124 1. Higher order Fourier analysis
modified set PW,b := {n N : Wn + b P}, where W :=
pw
p is the
product of all the primes less than w, and 1 b W is a residue class
coprime to W . In practice, w (and W ) are slowly growing functions of N,
e.g., one could take w = log log log N. By the pigeonhole principle, for any
given N and W there will exist a b for which PW,b is large (of cardinality
N
φ(W ) log N
, where φ(W ) is the number of residue classes coprime to W );
indeed, thanks to the prime number theorem in arithmetic progressions, any
such b would work (e.g. one can take b = 1). Note that every arithmetic
progression in PW,b is associated to a corresponding arithmetic progression
in P. Thus, for the task of locating arithmetic progressions at least, we
may as well work with PW,b; a similar claim also holds for more complicated
tasks, such as counting the number of linear patterns in P, though one now
has to work with several residue classes at once. The point of passing from
P to PW,b is that the latter set no longer has any particular bias to favour or
disfavour any residue class with modulus less than w; there are still biases
at higher moduli, but as long as w goes to infinity with N, the effect of such
biases will end up being negligible (ultimately contributing o(1) terms to
things like the linear forms condition).
To simplify the exposition a bit, though, let us ignore the W -trick and
pretend that we are working with the primes themselves rather than the
affine-shifted primes. We will also ignore the technical distinctions between
the interval [N] and the cyclic group Z/NZ.
The most natural candidate for the measure ν is the von Mangoldt func-
tion Λ: N
R+,
defined by setting Λ(n) := log p when n =
pj
is a prime
p or a power of a prime, and Λ(n) = 0 otherwise. One hint as to the
significance of this function is provided by the identity
log n =
d|n
Λ(d)
for all natural numbers n, which can be viewed as a generating function of
the fundamental theorem of arithmetic.
The prime number theorem tells us that Λ is indeed a measure:
En∈[N]Λ(n) = 1 + o(1),
and the primes have full density with respect to this function:
En∈[N]1P(n)Λ(n) = 1 + o(1).
Furthermore, the von Mangoldt function has good Fourier pseudorandom-
ness properties (after applying the W -trick), thanks to the classical tech-
niques of Hardy-Littlewood and Vinogradov. Indeed, to control exponential
sums such as En∈[N]Λ(n)e(ξn) for some ξ R, one can use tools such as the
Siegel-Walfisz theorem (a quantitative version of the prime number theorem
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