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 M¨ obius function μ enjoys better boundedness and equidistri-

bution properties than Λ. (For instance, Λ strongly favours odd numbers

over even numbers, whereas the M¨ 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.