{"paper":{"title":"Riemann and the logarithmic derivatives of zeta","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.HO","authors_text":"J. Arias de Reyna","submitted_at":"2026-05-26T18:18:47Z","abstract_excerpt":"In one of his posthumous papers, conserved in G\\\"ottingen, Riemann considers the derivatives of $\\log\\zeta(s)$ at the point $1/2$, giving explicit values for them. Around 2010 we shared Riemann's value of the second derivative with some mathematicians. From that time I have been asked several times for references. So I decided to write this. Specially explaining the wonderful formulas \\[\\frac{\\zeta'(\\frac12)}{\\zeta(\\frac12)}=\\frac{\\pi}{4}+\\frac{\\gamma}{2}+\\frac{\\log(8\\pi)}{2},\\quad \\frac{\\zeta''(\\frac12)}{\\zeta(\\frac12)}-\\Bigl(\\frac{\\zeta'(\\frac12)}{\\zeta(\\frac12)}\\Bigr)^2=8-\\frac{\\pi^2}{4}-2G"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.27552","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.27552/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}