{"paper":{"title":"Euler's $\\ell$-totients and Riemann hypothesis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Ahmed Gaber","submitted_at":"2026-07-28T15:34:39Z","abstract_excerpt":"This paper develops a new analytic framework for investigating the Riemann hypothesis. For each fixed integer $\\ell \\ge 1$, define Euler's $\\ell$-totient function $\\varphi_\\ell$ by \\[ \\varphi_\\ell(n):=n\\prod_{\\substack{p\\ \\mathrm{prime}\\\\ v_p(n)\\ge \\ell}}\\left(1-\\frac{1}{p}\\right), \\] and its summatory function by \\[ \\Phi_\\ell(x):=\\sum_{n\\le x}\\varphi_\\ell(n). \\] An analytic study of the generalized Euler $\\ell$-totient function $\\varphi_\\ell(n)$ is carried out, including its Euler product representation, meromorphic continuation, and pole structure. For each $\\ell$, necessary and sufficient c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26114","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/2607.26114/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"}