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 $$\underset{x\to \infty }{\lim }x\sum _{n=1}^{\infty }\frac{d^{\ast n}(x)}{n}=1,\text{where}{d}^{\ast n}:=\underset{n\text{ times}}{\underbrace{d\ast d\ast \cdots \ast d}}.$$ For a compactly supported distribution $d$, we show that if $c>0$ is a given constant and the function $f(k):=\hat{d}(k)-1$ does not vanish on the line $\{k\in \mathbb{C}:\Im k=-c\}$, where $\hat{d}$ is the Fourier transform of $d$, then one has the asymptotic expansion $$\sum _{n=1}^{\infty }\frac{d^{\ast n}(x)}{n}=\frac{1}{x}(1+\sum _{k}m(k)e^{-ikx}+O(e^{-cx}))(x\to +\infty ),$$ 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.

arXiv comment41 pages; some minor typos corrected; Section 5 expanded slightly