{"paper":{"title":"On the real zeroes of the Hurwitz zeta-function and Bernoulli polynomials","license":"","headline":"","cross_cats":["math.CA"],"primary_cat":"math.GM","authors_text":"A.P. Veselov, J.P. Ward","submitted_at":"2002-05-16T10:37:25Z","abstract_excerpt":"The behaviour of real zeroes of the Hurwitz zeta function $$\\zeta (s,a)=\\sum_{r=0}^{\\infty}(a+r)^{-s}\\qquad\\qquad a > 0$$ is investigated. It is shown that $\\zeta (s,a)$ has no real zeroes $(s=\\sigma,a)$ in the region $a >\\frac{-\\sigma}{2\\pi e}+\\frac{1}{4\\pi e}\\log (-\\sigma) +1$ for large negative $\\sigma$. In the region $0 < a < \\frac{-\\sigma}{2\\pi e}$ the zeroes are asymptotically located at the lines $\\sigma + 4a + 2m =0$ with integer $m$. If $N(p)$ is the number of real zeroes of $\\zeta(-p,a)$ with given $p$ then $$\\lim_{p\\to\\infty}\\frac{N(p)}{p}=\\frac{1}{\\pi e}.$$ As a corollary we have a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0205183","kind":"arxiv","version":1},"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"}