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Complements to Li’s criterion for the Riemann Hypothesis. (English) Zbl 0972.11079
It was shown by Xian-Jin Li [J. Number Theory 65, 325-333 (1997; Zbl 0884.11036)] that the Riemann Hypothesis is equivalent to the statement that $\lambda_n:= \left.\frac{1} {(n-1)!} \frac{d^n} {ds^n} [s^{n-1}\log \xi(s)] \right|_{s=1}> 0$ for every natural number $$n$$, where $$\xi(s)$$ is the usual Riemann $$\xi$$-function. One easily sees that, for the natural interpretation of the conditionally convergent sum, one has $\lambda_n= \sum\{1- (1-\rho^{-1})^n\}$ where $$\rho$$ runs over the non-trivial zeros of $$\zeta(s)$$.
In the present paper one takes $$\rho$$ to run over any multiset of complex numbers satisfying the usual 4-fold symmetry involving $$1-\rho$$ and $$\overline{\rho}$$, along with some other mild conditions. The principal result is then that the positivity of the number $$\lambda_n$$ is equivalent to the statement that $$\operatorname{Re}(\rho)= 1/2$$ for all $$\rho$$.
The paper goes on to express $$\lambda_n$$ via Weil’s explicit formula, and to interpret Li’s criterion in terms of Weil’s positivity condition.

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