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
and more generally (for higher complexity patterns)
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
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
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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