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Sharpenings of Li’s criterion for the Riemann hypothesis. (English) Zbl 1181.11055
Summary: Exact and asymptotic formulae are displayed for the coefficients \(\lambda_n\) used in Li’s criterion for the Riemann Hypothesis [see X.-J. Li, J. Number Theory, 65, No. 2, 325–333 (1997; Zbl 0884.11036), E. Bombieri and J.C. Lagarias, J. Number Theory 77, 274–287 (1999; Zbl 0972.11079)]. For \(n \rightarrow \infty\) we obtain that if (and only if) the Hypothesis is true, \(\lambda_n\sim n (A \log n + B)\) (with \(A>0\) and \(B\) explicitly given, also for the case of more general zeta or \(L\)-functions); whereas in the opposite case, \(\lambda_n\) has a non-tempered oscillatory form.

MSC:
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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