Prime Numbers

3.7 Counting primes 159
t =Im(s) ranging, we have
1
2πi
C
x
s ds
= θ(x − n). (3.26)
n s
It follows immediately from these observations that for a given contour (but
now with σ>1soastoavoidanylnζsingularity) we have:
π ∗ (x) = 1
x
2πi
s ln ζ(s) ds
. (3.27)
s
This last formula provides analytic means for evaluation of π(x), because if x
is not a prime power, say, we have from relation (3.25) the identity:
C
π ∗ (x) =π(x)+ 1
2 π
x 1/2
+ 1
3 π
x 1/3
+ ···,
whichseriesterminatesassoonasthetermπ x 1/n /n has 2 n >x.
It is evident that π(x) may be, in principle at least, computed from
a contour integral (3.27), and relatively easy side calculations of π x 1/n
starting with π ( √ x). One could also simply apply the contour integral relation
recursively, since the leading term of π ∗ (x) − π(x) isπ ∗ x 1/2 /2, and so on.
There is another alternative for extracting π if we can compute π ∗ , namely
by way of an inversion formula (again for x not a prime power)
π(x) =
∞
n=1
µ(n)
π∗ x
n 1/n
.
This analytic approach thus comes down to numerical integration, yet
such integration is the problematic stage. First of all, one has to evaluate ζ
with sufficient accuracy. Second, one needs a rigorous bound on the extent to
which the integral is to be taken along the contour. Let us address the latter
problem first. Say we have in hand a sharp computational scheme for ζ itself,
and we take x = 100,σ =3/2. Numerical integration reveals that for sample
integration limits T ∈{10, 30, 50, 70, 90}, respective values are
π ∗ (100) ≈ Re 1003/2
T
π 0
100it ln ζ(3/2+it) dt
3/2+it
≈ 30.14, 29.72, 27.89, 29.13, 28.3,
which values exhibit poor convergence of the contour integral: The true value
of π ∗ (100) can be computed directly, by hand, to be 428/15 ≈ 28.533 ... .
Furthermore, on inspection the value as a function of integration limit T is
rather chaotic in the way it hovers around the true value, and rigorous error
bounds are, as might be expected, nontrivial to achieve (see Exercise 3.37).
The suggestions in [Lagarias and Odlyzko 1987] address, and in principle
repair, the above drawbacks of the analytic approach. As for evaluation of ζ
itself, the Riemann–Siegel formula is often recommended for maximum speed;