{"paper":{"title":"Convolutions with probability distributions, zeros of L-functions, and the least quadratic nonresidue","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.NT","authors_text":"Konstantin A. Makarov, William D. Banks","submitted_at":"2014-11-07T19:18:20Z","abstract_excerpt":"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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.2009","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}