1.7. Linear equations in primes 127

is applied) by a moderately lengthy, but elementary and straightforward cal-

culation, based ultimately on the Chinese remainder theorem, an analysis

of the local problem (working mod q for small q), and the fundamental fact

that the Riemann zeta function ζ(s) is approximately equal to 1/(s − 1) for

s close to 1. See for instance [Ta2004] for more discussion.

If one uses (1.63), then we see that ν(n) is equal to log R when n is any

prime larger than R; if log R is comparable to log N, we thus see (from the

prime number theorem) that the primes in [N] do indeed have positive den-

sity relative to ν. This is then enough to be able to invoke the transference

principle and extend results such as Szemer´ edi’s theorem to the primes, es-

tablishing, in particular, that the primes contain arbitrarily long arithmetic

progressions; see [GrTa2008b] for details.

To use the Fourier-analytic approach, it turns out to be convenient to

replace the above measures ν by a slight variant which looks more compli-

cated in the spatial domain, but is easier to manipulate in the frequency

domain. More specifically, the expression (1.63) or (1.64) is replaced with a

variant such as

ν := log R

⎛

⎝

d|n;d≤R

μ(d)

d

φ(d)

q≤R/d;(q,d)=1

1

φ(q)

⎞2

⎠

where φ(d) is the Euler totient function (the number of integers from 1 to d

that are coprime to d). Some standard multiplicative number theory shows

that the weights

d

φ(d)

∑

q≤R/d;(q,d)=1

1

φ(q)

are approximately equal to log

R

d

in some sense. With such a representation, it turns out that the Fourier

coeﬃcients of ν can be computed more or less explicitly, and is essentially

supported on those frequencies of the form a/q with q ≤

R2.

This makes

it easy to verify the required Fourier-pseudorandomness hypothesis (1.55)

(once one applies the W -trick). As for the restriction estimate (1.54), the

first step is to use Exercise (1.7.3) and the Cauchy-Schwarz inequality to

reduce matters to showing an estimate of the shape

En|

ξ

g(ξ)e(ξnx/N)|2ν(n)

g

q

.

The right-hand side can be rearranged to be of the shape

ξ,ξ

g(ξ)g(ξ )ˆ(ξ ν − ξ ).

It is then possible to use the good pointwise control on the Fourier transform

ˆ ν of ν (in particular, the fact that it “decays” quite rapidly away from the

major arcs) to get a good restriction estimate. See [GrTa2006] for further

discussion.