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The first 21 zeros give rise to asymptotic harmonic behavior in $\\Phi(x)$ defined by the prime numbers up to one trillion."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1104.3617","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-04-19T00:59:18Z","cross_cats_sorted":[],"title_canon_sha256":"c1ba1511c8921809b09b8755b0518488124e04a64948c5920ed1a158be27b2e9","abstract_canon_sha256":"2bb8e2905ea2b873cbc5ecb21cdcec9a6c24f8e059912763191d6e73adc6a606"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:52:13.045590Z","signature_b64":"MU7g2NZkE9E/F3tW5rEolyXpJ315+V/X7+rZXkstOK+nlraLCoQV6koeLsX0dZ/NzTh2nfSvKSzPV9/S9McqAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"386bfd8753a97f5853aff3ddfa9a8294372a0910f88aef52c5ae7f51a41e32fb","last_reissued_at":"2026-05-18T02:52:13.045122Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:52:13.045122Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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~$z_k>\\frac{1}{2}$. 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