Number Theory: An Introduction to Mathematics

4 The Riemann Hypothesis 383

and to

ψ(x)=x+O(x^1 /^2 log^2 x).

Since it is still not known ifα∗<1, the error terms here are substantially smaller than
any that have actually been established.
It has been shown by Cram ́er (1922) that

(logx)−^1

∫x

2

(ψ(t)/t− 1 )^2 dt

has a finite limit asx→∞if the Riemann hypothesis holds, and is unbounded if it
does not. Similarly, for eachα<1,

x−^2 (^1 −α)

∫x

2

(ψ(t)−t)^2 t−^2 αdt

is bounded but does not have a finite limit asx→∞if the Riemann hypothesis holds,
and is unbounded otherwise.
For all values ofxlisted in Table 1 we haveπ(x)<Li(x), and at one time it
was conjectured that this inequality holds for allx>0. However, Littlewood(1914)
disproved the conjecture by showing that there exists a constantc>0suchthat

π(xn)−Li(xn)>cx

1 / 2

n log log logxn/logxn

for some sequencexn→∞and

π(ξn)−Li(ξn)<−cξn^1 /^2 log log logξn/logξn

for some sequenceξn→∞. This is a quite remarkable result, since no actual value
ofxis known for whichπ(x)>Li(x). However, it is known thatπ(x)>Li(x)for
somexbetween 1. 398201 × 10316 and 1. 398244 × 10316.
In this connection it may be noted that Rosser and Schoenfeld (1962) have shown
thatπ(x)>x/logxfor allx≥17. It had previously been shown by Rosser (1939)
thatpn>nlognfor alln≥1.
Not content with not being able to prove the Riemann hypothesis, Montgomery
(1973) has assumed it and made a further conjecture. For givenβ>0, letNT(β)be
the number of zeros 1/ 2 +iγ, 1 / 2 +iγ′ofζ(s)with 0<γ′<γ≤Tsuch that

γ−γ′≤ 2 πβ/logT.

Montgomery’s conjecture is that, for each fixedβ>0,

NT(β)∼(T/ 2 π)logT

∫β

0

{ 1 −(sinπu/πu)^2 }du asT→∞.

Goldston (1988) has shown that this is equivalent to

∫Tβ

1

{ψ(x+x/T)−ψ(x)−x/T}^2 x−^2 dx∼(β− 1 / 2 )log^2 T/T asT→∞,

for each fixedβ≥1, whereψ(x)is Chebyshev’s function.