Prime Numbers

8.6 Curious, anecdotal, and interdisciplinary references to primes 429
the research is indeed abstract, but it is modern computation that appears to
drive such interdisciplinary work.
Also, one should not think that the appearance of primes in physics is
relegated to studies of the Riemann ζ function. Indeed, [Vladimirov et al.
1994] authored an entire volume on the subject of p-adic field expansions in
theoretical physics. They say:
Elaboration of the formalism of mathematical physics over a p-adic number
field is an interesting enterprise apart from possible applications, as it
promotes deeper understanding of the formalism of standard mathematical
physics. One can think there is the following principle. Fundamental
physical laws should admit of formulation invariant under a choice of a
number field.
(The italics are theirs.) This quotation echoes the cooperative theme
of the present section. Within this interesting reference one can find
further references to p-adic quantum gravity and p-adic Einstein-relativistic
equations.
Physicists have from time to time even performed “prime number
experiments.” For example, [Wolf 1997] takes a signal, call it x =
(x0,x1,...,xN−1), where a component xj is the count of primes over some
interval. Specifically,
xj = π((j +1)M) − π(jM),
where M is some fixed interval length. Then is considered the DFT
Xk =
N−1
j=0
of which the zeroth Fourier component is
xje −2πijk/N ,
X0 = π(MN).
The interesting thing is that this particular signal exhibits the spectrum (the
behavior in the index k) of“1/f” noise—actually, we could call it “pink”
noise. Specifically, Wolf claims that
|Xk| 2 ∼ 1
k α
(8.6)
with exponent α ∼ 1.64 ... . This means that in the frequency domain (i.e.,
behavior in Fourier index k) the power law involves, evidently, a fractional
power. Wolf suggests that perhaps this means that the prime numbers are
in a “self-organized critical state,” pointing out that all possible (even) gaps
between primes conjecturally occur so that there is no natural “length” scale.
Such properties are also inherent in well-known complex systems that are
also known to exhibit 1/k α noise. Though the power law may be imperfect