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| In [[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.
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| ==Definition==
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| The [[Riemann Xi function|Riemann ξ function]] is given by
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| :<math>\xi (s)=\frac{1}{2}s(s-1) \pi^{-s/2} \Gamma \left(\frac{s}{2}\right) \zeta(s)</math>
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| where ζ is the [[Riemann zeta function]]. Consider the sequence
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| :<math>\lambda_n = \frac{1}{(n-1)!} \left. \frac{d^n}{ds^n}
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| \left[s^{n-1} \log \xi(s) \right] \right|_{s=1}.</math>
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| Li's criterion is then the statement that
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| :''the Riemann hypothesis is completely equivalent to the statement that <math>\lambda_n > 0</math> for every positive integer ''n''.''
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| The numbers <math>\lambda_n</math> may also be expressed in terms of the non-trivial zeros of the Riemann zeta function:
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| :<math>\lambda_n=\sum_{\rho} \left[1-
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| \left(1-\frac{1}{\rho}\right)^n\right]</math>
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| 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
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| :<math>\sum_\rho = \lim_{N\to\infty} \sum_{|\Im(\rho)|\le N}.</math>
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| ==A generalization==
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| 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'' = {''ρ''} be any collection of complex numbers ''ρ'', not containing ''ρ'' = 1, which satisfies
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| :<math>\sum_\rho \frac{1+\left|\Re(\rho)\right|}{(1+|\rho|)^2} < \infty.</math>
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| Then one may make several equivalent statements about such a set. One such statement is the following:
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| :''One has <math>\Re(\rho) \le 1/2</math> for every ρ if and only if''
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| ::<math>\sum_\rho\Re\left[1-\left(1-\frac{1}{\rho}\right)^{-n}\right]
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| \ge 0</math>
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| for all positive integers ''n''.
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| One may make a more interesting statement, if the set ''R'' obeys a certain [[functional equation]] under the replacement ''s'' ↦ 1 − ''s''. Namely, if, whenever ρ is in ''R'', then both the complex conjugate <math>\overline{\rho}</math> and <math>1-\rho</math> are in ''R'', then Li's criterion can be stated as:
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| :''One has'' Re(''ρ'') = 1/2 ''for every'' ''ρ'' ''if and only if''
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| ::<math>\sum_\rho\left[1-\left(1-\frac{1}{\rho}\right)^n \right] \ge 0.</math>
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| Bombieri and Lagarias also show that Li's criterion follows from [[Weil's criterion]] for the Riemann hypothesis.
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| ==References==
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| *{{cite journal
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| | author = [[Enrico Bombieri|Bombieri, Enrico]]; [[Jeffrey C. Lagarias|Lagarias, Jeffrey C.]]
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| | url = http://www.math.lsa.umich.edu/~lagarias/doc/bombieri.ps
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| | title = Complements to Li's criterion for the Riemann hypothesis
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| | journal = [[Journal of Number Theory]]
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| | volume = 77
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| | issue = 2
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| | year = 1999
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| | pages = 274–287
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| | id = {{MathSciNet | id = 1702145}}
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| | doi = 10.1006/jnth.1999.2392}}
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| *{{cite journal
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| | author = [[Jeffrey C. Lagarias|Lagarias, Jeffrey C.]]
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| | title = Li coefficients for automorphic L-functions
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| | year = 2004
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| | arxiv = archive = math.MG/id = 0404394}}
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| *{{cite journal
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| | author = Li, Xian-Jin
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| | title = The positivity of a sequence of numbers and the Riemann hypothesis
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| | journal = [[Journal of Number Theory]]
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| | volume = 65
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| | issue = 2
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| | year = 1997
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| | pages = 325–333
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| | id = {{MathSciNet | id = 1462847}}
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| | doi = 10.1006/jnth.1997.2137}}
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| [[Category:Zeta and L-functions]]
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