Li, Xian-Jin The positivity of a sequence of numbers and the Riemann hypothesis. (English) Zbl 0884.11036 J. Number Theory 65, No. 2, 325-333 (1997). Let \[ \xi(s)= s(s- 1)\pi^{-s/2} \Gamma\Biggl({s\over 2}\Biggr) \zeta(s)\quad\text{and}\quad\lambda_n= {1\over(n- 1)!} {d^n\over ds^n} [s^{n-1}\log \xi(s)]_{s= 1}. \] It is shown that the Riemann Hypothesis holds if and only if \(\lambda_n\geq 0\) for all \(n\geq 1\). An analogous result is also proved for the Dedekind zeta-function. Reviewer: D. R. Heath-Brown (Oxford) Cited in 19 ReviewsCited in 68 Documents MSC: 11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses 11M41 Other Dirichlet series and zeta functions 11R42 Zeta functions and \(L\)-functions of number fields Keywords:Riemann Hypothesis; Dedekind zeta-function; Riemann zeta-function PDFBibTeX XMLCite \textit{X.-J. Li}, J. Number Theory 65, No. 2, 325--333 (1997; Zbl 0884.11036) Full Text: DOI References: [1] Barner, K., On A. Weil’s explicit formula, J. Reine Angew. Math., 323, 139-152 (1981) · Zbl 0446.12013 [2] Edwards, H. M., Riemann’s Zeta Function (1974), Academic Press: Academic Press New York · Zbl 0315.10035 [3] Lang, S., Algebraic Number Theory (1970), Addison-Wesley: Addison-Wesley Reading · Zbl 0211.38404 [4] Weil, A., Basic Number Theory (1967), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 0176.33601 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.