{"paper":{"title":"Upper Bounds on the de Bruijn-Newman constant for Dirichlet $L$-functions with primitive real-valued characters","arxiv_id":"2610.27343","primary_cat":"math.NT","submitted_at":"2026-10-19T22:17:16+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. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"Let $\\chi_m$ be a primitive real-valued Dirichlet character of conductor $m\\ge3$, let $L(s)=L(s,\\chi_m)$ be the associated Dirichlet $L$-function, and let $\\xi_L(s)$ denote its completed form.","hedge":{"p":0.8,"label":"asserted"},"own_p":0.08,"main_claim_p":0.01,"quantitative":true,"quantitative_p":0.81,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":0.61,"id":"s1","own":false,"role":"background","text":"For each $t \\in \\mathbb R,$ we define the entire function \\[ H^L_t(z) := \\int_{-\\infty}^{\\infty} e^{tu^2}\\, \\Phi_L(u)\\, e^{izu}\\, du, \\] where $\\Phi_L$ is the superexponentially decaying density \\[ \\Phi_L(u) := \\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty} \\xi_L\\!\\left(\\frac{1+is}{2}\\right) e^{-isu}\\, ds.","hedge":{"p":0.97,"label":"asserted"},"own_p":0.37,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.15,"role_probabilities":{"gap":0.0,"method":0.36,"result":0.0,"background":0.61,"limitation":0.0,"future_work":0.0,"contribution":0.03,"interpretation":0.0}},{"i":2,"p":0.89,"id":"s2","own":false,"role":"background","text":"\\] The family $\\{H_t^L\\}_{t \\in \\mathbb R}$ describes the heat-flow evolution of $\\xi_L$ and solves a backward heat equation.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.1,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.09,"result":0.0,"background":0.89,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.02}},{"i":3,"p":1.0,"id":"s3","own":false,"role":"background","text":"By the work of Newman, there exists a finite constant $\\Lambda_L$, called the generalized de Bruijn--Newman constant, such that all the zeros of $H_t^L$ are real if and only if $t \\ge \\Lambda_L$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.02,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.4,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":4,"p":0.98,"id":"s4","own":false,"role":"background","text":"The Generalized Riemann Hypothesis (GRH) for $L$ is equivalent to the assertion that $\\Lambda_L \\le 0$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.04,"main_claim_p":0.03,"quantitative":true,"quantitative_p":0.91,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.98,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.02}},{"i":5,"p":0.98,"id":"s5","own":false,"role":"background","text":"Dobner established the matching lower bound $\\Lambda_L \\ge 0$, while de Bruijn proved the upper bound $\\Lambda_L \\le \\tfrac12$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.03,"main_claim_p":0.03,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.02,"method":0.0,"result":0.0,"background":0.98,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":6,"p":0.94,"id":"s6","own":true,"role":"result","text":"We prove the unconditional uniform bound $\\Lambda_L\\le953/3200=0.2978125$ for every primitive real-valued character of conductor $3\\le m\\le400$, together with stronger individual bounds.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.97,"main_claim_p":0.95,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.94,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.06,"interpretation":0.0}},{"i":7,"p":0.92,"id":"s7","own":true,"role":"method","text":"The proof extends the effective approximation of Polymath to Dirichlet $L$-functions, retaining the conductor dependence in the error estimates.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.75,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.92,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.08,"interpretation":0.0}},{"i":8,"p":0.99,"id":"s8","own":true,"role":"method","text":"We combine these estimates with Platt's numerical verification of GRH by Platt and rigorous interval computations.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.76,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.99,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":9,"p":0.67,"id":"s9","own":true,"role":"result","text":"We also give conditional bounds under explicit global initial zero-strip hypotheses, including $|\\operatorname{Im}z|\\le3/4$ for the zeros of $H_0^L$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.91,"main_claim_p":0.04,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.67,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.33,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":1.0,"text":"We prove the unconditional uniform bound $\\Lambda_L\\le953/3200=0.2978125$ for every primitive real-valued character of conductor $3\\le m\\le400$, together with stronger individual bounds.","location":"abstract s6","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per 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