Prime Numbers

36 Chapter 1 PRIMES!
This theorem is proved in [Hardy and Wright 1979]. The theorem is also a
corollary of the prime number Theorem 1.1.4, but it is simpler than the PNT
and predates it. The PNT still has something to offer, though; it gives smaller
error terms in (1.20) and (1.21). Incidentally, the computation of the Mertens
constant B is an interesting challenge (Exercise 1.90).
We have seen that certain facts about the primes can be thought of as
facts about the Riemann zeta function. As one penetrates more deeply into
the “critical strip,” that is, into the region 0 < Re(s) < 1, one essentially gains
more and more information about the detailed fluctuations in the distribution
of primes. In fact it is possible to write down an explicit expression for π(x)
that depends on the zeros of ζ(s) in the critical strip. We illustrate this for
a function that is related to π(x), but is more natural in the analytic theory.
Consider the function ψ0(x). This is the function ψ(x) defined as
ψ(x) =
p m ≤x
ρ
ln p =
ln x
ln p , (1.22)
ln p
except if x = pm , in which case ψ0(x) =ψ(x) − 1
2 ln p. Then (see [Edwards
1974], [Davenport 1980], [Ivić 1985]) for x>1,
ψ0(x) =x − xρ 1
− ln(2π) −
ρ 2 ln 1 − x −2 , (1.23)
where the sum is over the zeros ρ of ζ(s) withRe(ρ) > 0. This sum is not
absolutely convergent, and since the zeros ρ extend infinitely in both (vertical)
directions in the critical strip, we understand the sum to be the limit as T →∞
of the finite sum over those zeros ρ with |ρ|