Prime Numbers

1.4 Analytic number theory 37
valid certainly for Re(s) > 1. It is interesting that the behavior of the Mertens
function runs sufficiently deep that the following equivalences are known (in
this and subsequent such uses of big-O notation, we mean that the implied
constant depends on ɛ only):
Theorem 1.4.3. The PNT is equivalent to the statement
M(x) =o(x),
while the Riemann hypothesis is equivalent to the statement
M(x) =O x 1
2 +ɛ
for any fixed ɛ>0.
What a compelling notion, that the Mertens function, which one might
envision as something like a random walk, with the Möbius µ contributing to
the summation for M in something like the style of a random coin flip, should
be so closely related to the great theorem (PNT) and the great conjecture
(RH) in this way. The equivalences in Theorem 1.4.3 can be augmented with
various alternative statements. One such is the elegant result that the PNT
is equivalent to the statement
∞
n=1
µ(n)
n =0,
as shown by von Mangoldt. Incidentally, it is not hard to show that the sum
in relation (1.24) converges absolutely for Re(s) > 1; it is the rigorous sum
evaluation at s = 1 that is difficult (see Exercise 1.19). In 1859, Riemann
conjectured that for each fixed ɛ>0,
π(x) = li (x)+O
x 1/2+ɛ
, (1.25)
which conjecture is equivalent to the Riemann hypothesis, and perforce to the
second statement of Theorem 1.4.3. In fact, the relation (1.25) is equivalent
to the assertion that ζ(s) has no zeros in the region Re(s) > 1/2 +ɛ. The
estimate (1.25) has not been proved for any ɛ0. Remarkably, we know rigorously that pn+1 − pn =
O p0.525
n , a result of R. Baker, G. Harman, and J. Pintz. But much more is