{"paper":{"title":"Complete Asymptotic Expansion of the Additive Mertens Sum $S_k(x)$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Daoyi Peng, Hao Liu","submitted_at":"2026-07-05T15:50:33Z","abstract_excerpt":"Let $p_1, \\dotsc, p_k$ be primes not exceeding $x$ ($k \\geqslant 2$), and define the additive Mertens sum \\[\n  S_k(x) = \\sum_{p_1 \\leqslant x} \\cdots \\sum_{p_k \\leqslant x} \\frac{1}{p_1 + \\dotsm + p_k}. \\] In contrast to Tenenbaum's generalized (multiplicative) Mertens sum, whose leading term has order $(\\log \\log x)^k$, the sum $S_k(x)$ has leading term of order $x^{k-1}/\\log^k x$. We establish the complete asymptotic expansion \\[ S_k(x) = \\frac{x^{k-1}}{\\log^k x} \\sum_{n=0}^{N} \\frac{E_{k,n}}{\\log^n x} + O\\left(\\frac{x^{k-1}}{\\log^{k+N+1} x}\\right) \\quad (\\forall\\, N \\geqslant 0), \\] where t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04366","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/2607.04366/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"}