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The positivity of a sequence of numbers and the Riemann hypothesis. (English) Zbl 0884.11036
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.

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
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References:
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