{"paper":{"title":"Even and Odd Integer Values of the Secondary Zeta Function","arxiv_id":"2610.19175","primary_cat":"math.GM","submitted_at":"2026-10-13T20:40:46+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":"The secondary zeta function is defined by $$ Z(s)=\\sum_{\\gamma>0}\\frac{1}{\\gamma^{s}}, $$ where $\\tfrac12+i\\gamma$ runs over the non-trivial zeros of the Riemann zeta function with positive ordinate, and the Riemann hypothesis is assumed throughout.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.14,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.37,"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":0.97,"id":"s1","own":true,"role":"contribution","text":"In this article, we derive convergent integral representations of $Z(s)$ at positive even and odd integer argument, expressed through a differential operator acting on the logarithm of the Riemann $\\xi$-function.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.97,"main_claim_p":0.8,"quantitative":false,"quantitative_p":0.1,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.03,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.97,"interpretation":0.0}},{"i":2,"p":0.83,"id":"s2","own":true,"role":"method","text":"Define $$ L(x)=\\log\\left(\\frac{\\xi(x+\\tfrac12)}{\\xi(\\tfrac12)}\\right), \\qquad \\mathcal{E}=x\\frac{\\mathrm{d}}{\\mathrm{d}x}, \\qquad \\mathcal{D}_k=\\mathcal{E}\\prod_{j=1}^{k}(\\mathcal{E}-2j).","hedge":{"p":0.73,"label":"asserted"},"own_p":0.64,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.21,"role_probabilities":{"gap":0.0,"method":0.83,"result":0.01,"background":0.02,"limitation":0.0,"future_work":0.0,"contribution":0.14,"interpretation":0.0}},{"i":3,"p":0.93,"id":"s3","own":true,"role":"result","text":"$$ Then, for $k\\geq1$, $$ Z(2k)=\\frac{(-1)^k}{2^k(k-1)!}\\int_{0}^{\\infty}\\frac{\\mathcal{D}_k L(x)}{x^{2k+1}}\\,\\mathrm{d}x $$ and $$ Z(2k+1)=\\frac{(-1)^k}{\\pi\\,(2k-1)!!}\\int_{0}^{\\infty}\\frac{\\mathcal{D}_k L(x)}{x^{2k+2}}\\,\\mathrm{d}x.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.75,"quantitative":true,"quantitative_p":0.61,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.93,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.06,"interpretation":0.0}},{"i":4,"p":0.91,"id":"s4","own":true,"role":"contribution","text":"$$ In the last part, we write these formulas out explicitly for $Z(k)$, $2\\leq k\\leq 7$, both in terms of $L(x)$ and in terms of $\\xi(x+\\tfrac12)$ for reference.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.96,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.84,"role_probabilities":{"gap":0.0,"method":0.04,"result":0.05,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.91,"interpretation":0.0}},{"i":5,"p":0.64,"id":"s5","own":true,"role":"contribution","text":"We also derive a limit formula for $\\gamma_1$, the ordinate of the first nontrivial zero, and demonstrate its convergence numerically.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.1,"quantitative":false,"quantitative_p":0.25,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.36,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.64,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.75,"text":"In this article, we derive convergent integral representations of $Z(s)$ at positive even and odd integer argument, expressed through a differential operator acting on the logarithm of the Riemann 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