|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'506'133
Articles rated: 2609
26 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
Convolutions with probability distributions, zeros of L-functions, and the least quadratic nonresidue | | William D. Banks
; Konstantin A. Makarov
; | | Date: |
7 Nov 2014 | | Abstract: | Let $d$ be a probability distribution. Under certain mild conditions we show
that $$ lim_{x oinfty}xsum_{n=1}^infty
frac{d^{*n}(x)}{n}=1,qquad ext{where}quad
d^{*n}:=underbrace{,d*d*cdots*d,}_{n ext{ times}}. $$ 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 ${kin{mathbb
C}:Im,k=-c}$, where $widehat d$ is the Fourier transform of $d$, then one
has the asymptotic expansion $$
sum_{n=1}^inftyfrac{d^{*n}(x)}{n}=frac{1}{x}igg(1+sum_k m(k)
e^{-ikx}+O(e^{-c x})igg)qquad (x o +infty), $$ where the sum is taken over
those zeros $k$ of $f$ that lie in the strip ${kin{mathbb C}:-c<Im,k<0}$,
$m(k)$ 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 ${kin{mathbb
C}:Im,k<0}$.
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 $n_0(p)ll
p^{1/(4sqrt{e})+epsilon}$ cannot be improved. | | Source: | arXiv, 1411.2009 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER