{"paper":{"title":"Discorrelation of the M\\\"obius function with linear phases in short intervals","arxiv_id":"2610.11487","primary_cat":"math.NT","submitted_at":"2026-10-08T08:29:23+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. 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x$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.59,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":1.0,"id":"s2","own":true,"role":"result","text":"In the same range of $H$ we also show that \\[ \\Big|\\sum_{x<n \\leq x+H} \\Lambda (n) e(n\\alpha)\\Big| \\ll H(\\log x)^{-A} \\] for arbitrarily large $A>0$ unless there exists an integer $1\\leq q\\ll (\\log x)^{O_A(1)}$ such that $\\|q\\alpha \\| \\ll x (\\log x)^{O_A(1)}/H^2$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.17,"quantitative":true,"quantitative_p":0.96,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":0.57,"id":"s3","own":true,"role":"interpretation","text":"This breaks the $3/5$ barrier in Theorem 1.5 of arXiv:1911.09076v2 and Theorem 2 of [T.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.83,"main_claim_p":0.1,"quantitative":true,"quantitative_p":0.92,"role_probabilities":{"gap":0.01,"method":0.0,"result":0.17,"background":0.02,"limitation":0.0,"future_work":0.0,"contribution":0.23,"interpretation":0.57}},{"i":4,"p":0.48,"id":"s4","own":false,"role":"background","text":"Zhan, \"On the representation of large odd integer as a sum of three almost equal primes,\" Acta Mathematica Sinica 7.3 (1991), 259-272].","hedge":{"p":0.75,"label":"unclear"},"own_p":0.08,"main_claim_p":0.03,"quantitative":true,"quantitative_p":0.54,"role_probabilities":{"gap":0.02,"method":0.0,"result":0.13,"background":0.48,"limitation":0.0,"future_work":0.0,"contribution":0.12,"interpretation":0.25}},{"i":5,"p":0.93,"id":"s5","own":true,"role":"interpretation","text":"Moreover, by our method, any improvement in large value estimates for character-twisted Dirichlet polynomials would lead to a corresponding 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