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.
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