Prime Numbers

46 Chapter 1 PRIMES!
Vinogradov’s proof was an extension of the earlier work of Hardy and
Littlewood (see the monumental collection [Hardy 1966]), whose “circle
method” was a tour de force of analytic number theory, essentially connecting
exponential sums with general problems of additive number theory such as,
but not limited to, the Goldbach problem.
Let us take a moment to give an overview of Vinogradov’s method for
estimating the integral (1.36). The guiding observation is that there is a
strong correspondence between the distribution of primes and the spectral
information embodied in En(t). Assume that we have a general estimate
on primes not exceeding n and belonging to an arithmetic progression
{a, a + d, a +2d,...} with gcd(a, d) = 1, in the form
π(n; d, a) = 1
π(n)+ɛ(n; d, a),
ϕ(d)
which estimate, we assume, will be “good” in the sense that the error
term ɛ will be suitably small for the problem at hand. (We have given a
possible estimate in the form of the ERH relation (1.32) and the weaker,
but unconditional Theorem 1.4.6.) Then for rational t = a/q we develop an
estimate for the sum (1.35) as
En(a/q) =
q−1
f=0
=
gcd(f,q)=1
=
gcd(f,q)=1
p≡f (mod q), p≤n
e 2πipa/q
π(n; q, f)e 2πifa/q +
p|q, p≤n
π(n; q, f)e 2πifa/q + O(q),
e 2πipa/q
where it is understood that the sums involving gcd run over the elements
f ∈ [1,q− 1] that are coprime with q. It turns out that such estimates are of
greatest value when the denominator q is relatively small. In such cases one
may use the chosen estimate on primes in a residue class to arrive at
En(a/q) = cq(a)
π(n)+O(q + |ɛ|ϕ(q)),
ϕ(q)
where |ɛ| denotes the maximum of |ɛ(n; q, f)| taken over all residues f coprime
to q, andcqis the well-studied Ramanujan sum
cq(a) =
e 2πifa/q . (1.37)
gcd(f,q)=1
We shall encounter this Ramanujan sum later, during our tour of discrete
convolution methods, as in equation (9.26). For the moment, we observe that
[Hardy and Wright 1979]
cq(a) = µ(q/g)ϕ(q)
, g =gcd(a, q). (1.38)
ϕ(q/g)