authors reduce the problem to covariance calculations for Dirichlet polynomials and combine sharp zeta-function bounds (Vinogradov–Korobov) with L^2 averages of Dirichlet polynomials to obtain an explicit asymptotic: E N(T) = (1/(π√3)) T log T − (γ/(2π√3)) T + O(T/log T). They further extend the analysis to the k-th derivative S_T^{(k)}(t), deriving E N^{(k)}(T) with leading coefficient √((2k+1)/(2k+3)) and a secondary term involving the k-th Stieltjes constants γ_{2k}. The paper also discusses extensions to alternative distributions, potential generalizations with a σ-parameter, and open questions regarding zeros of the full complex sum and related models, underscoring a deep link between random models and ζ-zero statistics in a probabilistic setting.

Abstract

Let $T>0$ and consider the random Dirichlet polynomial $S_T(t)=Re\, \sum_{n\leq T} X_n n^{-1/2-it}$, 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 $ζ$ 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 $ζ$ function known up to date, and also estimates for the $L^2$ averages of certain Dirichlet polynomials.

Theorems & Definitions (16)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Lemma 3.1
• proof
• Lemma 3.2: Yang daodao_BLMS, p. 8, Lemma 4, and tenenbaumlivro, pg. 236
• Lemma 3.3
• Lemma 3.4
• proof
• Lemma 3.5