Convolutions with probability distributions, zeros of $L$-functions, and the least quadratic nonresidue

Abstract

Let ${\tt d}$ be the density of a probability distribution that is compactly supported in the positive semi-axis. Under certain mild conditions we show that $$ \lim_{x\to\infty}x\sum_{n=1}^\infty \frac{{\tt d}^{*n}(x)}{n}=1,\qquad\text{where}\quad {\tt d}^{*n}:=\underbrace{{\tt d} *{\tt d}*\cdots*{\tt d}}_{n\text{~times}}. $$ We also show that if $c>0$ is a given constant for which the function $f(k):=\widehat{\tt d}(k)-1$ does not vanish on the line $\{k\in\mathbb{C}:\Im k=-c\}$, where $\widehat{\tt d}$ is the Fourier transform of ${\tt d}$, then one has the asymptotic expansion $$ \sum_{n=1}^\infty\frac{{\tt d}^{*n}(x)}{n}=\frac{1}{x}\bigg(1+\sum_k m(k) e^{-ikx}+O(e^{-c x})\bigg)\qquad (x\to +\infty), $$ where the sum is taken over those zeros $k$ of $f$ that lie in the strip $\{k\in\mathbb{C}:-c<\Im k<0\}$, $m(k)$ is the multiplicity of any such zero, and the implied constant depends only on $c$. For a given distribution of this type, we briefly describe the location of the zeros $k$ of $f$ in the lower half-plane $\{k\in\mathbb{C}:\Im k<0\}$. For an odd prime $p$, let $n_0(p)$ be the least natural number such that $(n|p)=-1$, where $(\cdot|p)$ is the Legendre symbol. As an application of our work on probability distributions, we generalize a well known result of Heath-Brown concerning the exhibited behavior of the Dirichlet $L$-function $L(s,(\cdot|p))$ under the assumption that the Burgess bound $n_0(p)\ll p^{1/(4\sqrt{e})+\varepsilon}$ cannot be improved.