{"paper":{"title":"Unshared zeros of Dirichlet $L$-functions","arxiv_id":"2607.22930","primary_cat":"math.NT","submitted_at":"2026-07-24T22:05:47+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":0.96,"id":"s0","own":true,"role":"result","text":"We prove that no Dirichlet $L$-function (and more generally, no nontrivial finite linear combination of Dirichlet $L$-functions) can vanish at every zero of a fixed $L(s,\\chi_0)$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.96,"quantitative":false,"quantitative_p":0.15,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.96,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.04,"interpretation":0.0}},{"i":1,"p":0.86,"id":"s1","own":true,"role":"method","text":"At the heart of the proof is a short-window asymptotic for the twisted discrete moment $\\sum_{\\rho} x^{\\rho} L(\\rho,\\chi_1)$, where $\\chi_0\\ne\\chi_1$ are primitive Dirichlet characters, $\\rho=\\beta+i\\gamma$ runs over zeros of $L(s,\\chi_0)$ with $T-\\Delta<\\gamma\\le T$, and $x\\in\\mathbb{Z}$.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.92,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.3,"role_probabilities":{"gap":0.0,"method":0.86,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.14,"interpretation":0.0}},{"i":2,"p":0.81,"id":"s2","own":true,"role":"result","text":"The asymptotic is unconditional, assuming no hypothesis of GRH type, and it holds for every window width $\\Delta\\in[T e^{-C\\sqrt{\\log T}},\\,T/\\log T]$, thus reaching windows shorter than $T(\\log T)^{-A}$ for any fixed $A$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.04,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.04,"result":0.81,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.14,"interpretation":0.0}},{"i":3,"p":0.9,"id":"s3","own":true,"role":"result","text":"Notably, the main term $\\frac{\\chi_1(x)}{2\\pi}\\,\\Delta\\log T$ depends on $x$ only through the single value $\\chi_1(x)$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.89,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.22,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.9,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.09}},{"i":4,"p":0.63,"id":"s4","own":true,"role":"result","text":"Since distinct primitive characters are distinguished by their values, varying $x$ isolates the contribution of each $L$-function within a linear combination, and we deduce that for a positive density of $x\\in\\mathbb{N}$, every nontrivial combination is nonzero at some zero of $L(s,\\chi_0)$ in any sufficiently high short window.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.42,"quantitative":true,"quantitative_p":0.54,"role_probabilities":{"gap":0.0,"method":0.02,"result":0.63,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.34}},{"i":5,"p":0.99,"id":"s5","own":true,"role":"method","text":"The proof combines contour integration of $-\\frac{L'}{L}(1-s,\\overline{\\chi}_0)\\,L(s,\\chi_1)$ with short-interval estimates for the Dirichlet convolution $(\\chi_0\\Lambda)*\\chi_1$, which derive from the classical de la Vall\\'ee Poussin zero-free region.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.8,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.05,"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}}]},"compiled":{"main_claim":{"p":1.0,"text":"We prove that no Dirichlet $L$-function (and more generally, no nontrivial finite linear 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