Prime Numbers

1.4 Analytic number theory 45
bound on ζ(σ + it) could be given as a nontrivial power of t. For example, the
Riemann zeta function can be bounded on the critical line σ =1/2, as
ζ(1/2+it) =O(t 1/6 ),
when t ≥ 1; see [Graham and Kolesnik 1991]. The exponent has been
successively reduced over the years; for example, [Bombieri and Iwaniec 1986]
established the estimate O t9/56+ɛ and [Watt 1989] obtained O t89/560+ɛ .
The Lindelöf hypothesis is the conjecture that ζ(1/2 +it) =O(tɛ ) for any
ɛ>0. This conjecture also has consequences for the distribution of primes,
such as the following result in [Yu 1996]: If pn denotes the n-th prime, then
on the Lindelöf hypothesis,
pn≤x
(pn+1 − pn) 2 = x 1+o(1) .
The best that is known unconditionally is that the sum is O x23/18+ɛ for any
ɛ>0, a result of D. Heath-Brown. A consequence of Yu’s conditional theorem
is that for each ɛ>0, the number of integers n ≤ x such that the interval
(n, n+n ɛ ) contains a prime is ∼ x. Incidentally, there is a connection between
the Riemann hypothesis and the Lindelöf hypothesis: The former implies the
latter.
Though not easy, it is possible to get numerically explicit estimates via
exponential sums. A recent tour de force is the paper [Ford 2002], where it is
shown that
|ζ(σ + it)| ≤76.2t 4.45(1−σ)3/2
ln 2/3 t,
for 1/2 ≤ σ ≤ 1andt≥2. Such results can lead to numerically explicit zerofree
regions for the zeta function and numerically explicit bounds relevant to
various prime-number phenomena.
As for additive problems with primes, one may consider another important
class of exponential sums, defined by
En(t) =
e 2πitp , (1.35)
p≤n
where p runs through primes. Certain integrals involving En(t) over finite
domains turn out to be associated with deep properties of the prime numbers.
In fact, Vinogradov’s proof that every sufficiently large odd integer is the
sum of three primes starts essentially with the beautiful observation that the
number of three-prime representations of n is precisely
R3(n) =
1
=
0
n≥p,q,r ∈P
1
E
0
3 n(t)e −2πitn dt.
e 2πit(p+q+r−n) dt (1.36)