Li's criterion

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In mathematics, in the area of number theory, Li's criterion is a particular statement about the positivity of a certain series that is completely equivalent to the Riemann hypothesis. The criterion is named after Xian-Jin Li, who presented it in 1997. Recently, Enrico Bombieri and Jeffrey C. Lagarias provided a generalization, showing that Li's positivity condition applies to any collection of points that lie on the Re s=1/2 axis.

Definition

The Riemann ξ function is given by

[\xi (s)=\fracs(s-1) \pi^ \Gamma \left(\frac\right) \zeta(s)]

where ζ is the Riemann zeta function. Consider the sequence

[\lambda_n = \frac \left. \frac \left[s^ log xi(s) right] \right|_]

Li's criterion is then the statement that

the Riemann hypothesis is completely equivalent to the statement that [\lambda_n > 0] for every positive integer n.

The numbers [\lambda_n] may also be expressed in terms of the non-trivial zeros of the Riemann zeta function:

[\lambda_n=\sum_ \left[1- left(1-fracright)^nright]]

where the sum extends over ρ, the non-trivial zeros of the zeta function. This conditionally convergent sum should be understood in the sense that is usually used in number theory, namely, that

[\sum_\rho = \lim_ \sum_

A generalization

Bombieri and Lagarias demonstrate that a similar criterion holds for any collection of complex numbers, and is thus not restricted to the Riemann hypothesis. More precisely, let [R=\] be any collection of complex numbers ρ, not containing [\rho=1], which satisfies

[\sum_\rho \frac < \infty]

Then one may make several equivalent statements about such a set. One such statement is the following:

One has [\Re(\rho) \le 1/2] for every ρ if and only if

:[\sum_\rho\Re\left[1-left(1-fracright)^right]\ge 0]

One may make a more interesting statement, if the set R obeys a certain functional equation under the replacement of [s \mapsto (1-s)]. Namely, if, whenever ρ is in R, then both the complex conjugate [\overline] and [1-\rho] are in R, then Li's criterion can be stated as:

One has [\Re(\rho) = 1/2] for every ρ if and only if

:[\sum_\rho\left[1-left(1-fracright)^right] \ge 0]

Bombieri and Lagarias also show that Li's criterion follows from Weil's criterion for the Riemann hypothesis.

References

Xian-Jin Li, "The Positivity of a Sequence of Numbers and the Riemann Hypothesis", Journal of Number Theory, 65 (1997), 325-333.
Enrico Bombieri and Jeffery C. Lagarias, "[Complements to Li's Criterion for the Riemann Hypothesis]", Journal of Number Theory 77 (1999) 274-287.

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