{"paper":{"title":"On divergence related to Riemann--von Mangoldt's explicit formula of the prime-counting function","arxiv_id":"2609.02713","primary_cat":"math.NT","submitted_at":"2026-09-02T15:18:41+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":"An explicit formula for the prime-counting function $\\pi(x)$, usually attributed to Riemann and von Mangoldt, is prominently stated as the equation $\\pi(x)=R(x)-\\sum_\\rho R(x^\\rho)$, where the sum runs over all zeros $\\rho$ of the Riemann $\\zeta$-function, the non-trivial ones being ordered by increasing absolute value of their imaginary parts and counted with multiplicity.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.04,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.16,"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":0.95,"id":"s1","own":false,"role":"background","text":"This particularly entails the claim that the partial sums over the non-trivial zeros, $\\Sigma R_T(x):=\\sum_{0<|\\Im m(\\rho)|\\le T} R(x^\\rho)$ converge as ${T\\to\\infty}$.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.06,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.2,"role_probabilities":{"gap":0.01,"method":0.0,"result":0.0,"background":0.95,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.04}},{"i":2,"p":0.95,"id":"s2","own":true,"role":"result","text":"Writing $\\Theta:=\\sup\\{\\Re e(\\rho):\\ \\zeta(\\rho)=0,\\ 0<\\Re e(\\rho)<1\\},$ for what has recently been called ``Riemann's constant'', we prove that, for every fixed $x>1$ and every $\\theta<\\Theta$, the sums $\\Sigma R_T(x)$ are not $O(T^\\theta)$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.21,"quantitative":true,"quantitative_p":0.92,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.95,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.05,"interpretation":0.0}},{"i":3,"p":0.9,"id":"s3","own":true,"role":"result","text":"As a consequence, $\\limsup_{{T\\to\\infty}}|\\Sigma R_T(x)|=\\infty$ and $\\sum_\\rho R(x^\\rho)$ diverges.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.81,"quantitative":true,"quantitative_p":0.57,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.9,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.1}},{"i":4,"p":0.86,"id":"s4","own":true,"role":"result","text":"We conclude the paper by showing that an adapted, but simpler strategy also gives the divergence of the contribution of the trivial zeros to $\\sum_\\rho R(x^\\rho)$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.86,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.13,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":1.0,"text":"As a consequence, $\\limsup_{{T\\to\\infty}}|\\Sigma R_T(x)|=\\infty$ and $\\sum_\\rho R(x^\\rho)$ diverges.","location":"abstract s3","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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0<\\Re e(\\rho)<1\\},$ for what has recently been called ``Riemann's constant'', we prove that, for every fixed $x>1$ and every $\\theta<\\Theta$, the sums $\\Sigma R_T(x)$ are not $O(T^\\theta)$.","location":"abstract s2"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v1"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":5,"sentences_in_abstract":5},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; the figure is how often the published label matches their 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