Li's criterion

In mathematics, in the area of number theory, Li's criterion is a particular statement about the positivity of a certain sequence 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:$xi\; (s)=frac\{1\}\{2\}s(s-1)\; pi^\{-s/2\}\; Gamma\; left(frac\{s\}\{2\}\; ight)\; zeta(s)$

where ζ is the Riemann zeta function. Consider the sequence

:$lambda\_n\; =\; frac\{1\}\{(n-1)!\}\; left.\; frac\{d^n\}\{ds^n\}\; left\; [s^\{n-1\}\; log\; xi(s)\; ight]\; ight|\_\{s=1\}.$

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:

:$lambda\_n=sum\_\{\; ho\}\; left\; [1-\; left(1-frac\{1\}\{\; ho\}\; ight)^n\; ight]$

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

:

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

:"One has $Re(\; ho)\; le\; 1/2$ for every ρ if and only if"::$sum\_\; hoReleft\; [1-left(1-frac\{1\}\{\; ho\}\; ight)^\{-n\}\; ight]\; ge\; 0$for all positive integers $n$.

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 $overline\{\; ho\}$ and $1-\; ho$ are in "R", then Li's criterion can be stated as:

:"One has $Re(\; ho)\; =\; 1/2$ for every ρ if and only if"::$sum\_\; holeft\; [1-left(1-frac\{1\}\{\; ho\}\; ight)^\{n\}\; ight]\; ge\; 0.$

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

References

*cite journal
author = Bombieri, Enrico; Lagarias, Jeffrey C.
url = http://www.math.lsa.umich.edu/~lagarias/doc/bombieri.ps
title = Complements to Li's criterion for the Riemann hypothesis
journal = Journal of Number Theory
volume = 77
issue = 2
year = 1999
pages = 274–287
id = MathSciNet | id = 1702145
doi = 10.1006/jnth.1999.2392

*cite journal
author = Lagarias, Jeffrey C.
title = Li coefficients for automorphic L-functions
year = 2004
id = arxiv | archive = math.MG | id = 0404394

*cite journal
author = Li, Xian-Jin
title = The positivity of a sequence of numbers and the Riemann hypothesis
journal = Journal of Number Theory
volume = 65
issue = 2
year = 1997
pages = 325–333
id = MathSciNet | id = 1462847
doi = 10.1006/jnth.1997.2137