128 1. Higher order Fourier analysis
As discussed in the previous section, to get asymptotics for patterns in
the primes we also need to control exponential sums such as
p≤N
e(ξp)
and more generally (for higher complexity patterns)
p≤N
F (g(p)Γ)
for various nilsequences n F (g(n)Γ). Again, it is convenient to use the von
Mangoldt function Λ as a proxy for the primes, thus leading to expressions
such as
n≤N
Λ(n)F (g(n)Γ).
Actually, for technical reasons it is convenient to use identities such as (1.62)
to replace this type of expression with expressions such as
n≤N
μ(n)F (g(n)Γ),
because the obius function μ enjoys better boundedness and equidistri-
bution properties than Λ. (For instance, Λ strongly favours odd numbers
over even numbers, whereas the obius function has no preference.) It
turns out that these expressions can be controlled by a generalisation of the
method of Vinogradov used to compute exponential sums over primes, using
the equidistribution theory of nilsequences as a substitute for the classical
theory of exponential sums over integers. See [GrTa2008c] for details.
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