{"paper":{"title":"Asymptotic harmonic behavior in the prime number distribution","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Maurice H.P.M. van Putten","submitted_at":"2011-04-19T00:59:18Z","abstract_excerpt":"We consider $\\Phi(x)=x^{-\\frac{1}{4}}\\left[1-2\\sqrt{x}\\Sigma e^{-p^2\\pi x}\\ln p\\right]$ on $x>0$, where the sum is over all primes $p$. If $\\Phi$ is bounded on $x>0$, then the Riemann hypothesis is true or there are infinitely many zeros Re~$z_k>\\frac{1}{2}$. The first 21 zeros give rise to asymptotic harmonic behavior in $\\Phi(x)$ defined by the prime numbers up to one trillion."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.3617","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"}