{"paper":{"title":"Explicit formula for the discrete Laplace transform of the M\\\"obius function, related special functions, and a criterion for the Riemann hypothesis","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Sergey Liflandsky","submitted_at":"2026-07-09T12:12:21Z","abstract_excerpt":"In this paper, we assume that all the zeros of the Riemann zeta function are simple. Under this assumption we give an explicit formula for the function $\\Phi(e^{-t})=\\sum_{n=1}^{\\infty}\\mu(n)e^{-nt}$, as a function of the values of $\\zeta(s)$ and $\\zeta'(s)$ at the odd integers and as a function of the zeros of $\\zeta(s)$. A structural feature distinguishes this formula from the classical explicit formula for the Mertens function: the poles of $\\Gamma(s)$ collide with the trivial zeros of $\\zeta(s)$, producing double poles whose residues contain a logarithmic term. Using this formula, we give "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.09797","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/2607.09797/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"}