{"paper":{"title":"Improved first moments of primitive quartic and sextic Dirichlet $L$-functions","arxiv_id":"2611.01452","primary_cat":"math.NT","submitted_at":"2026-11-01T02:40:39+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":0.97,"id":"s0","own":true,"role":"contribution","text":"We study the smoothed first moments of primitive quartic and sextic Dirichlet $L$-functions for which every nontrivial power of the character remains primitive.","hedge":{"p":0.86,"label":"asserted"},"own_p":0.91,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.21,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.03,"limitation":0.0,"future_work":0.0,"contribution":0.97,"interpretation":0.0}},{"i":1,"p":0.99,"id":"s1","own":true,"role":"result","text":"Assuming the generalized Riemann hypothesis for the Dirichlet and finite order Hecke $L$-functions occurring in the proof, we obtain asymptotic formulas with errors $O(Q^{\\frac{49}{58}+\\eps})$ and $O(Q^{\\frac{23}{26}+\\eps})$, respectively.","hedge":{"p":0.96,"label":"asserted"},"own_p":0.94,"main_claim_p":0.83,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.99,"background":0.0,"limitation":0.01,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":1.0,"id":"s2","own":true,"role":"method","text":"We follow the double Dirichlet series approach of Gao and Zhao.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.57,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":1.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":0.61,"id":"s3","own":true,"role":"method","text":"The main ingredient is a joint estimate for the two sums arising from the removal of rational prime divisors, with the M\\\"obius coefficients retained.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.84,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.61,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.39,"interpretation":0.0}},{"i":4,"p":0.36,"id":"s4","own":true,"role":"method","text":"This gives a quotient of Dirichlet $L$-functions.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.74,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.36,"result":0.35,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.22,"interpretation":0.07}},{"i":5,"p":0.93,"id":"s5","own":true,"role":"method","text":"Under GRH, we bound both the Dirichlet $L$-functions and their reciprocals.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.04,"quantitative":true,"quantitative_p":0.53,"role_probabilities":{"gap":0.0,"method":0.93,"result":0.01,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.06,"interpretation":0.0}},{"i":6,"p":0.42,"id":"s6","own":true,"role":"result","text":"Combining the joint estimate with estimates for the generating series of Gauss sums gives larger regions of meromorphic continuation.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.89,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.39,"result":0.42,"background":0.0,"limitation":0.01,"future_work":0.0,"contribution":0.17,"interpretation":0.01}},{"i":7,"p":0.78,"id":"s7","own":true,"role":"limitation","text":"These regions do not separate the possible secondary 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