{"paper":{"title":"Zeta functions and asymptotic additive bases with some unusual sets of primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"William D. Banks","submitted_at":"2015-08-28T22:24:35Z","abstract_excerpt":"Fix $\\delta\\in(0,1]$, $\\sigma_0\\in[0,1)$ and a real-valued function $\\varepsilon(x)$ for which $\\limsup_{x\\to\\infty}\\varepsilon(x)\\le 0$. For every set of primes ${\\mathcal P}$ whose counting function $\\pi_{\\mathcal P}(x)$ satisfies an estimate of the form $$\\pi_{\\mathcal P}(x)=\\delta\\,\\pi(x)+O\\bigl(x^{\\sigma_0+\\varepsilon(x)}\\bigr),$$ we define a zeta function $\\zeta_{\\mathcal P}(s)$ that is closely related to the Riemann zeta function $\\zeta(s)$. For $\\sigma_0\\le\\frac12$, we show that the Riemann hypothesis is equivalent to the non-vanishing of $\\zeta_{\\mathcal P}(s)$ in the region $\\{\\sigma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.07367","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"}