Prime Numbers

66 Chapter 1 PRIMES!
implies that ζ(s) has no zeros in the half-plane Re(s) >α. This shows the
connection between the essential error in the PNT estimate and the zeros of ζ.
For the other (harder) direction, assume that ζ has no zeros in the halfplane
Re(s) >α. Looking at relation (1.23), prove that
Im(ρ) ≤ T
x ρ
|ρ| = O(xα ln 2 T ),
which proof is nontrivial and interesting in its own right [Davenport 1980].
Finally, conclude that
ψ(x) =x + O x α+ɛ
for any ɛ>0. These arguments reveal why the Riemann conjecture
π(x) = li (x)+O(x 1/2 ln x)
is sometimes thought of as “the PNT form of the Riemann hypothesis.”
1.61. Here we show how to evaluate the Riemann zeta function on the
critical line, the exercise being to implement the formula and test against
some high-precision values given below. We describe here, as compactly as we
can, the celebrated Riemann–Siegel formula. This formula looms unwieldy on
the face of it, but when one realizes the formula’s power, the complications
seem a small price to pay! In fact, the formula is precisely what has been used
to verify that the first 1.5 billion zeros (of positive imaginary part) lie exactly
on the critical line (and parallel variants have been used to push well beyond
this; see the text and Exercise 1.62).
A first step is to define the Hardy function
where the assignment
ϑ(t) = Im
Z(t) =e iϑ(t) ζ(1/2+it),
1 it
ln Γ +
4 2
− 1
t ln π
2
renders Z a real-valued function on the critical line (i.e., for t real). Moreover,
the sign changes in Z correspond to the zeros of ζ. ThusifZ(a),Z(b) have
opposite sign for reals a