Prime Numbers

1.4 Analytic number theory 47
In particular, when a, q are coprime, we obtain a beautiful estimate of the
form
En(a/q) =
e 2πipa/q = µ(q)
ϕ(q) π(n)+ɛ′ , (1.39)
p≤n
where the overall error ɛ ′ depends in complicated ways on a, q, n, and, of
course, whatever is our theorem of choice on the distribution of primes in
a residue class. We uncover thus a fundamental spectral property of primes:
When q is small, the magnitude of the exponential sum is effectively reduced,
by an explicit factor µ/ϕ, below the trivial estimate π(n). Such reduction
is due, of course, to cancellation among the oscillating summands; relation
(1.39) quantifies this behavior.
Vinogradov was able to exploit the small-q estimate above in the following
way. One chooses a cutoff Q = ln B n for appropriately large B, thinking
of q as “small” when 1 ≤ q ≤ Q. (It turns out to be enough to consider
only the range Q < q < n/Q for “large” q.) Now, the integrand in (1.36)
exhibits “resonances” when the integration variable t lies near to a rational
a/q for the small q ∈ [1,Q]. These regions of t are traditionally called
the “major arcs.” The rest of the integral—over the “minor arcs” having
t ≈ a/q with q ∈ (Q, n/Q)—can be thought of as “noise” that needs to
be controlled (bounded). After some delicate manipulations, one achieves an
integral estimate in the form
R3(n) = n2
2ln 3 n
Q
q=1
µ(q)cq(n)
ϕ 3 (q) + ɛ′′ , (1.40)
where we see a resonance sum from the major arcs, while ɛ ′′ now contains
all previous arithmetic-progression errors plus the minor-arc noise. Already in
the above summation over q ∈ [1,Q] one can, with some additional algebraic
effort, see how the final ternary-Goldbach estimate (1.12) results, as long as
the error ɛ ′′ and the finitude of the cutoff Q and are not too troublesome (see
Exercise 1.68).
It was the crowning achievement of Vinogradov to find an upper bound on
the minor-arc component of the overall error ɛ ′′ . The relevant theorem is this:
If gcd(a, q) =1,q ≤ n, andarealtis near a/q in the sense |t − a/q| ≤1/q2 ,
then
n
|En(t)|