{"paper":{"title":"A Tsang-range high-moment bound for $\\operatorname{Im}\\log L(\\tfrac12+it,\\chi)$ under GRH","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Scott D. Hughes","submitted_at":"2026-06-08T23:09:36Z","abstract_excerpt":"Conditional on the Generalized Riemann Hypothesis for $L(s,\\chi)$, we prove the Selberg--Tsang high-moment bound for $X_\\chi(t) = \\operatorname{Im}\\log L(\\tfrac12+it,\\chi)$ at fixed squarefree odd conductor $q \\ge 3$ and primitive non-principal character $\\chi$. Writing $L_T = \\log\\log(qT)$: for every $K > 0$ there exist constants $C_K$ and $T_0$ such that $\\frac{1}{T}\\int_T^{2T} |X_\\chi(t)|^{2k}\\,dt \\le (C_K\\,k\\,L_T)^k$ for all $T \\ge T_0$ and every integer $1 \\le k \\le K L_T$. The proof ports Selberg's pointwise approximate formula for $S(t)$ to $L(s,\\chi)$ at fixed conductor under GRH, spli"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.10242","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/2606.10242/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"}