Prime Numbers

8.6 Curious, anecdotal, and interdisciplinary references to primes 427
has mean value 1. But computer plots of the histogram of δ values show
a remarkable agreement for the same (theoretically known) statistic on
eigenvalues of a GUE matrix. Such comparisons have been done on over
10 8 zeros neighboring zN where N ≈ 10 20 (though the work of [Odlyzko
2005] involves 10 10 zeros of even greater height). The situation is therefore
compelling: There may well be an operator whose eigenvalues are precisely the
Riemann critical zeros (scaled by the logarithmic factor). But the situation is
not as clean as it may appear. For one thing, Odlyzko has plotted the Fourier
transform
N+40000
N+1
e ixzn ,
and it does not exhibit the decay (in x) expected of GUE eigenvalues. In
fact, there are spikes reported at x = p k , i.e., at prime-power frequencies.
This is expected from a number-theoretical perspective. But from the
physics perspective, one can say that the critical zeros exhibit “long-range
correlation,” and it has been observed that such behavior would accrue if the
critical zeros were not random GUE eigenvalues per se, but eigenvalues of
some unknown Hamiltonian appropriate to a chaotic-dynamical system. In
this connection, a great deal of fascinating work—by M. Berry and others—
under the rubric of “quantum chaology” has arisen [Berry 1987].
There are yet other connections between the Riemann ζ and concepts from
physics. For example, in [Borwein et al. 2000] one finds mention of an amusing
connection between the Riemann ζ and quantum oscillators. In particular, as
observed by Crandall in 1991, there exists a quantum wave function ψ(x, 0)—
smooth, devoid of any zero crossings on the x axis—that after a finite time T of
evolution under the Schrödinger equation becomes a “crinkly” wave function
ψ(x, T ) with infinitely many zero crossings, and these zeros are precisely the
zeros of ζ(1/2+ix) on the critical line. In fact, for the wave function at the
special time T in question, the specific eigenfunction expansion evaluates as
ψ(x, T )=f
1 1
+ ix ζ + ix
2 2
= e −x2 /(2a 2 )
∞
cn(−1) n H2n(x/a), (8.5)
for some positive real a and a certain sequence (cn) of real coefficients
depending on a, withHm being the standard Hermite polynomial of order
m. Here, f(s) is an analytic function of s having no zeros. It is amusing that
one may truncate the n-summation at some N, say, and numerically obtain—
now as zeros of a degree-2N polynomial—fairly accurate critical zeros. For
example, for N = 27 (so polynomial degree is 54) an experimental result
appears in [Borwein et al. 2000] in which the first seven critical zeros are
obtained, the first of which being to 10 good decimals. In this way one can
in principle approximate arbitrarily closely the Riemann critical zeros as the
eigenvalues of a Hessenberg matrix (which in turn are zeros of a particular
polynomial). A fascinating phenomenon occurs in regard to the Riemann
hypothesis, in the following way. If one truncates the Hermite sum above,
n=0