Prime Numbers

1.4 Analytic number theory 33
particular can be calculated using various enhancements, such as arithmetic
progression-based products and polynomial evaluation, as discussed in
Chapter 8.8. For example, it is known that for p =2 40 +5,
(p − 1)! ≡−1 − 533091778023p (mod p 2 ),
as obtained by polynomial evaluation of the relevant factorial [Crandall et al.
1997]. This p is therefore not a Wilson prime, yet it is of interest that in this
day and age, machines can validate at least 12-digit primes via application of
Lagrange’s converse of the classical Wilson theorem.
In searches for these rare primes, some “close calls” have been encountered.
Perhaps the only importance of a close call is to verify heuristic beliefs about
the statistics of such as the Fermat and Wilson quotients. Examples of the
near misses with their very small (but alas nonzero) quotients are
p = 76843523891, qp(2) ≡−2(modp),
p = 12456646902457, qp(2) ≡ 4(modp),
p = 56151923, wp ≡−1(modp),
p = 93559087, wp ≡−3(modp),
and we remind ourselves that the vanishing of any Fermat or Wilson quotient
modulo p would have signaled a successful “strike.”
1.4 Analytic number theory
Analytic number theory refers to the marriage of continuum analysis with the
theory of the (patently discrete) integers. In this field, one can use integrals,
complex domains, and other tools of analysis to glean truths about the natural
numbers. We speak of a beautiful and powerful subject that is both useful in
the study of algorithms, and itself a source of many interesting algorithmic
problems. In what follows we tour a few highlights of the analytic theory.
1.4.1 The Riemann zeta function
It was the brilliant leap of Riemann in the mid-19th century to ponder an
entity so artfully employed by Euler,
ζ(s) =
∞
n=1
1
, (1.17)
ns but to ponder with powerful generality, namely, to allow s to attain complex
values. The sum converges absolutely for Re(s) > 1, and has an analytic
continuation over the entire complex plane, regular except at the single point
s = 1, where it has a simple pole with residue 1. (That is, (s−1)ζ(s) is analytic
in the entire complex plane, and its value at s = 1 is 1.) It is fairly easy to
see how ζ(s) can be continued to the half-plane Re(s) > 0: For Re(s) > 1we