{"paper":{"title":"Rank Amplification for Shifted Equal Values of Euler's Totient Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Eric Li (Trinity College, University of Cambridge)","submitted_at":"2026-06-22T17:59:25Z","abstract_excerpt":"Let $S_h^\\varphi(x)$ denote the number of integers $n\\le x$ for which $\\varphi(n)=\\varphi(n+h)$. For the unit shift, we prove $S_1^\\varphi(x)\\ll x\\exp{-(1/2-o(1))\\sqrt{\\log x,\\log_2 x}}$. More generally, put $A=\\log_3 x+\\log_4 x-\\log 2$, $G=\\sqrt{\\log x,A}$, and $V=\\log x/G$. For every fixed integer $J\\ge 1$, uniformly for $1\\le h\\le \\exp{G/\\sqrt{J}}$, we obtain $S_h^\\varphi(x)=D_{h,>Y_J}^\\varphi(x)+O_J(x\\exp{-\\sqrt{J},G+o_J(V)})$, where $Y_J=\\exp{\\sqrt{J},G}$. Here $D_{h,>Y_J}^\\varphi(x)$ is the above-cutoff part of the classical Graham--Holt--Pomerance same-support family; it is empty for od"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.23681","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/2606.23681/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"}