n-Point Correlation

Contents

  1. 2-Point (or Pair) Correlation
    1. Montgomery-Odlyzko Conjecture
      1. Random Matrix Theory analog
  2. n-Point Correlation
  3. Known Results
    1. Riemann Zeta-function
      1. Theoretical Knowledge
      2. Numerical Evidence
      3. Heuristics
  4. References

2-Point (or Pair) Correlation

Montgomery-Odlyzko Conjecture

If 1/2+ix_n are the unfolded, non-trivial zeros of an L-function, then for test functions f such that f(x)\rightarrow 0 as |x|\rightarrow \infty,

\lim_{N\rightarrow\infty} \frac{1}{N} \sum_{m,n\le N} f(x_n-x_m)= \int_{-\infty}^\infty f(x)\left[\delta(x)+1-\frac{\sin^2(\pi x)}{\pi^2 x^2}\right]dx

Random Matrix Theory analog

The above conjecture is analogous to the pair correlation statistic for the eigenvalues of a Random Matrix with unitary symmetry. The original conjecture was made for the Riemann \zeta-function in comparison to the Gaussian Unitary Ensemble (GUE), however as current thinking sees the zeros of L-functions compared to ensembles over the Classical Compact Groups, we present the pair correlation for U(N).

If \phi_n, n\in \{1,\dots,N\}, are the unfolded eigenangles of a unitary matrix A, then for test functions f such that f(x)\rightarrow 0 as |x|\rightarrow \infty,

\lim_{N\rightarrow\infty} \int_{U(N)}\frac{1}{N} \sum_{m,n\le N} f(\phi_n-\phi_m) dA= \int_{-\infty}^\infty f(x)\left[\delta(x)+1-\frac{\sin^2(\pi x)}{\pi^2 x^2}\right]dx

n-Point Correlation

Known Results

Riemann Zeta-function

Theoretical Knowledge

Montgomery \cite{Montgomery} has proven the conjecture for test functions f with fourier transform having support in [-1,1].

Hejhal has calculated the 3-point correlation.

Rudnick & Sarnak generalised Montgomery's approach to all n-point correlations.

Numerical Evidence

Strong numerical evidence has been obtained by Odlyzko that suggests the result is true for general test functions,

Heuristics

Keating, Bogomolny

References

D.A. Hejhal. On the triple correlation of zeros of the zeta function. Int. Math. Res. Not., 7:293,1994.

H.L. Montgomery. The pair correlation of zeros of the Riemann zeta-function. Proc. Symp. Pure Math., 24:181-93, 1973.

A.M. Odlyzko. The 10^{20}th zero of the Riemann zeta function and 70 million of its neighbours. Preprint, 1989.

L-functions/Statistics/SingleL-function/nPointCorrelation (last edited 2009-03-01 00:55:50 by localhost)