Prime Numbers

428 Chapter 8 THE UBIQUITY OF PRIME NUMBERS
say at n = N, then one expects 2N complex zeros of the resulting, degree-
2N polynomial in x. But in practice, only some of these 2N zeros are real
(i.e., such that 1
2 + ix is on the Riemann critical line). For large N, and
again experimentally, the rest of the polynomial’s zeros are “expelled” a good
distance away from the critical line. The Riemann hypothesis, if it is to be
cast in language appropriate to the Hermite expansion, must somehow address
this expulsion of nonreal polynomial zeros away from the real axis. Thus
the Riemann hypothesis can be cast in terms of quantum dynamics in some
fashion, and it is not out of the question that this kind of interdisciplinary
approach could be fruitful.
An anecdote cannot be resisted here; this one concerns the field of
engineering. Peculiar as it may seem today, the scientist and engineer van
der Pol did, in the 1940s, exhibit tremendous courage in his “analog”
manifestation of an interesting Fourier decomposition. An integral used by
van der Pol was a special case (σ =1/2) of the following relation, valid for
s = σ + it, σ ∈ (0, 1) [Borwein et al. 2000]:
∞
ζ(s) =s e
−∞
−σω (⌊e ω ⌋−e ω ) e −iωt dω.
Van der Pol actually built and tested an electronic circuit to carry out the
requisite transform in analog fashion for σ =1/2, [van der Pol 1947]. In today’s
primarily digital world it yet remains an open question whether the van der
Pol approach can be effectively used with, say, a fast Fourier transform to
approximate this interesting integral. In an even more speculative tone, one
notes that in principle, at least, there could exist an analog device—say an
extremely sophisticated circuit—that sensed the prime numbers, or something
about such numbers, in this fashion.
At this juncture of our brief interdisciplinary overview, a word of caution
is in order. One should not be led into a false presumption that theoretical
physicists always endeavor to legitimize the prevailing conjectural models of
the prime numbers or of the Riemann ζ function. For example, in the study
[Shlesinger 1986], it is argued that if the critical behavior of ζ corresponds to
a certain “fractal random walk” (technically, if the critical zeros determine a
Levy flight in a precise, stochastic sense), then fundamental laws of probability
are violated unless the Riemann hypothesis is false.
In recent years there has been a flurry of interdisciplinary activity—
largely computational—relating the structure of the primes to the world
of fractals. For example, in [Ares and Castro 2004] an attempt is made to
explain hidden structure of the primes in terms of spin-physics systems and
the Sierpiński gasket fractal; see also Exercise 8.26. A fascinating approach
to a new characterization of the primes is that of [van Zyl and Hutchinson
2003], who work out a quantum potential whose eigenvalues (energy levels)
are the prime numbers. Then they find that the fractal dimension of said
potential is about 1.8, which indicates surprising irregularity. We stress that
such developments certainly sound theoretical on the face of it, and some of