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In this article, we discover a relation between the normality of algebraic numbers and a mean of the Riemann zeta function on vertical arithmetic progressions. Consequently, we reveal that a positive algebraic irrational number $\\alpha$ is normal to base $b$ if and only if we have \\[ \\lim_{N\\to \\infty}\\frac{1}{\\log N} \\sum_{1\\leq |n|\\leq N} \\zeta"},"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":"2412.02337","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-12-03T09:59:36Z","cross_cats_sorted":[],"title_canon_sha256":"ead014bc3d27c955a191c0891e852a8a1d71aa3772c74350b802c71d3e55bb82","abstract_canon_sha256":"c8a73f354363a448f20e944259687592a71c3356bcb277e42617e43ff6997cc7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:50:07.774917Z","signature_b64":"pKpVbdN2fUw0VP2J+lkQlsJ77E1c51zp+SurbyGijFYbTFS72v9ZCsygrhCMXcVysR3U+1hF64NXHb0AicdoDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5a4dbe2ed4e5598bcf692fcd51d36b9e246b64d7e4902377b74fcde48c4d3c11","last_reissued_at":"2026-07-05T09:50:07.774395Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:50:07.774395Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Normality of algebraic numbers and the Riemann zeta function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kota Saito, Yuya Kanado","submitted_at":"2024-12-03T09:59:36Z","abstract_excerpt":"A real number is called simply normal to base $b$ if every digit $0,1,\\ldots ,b-1$ should appear in its $b$-adic expansion with the same frequency $1/b$. 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