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

Abstract

Let $d$ be a probability distribution. Under certain mild conditions we show that $$ \lim_{x\to\infty}x\sum_{n=1}^\infty \frac{d^{*n}(x)}{n}=1,\qquad\text{where}\quad d^{*n}:=\underbrace{\,d*d*\cdots*d\,}_{n\text{ times}}. $$ For a compactly supported distribution $d$, we show that if $c>0$ is a given constant and the function $f(k):=\widehat d(k)-1$ does not vanish on the line $\{k\in{\mathbb C}:\Im\,k=-c\}$, where $\widehat d$ is the Fourier transform of $d$, then one has the asymptotic expansion $$ \sum_{n=1}^\infty\frac{d^{*n}(x)}{n}=\frac{1}{x}\bigg(1+\sum_k m(k) e^{-ikx}+O(e^{-c x})\bigg)\qquad (x\to +\infty), $$ 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 $d$ 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, in this paper we generalize a well known result of Heath-Brown concerning the behavior of the Dirichlet $L$-function $L(s,(\cdot|p))$ under the assumption that the Burgess bound $n_0(p)\ll p^{1/(4\sqrt{e})+\epsilon}$ cannot be improved.