Prime Numbers

426 Chapter 8 THE UBIQUITY OF PRIME NUMBERS
Another appearance of the noble primes—this time in connection with
molecular biology—is in [Yan et al. 1991]. These authors infer that certain
amino acid sequences in genetic matter exhibit patterns expected of (binary
representations of) prime numbers. In one segment they say:
Additively generated numbers can be primes or nonprimes. Multiplicatively
generated numbers are nonprimes (“composites” in number theory
terminology). Thus, prime numbers are more creative than nonprimes ....
The creativeness and indivisibility of prime numbers leads one to infer that
primes smaller than 64 are the number equivalents of amino acids; or that
amino acids are such Euclid units of living molecules.
The authors go on to suggest Diophantine rules for their theory. The present
authors do not intend to critique the interdisciplinary notion that composite
numbers somehow contain less information (are less profound) than the
primes. Rather, we simply point out that some thought has gone into this
connection with genetic codes.
Let us next mention some involvements of prime numbers in the particular
field of physics. We have already touched upon the connection of quantum
computation and number-theoretical problems. Aside from that, there is the
fascinating history of the Hilbert–Pólya conjecture, saying in essence that
the behavior of the Riemann zeta function on the critical line Re(s) =1/2
depends somehow on a mysterious (complex) Hermitian operator, of which
the critical zeros would be eigenvalues. Any results along these lines—even
partial results—would have direct implications about prime numbers, as we
saw in Chapter 1. The study of the distribution of eigenvalues of certain
matrices has been a strong focus of theoretical physicists for decades. In the
early 1970s, a chance conversation between F. Dyson, one of the foremost
researchers on the physics side of random matrix work, and H. Montgomery,
a number theorist investigating the influence of critical zeros of the zeta
function on primes, led them to realize that some aspects of the distribution
of eigenvalues of random matrices are very close to those of the critical zeros.
As a result, it is widely conjectured that the mysterious operator that would
give rise to the properties of ζ is of the Gaussian unitary ensemble √ (GUE)
class. A relevant n × n matrix G in such a theory has Gaa = xaa 2 and for
a>b, Gab = xab + iyab, together with the Hermitian condition Gab = G∗ ba ;
where every xab,yab is a Gaussian random variable with unit variance, mean
zero. The works of [Odlyzko 1987, 1992, 1994, 2005] show that the statistics
of consecutive critical zeros are in many ways equivalent—experimentally
speaking—to the theoretical distribution of eigenvalues of a large such matrix
G. In particular, let {zn : n =1, 2,...} be the collection of the (positive)
imaginary parts of the critical zeros of ζ, in increasing order. It is known from
the deeper theory of the ζ function that the quantity
δn = zn+1 − zn
2π
ln zn
2π