Complements to Li’s criterion for the Riemann Hypothesis. (English) Zbl 0972.11079

Bombieri, Enrico; Lagarias, Jeffrey C.

J. Number Theory 77, No. 2, 274-287 (1999).

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.

Reviewer: Roger Heath-Brown (Oxford)

MSC:

11M26

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

Keywords:

Riemann hypothesis; zeros; Weil’s explicit formula; Li’s criterion; Weil’s positivity condition