标题： 关于随机狄利克雷多项式的根的期望数量 显示英文标题

标题： On the expected number of roots of a random Dirichlet polynomial

摘要： 设$T>0$并考虑随机狄利克雷多项式$S_T(t)=Re\, \sum_{n\leq T} X_n n^{-1/2-it}$，其中$(X_n)_{n}$是均值为$0$和方差为$1$的独立同分布高斯随机变量。 我们证明了在二进制区间$[T,2T]$，比如说$\mathbb{E} N(T)$，$S_T(t)$的预期根的数量大约是黎曼$\zeta$函数在临界带中到高度$T$的零点数量的$2/\sqrt{3}$倍。 此外，我们还计算了在相同的二进制区间内，$k$-th 阶导数的零期望数量，其中$S_T(t)$保持不变。 我们的证明需要目前所知关于黎曼函数$\zeta$的最佳上界，以及某些狄利克雷多项式的$L^2$平均值的估计。 显示英文摘要

摘要： 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 $\zeta$ 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.