Nontrivial zeros and the eigenvalues of random matrices

According to the article written by Meier and Steuding[5], one of the most interesting and aesthetic of all possible scenarios proposed by Riemann Hypothesis (RH) is regularity in the distribution of zeros, and thus in prime number distribution as well. There are numerous articles and papers which attempt to prove RH and to investigate its consequence. One of them is to relate the zeros of the Riemann zeta function and the eigenvalues of random matrices.

The Riemann zeta function and its Euler’s expression can be defined as :

$\zeta(s)= \sum_{n=1}^{\infty}\frac{1}{n^{s}}= \prod_{p}\left ( 1-\frac{1}{p^{s}} \right )^{-1}$

here the product on the right side is taken over all prime numbers p. The Euler-product on the right side may be ragarded as an analytic version of the unique prime factorization of integers, and it converges for $Re(s)>1$. Riemann showed that $\zeta(s)$ has a continuation to the complex plane and satisfi es a functional equation :

$\xi(s):=\pi^{-s/2} \Gamma \left ( s/2 \right )\zeta(s)= \xi \left ( 1-s \right )$

$\xi(s)$ is entire except for simple poles at s = 0 and 1. The zeros of $\xi(s)$ is

$\frac{1}{2}+ i \gamma$

It’s clear that $\left |Im(\gamma) \right | \leq \frac{1}{2}$. Hadamard and de la Vallée Poussin in their proofs of Prime Number Theorem established that $\left |Im(\gamma) \right | < \frac{1}{2}$.

The Riemann hypothesis (RH) states that all nontrivial zeros of $\zeta(s)$ lie on the critical line $Re(s) =\frac{1}{2}$. So here $\gamma \in \mathbb{R}$.

In 1973 Montgomery[1] conjectured that the number of nontrivial zeros $\frac{1}{2}+ i\gamma$, $\frac{1}{2}+i\gamma '$ of $\zeta(s)$ satisfying the inequalities :

$0<\alpha\leq \frac{\log T}{2\pi}(\gamma-\gamma ')\leq\beta$

is asymptotically equal to

$N(T)\displaystyle\int_{\alpha }^{\beta }\left ( 1-\left ( \frac{\sin\pi u}{\pi u} \right )^2 \right )du$

as $T\rightarrow\infty$.

This conjecture is now called Montgomery’s pair correlation and plays a complementary role to the Riemann hypothesis; i.e., vertical vs. horizontal distribution of the nontrivial zeros.

Montgomery’s pair correlation conjecture claims that the two-point correlation for the zeros of the zeta function on the critical line is (in the limit) equal to the two-point correlation for the eigenangles of random Hermitian matrices from the Gaussian Unitary Ensemble (GUE). This conjecture implies that almost all zeros of the zeta function are simple; the predicted pair correlation matches to the one of the eigenangles of certain random matrix ensembles. Here because the corresponding asymptotics is a theorem in random matrix theory, not only a conjecture[5].

Let’s assume RH and order the ordinates $\gamma$ :

$...\gamma_{-1}\leq 0\leq \gamma_{1}\leq \gamma_{2}...$

Then $\gamma_{j}= -\gamma_{-j}, j= 1,2,...$, Riemann computed $\gamma_{1}$, the one with smallest positive imaginary part being 14,43725. Riemann noted that :

$\left \{ j:0\leq \gamma_{j}\leq T \right \}\sim \frac{T \log T}{2 \pi}$, as $T\rightarrow \infty$

In particular, the mean spacing between the $\gamma^{'}_{j}s$ tends to zero as $j\rightarrow \infty$.

Local spacings law between these numbers can be represented by re-normalization as follows :

$\hat{\gamma_{j}}= \frac{\gamma_{j}\log \gamma_{j}}{2\pi}$ for $j \geq 1$

The consecutive spacings $\delta_j$ are de fined to be :

$\delta_j=\hat{\gamma_{j+1}}-\hat{\gamma_{j}}, j=1,2,...$

More generally, the k-th consecutive spacings are :

$\delta^{(k)}_j=\hat{\gamma_{j+k}}-\hat{\gamma_{j}}, j=1,2,...$

Odlyzko [2] has made an extensive and profound numerical study of the zeros and in particular their local spacings. He finds that they obey the laws for the (scaled) spacings between the eigenvalues of a typical large unitary matrix. That is they obey the laws of the Gaussian Unitary Ensemble (GUE). Computations of Odlyzko[2] showed that also the nearest neighbor spacing for the nontrivial zeros of the zeta-function seems to be amazingly close to those for the eigenangles of the GUE.

The upper figure shows Odlyzko’s pair correlation for $2 \times 10^8$ zeros of $\zeta (s)$ near the $10^{23}$rd zero. The lower figure shows the difference between the histogram in the first graph and $1-\left( \frac{\sin\pi t}{\pi t}\right )^2$. In the interval displayed, the two agree to within about 0.002 …. (pictures credited to Meier and Steuding[5]).

The Montgomery-Odlyzko law claims that these distributions are, statistically, the same. But the only proved cases of pair correlation asymptotics are those of Katz and Sarnak[4] for certain local zeta functions. (I found references below for comprehensive chronology of these developments or here [6] for the review).

References :

[1] H.L. Montgomery, The pair correlation of zeros of the Riemann zeta-function on the critical line, Proc. Symp. Pure Math. Providence 24 (1973), 181-193.

[2] A.M. Odlyzko, The 10^{20}th zero of the Riemann zeta-function and 70 million of its neighbors, in ’Dynamical, spectral, and arithmetic zeta functions’ (San Antonio, TX, 1999), 139–144, Contemp. Math. 290, Amer. Math. Soc., Providence 2001.

[3] N.M. Katz, P. Sarnak, Zeros of The Zeta Function and Symmetry, Bulletin of The AMS, Vol. 36, Number 1, January 1999, Pages 1-26.

[4] N.M. Katz, P. Sarnak, Random matrices, Frobenius eigenvalues, and monodromy, AMS, Providence 1999.

[5] P. Meier, J. Steuding, The Riemann Hypothesis, available at claymath.org.

[6] J. Steuding, The Riemann Zeta Function and Predictions from Random Matrix Theory, AMS, subject classification numbers: 11M06.