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In mathematics, '''Montgomery's pair correlation conjecture''' is a conjecture made by {{harvs|txt|authorlink=Hugh Montgomery (mathematician)|year=1973|last=Montgomery|first=Hugh}} that the pair correlation between pairs of zeros of the [[Riemann zeta function]] (normalized to have unit average spacing) is
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:<math>1-\left(\frac{\sin(\pi u)}{\pi u}\right)^2 +\delta(u),</math>
which, as [[Freeman Dyson]] pointed out to him, is the same as the pair [[correlation function]] of [[Gaussian unitary ensemble|random Hermitian matrices]]. Informally, this means that the chance of finding a zero in a very short interval of length 2π''L''/log(''T'') at a distance 2π''u''/log(''T'') from a zero 1/2+''iT'' is about ''L'' times the expression above. (The factor 2π/log(''T'') is a normalization factor that can be thought of informally as the average spacing between zeros with imaginary part about ''T''.) {{harvs|txt|authorlink=Andrew Odlyzko|year=1987|last=Odlyzko|first=Andrew}} showed that the conjecture was supported by large-scale computer calculations of the zeros. The conjecture has been extended to correlations of more than 2 zeros, and also to zeta functions of automorphic representations {{harv|Rudnick|Sarnak|1996}}. In 1982 a student of Montgomery's, Ali Erhan Özlük, ''proved'' the pair correlation conjecture for some of Dirichlet's L-functions.{{harvs|txt|authorlink=A.E.Ozluk |year=1982|last=Ozluk |first=A.E.}}
 
The connection with random unitary matrices could lead to a proof of the [[Riemann hypothesis]]. The [[Hilbert–Pólya conjecture]] asserts that the zeros of the Riemann Zeta function correspond to the eigenvalues of a linear operator, and implies RH. Some people think this is a promising approach ({{harvs|txt|authorlink=Andrew Odlyzko|year=1987|last=Odlyzko|first=Andrew}}).
 
Montgomery was studying the [[Fourier transform]] ''F''(''x'') of the pair correlation function, and showed (assuming the Riemann hypothesis) that it was
equal to |''x''| for |''x''|<1. His methods were unable to determine it for |''x''|≥1, but he  conjectured that it was equal to 1 for these ''x'', which implies that the pair correlation function is as above.
 
== Numerical calculation by Odlyzko ==
[[File:Montgomery-Odlyzko law.png|right|thumb|The real line describes the two-point correlation function of the random matrix of type GUE. Blue dots describe the normalized spacings of the non trivial zeros of Riemann zeta function, the first 10<sup>5</sup> zeros.]]
 
In the 1980s, motivated by the Montgomery's conjecture, Odlyzko began an intensive numerical　study of the statistics of the zeros of ζ(s).　He confirmed that the distribution of the spacings between non-trivial zeros using detail numerical calculation and demonstrated that the Montgomery's conjecture would be true and the distribution would agree with the distribution of spacings of GUE random matrics eigenvalues using [[Cray X-MP]]. In 1987 he reported them on the paper {{harvs|txt|authorlink=Andrew Odlyzko|year=1987|last=Odlyzko|first=Andrew}}.
 
For non trivial zero, 1/2+i&gamma;<sub>n</sub>, let the normalized spacings be
:<math>
\delta_n = (\gamma_{n+1}-\gamma_{n}) \frac{ \log{ \frac{\gamma_n}{2 \pi} }}{2 \pi}
.</math>
Then we would expect the following formula as the limit, M, N →∞. Then
 
:<math>\frac{1}{M} \{(n,k) | N \leq n \leq N+M, \,</math>
:::<math>k \geq 0, \,\delta_{n}+ \delta_{n+1}+ \cdots +\delta_{n+k} \in [ \alpha,  \beta]  \}</math>
:::::<math>\sim  \int_{\alpha}^{\beta} \left (
1- \biggl ( \frac{\sin{\pi u}}{\pi u}  \biggr )^2  \right ) du
</math>
 
Based on a new algorithm developed by Odlyzko　and Schönhage that allowed them to compute a　value of ζ(1/2 + it) in an average time of t<sup>ε</sup> steps, he computed millions of zeros at heights around　10<sup>20</sup> and spectacularly confirmed the GUE conjecture.<ref name ="odlyzko1989">A. M. Odlyzko, "The 10<sup>20</sup>-th zero of the Riemann zeta function and 70 million of its neighbors," AT&T Bell Lab. preprint (1989)</ref><ref name ="metha">M. Mehta (1990), chap.1</ref>
 
Here, the upper figure contains the first 10<sup>5</sup> non-trivial zeros of the Riemann zeta function. The bellow figure contains the first 10<sup>20</sup> zeros. Then they show that as the more the sampling zeros, the more approximately their distribution shapes to the GUE random matrix.
 
==References==
{{reflist}}
*{{Citation | last1=Ozluk |first1=A.E. | title=Pair Correlation of Zeros of Dirichlet's L-functions | series=Ph. D. Dissertation |publisher=Univ. of Michigan |location=Ann Arbor |year=1982 }}
*{{Citation | author1-link=Nicholas M. Katz | author2-link=Peter Sarnak | last1=Katz | first1=Nicholas M. | last2=Sarnak | first2=Peter | title=Zeroes of zeta functions and symmetry | url=http://www.ams.org/bull/1999-36-01/S0273-0979-99-00766-1/home.html | doi=10.1090/S0273-0979-99-00766-1  | mr=1640151  | year=1999 | journal=American Mathematical Society. Bulletin. New Series | issn=0002-9904 | volume=36 | issue=1 | pages=1–26}}
*{{Citation | last1=Montgomery | first1=Hugh L. | title=Analytic number theory |series=Proc. Sympos. Pure Math.|volume= XXIV | publisher=[[American Mathematical Society]] | location=Providence, R.I. | mr=0337821  | year=1973 | chapter=The pair correlation of zeros of the zeta function | pages=181–193}}
*{{Citation | authorlink=Odlyzko | last1=Odlyzko | first1=A. M. | title=On the distribution of spacings between zeros of the zeta function | jstor=2007890 | mr=866115  | year=1987 | journal=[[Mathematics of Computation]] | issn=0025-5718 | volume=48 | issue=177 | pages=273–308 | doi=10.2307/2007890 | publisher=American Mathematical Society}}
*{{Citation | author2-link=Peter Sarnak | last1=Rudnick | first1=Zeév | last2=Sarnak | first2=Peter | title=Zeros of principal L-functions and random matrix theory | url=http://projecteuclid.org/euclid.dmj/1077245671 | doi=10.1215/S0012-7094-96-08115-6 | mr=1395406  | year=1996 | journal=[[Duke Mathematical Journal]] | issn=0012-7094 | volume=81 | issue=2 | pages=269–322}}
 
[[Category:Zeta and L-functions]]
[[Category:Conjectures]]
[[Category:Analytic number theory]]

Latest revision as of 21:29, 22 December 2014

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