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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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The author is called Irwin. It's not a typical thing but what she likes doing is base jumping and now she is trying to earn cash with it. Her husband and her reside in Puerto Rico but she will have to transfer one day or another. I am a meter reader but I plan on changing it.
my web-site: over the counter std test (try what he says)