1.7. Linear equations in primes 125
in arithmetic progressions) to control such sums in the “major arc” case
when ξ is close to a rational of small height, while in the “minor arc” case
when ξ behaves irrationally, one can use the standard identity
(1.62) Λ(n) =
d|n
μ(d) log
n
d
,
where μ is the obius
function24,
to re-express such a sum in terms of
expressions roughly of the form
d,m
μ(d) log me(ξdm)
where we are intentionally vague as to what range the d, m parameters are
being summed over. The idea is then to eliminate the μ factor by tools
such as the triangle inequality or the Cauchy-Schwarz inequality, leading to
expressions such as
d
|
m
log me(ξdm)|;
the point is that the inner sum does not contain any number-theoretic factors
such as Λ or μ, but is still oscillatory (at least if ξ is sufficiently irrational),
and so one can extract useful cancellation from here. Actually, the situation
is more complicated than this, because there are regions of the range of (d, m)
for which this method provides insufficient cancellation, in which case one
has to rearrange the sum further using more arithmetic identities such as
(1.62) (for instance, using a truncated version of (1.62) known as Vaughan’s
identity). We will not discuss this further here, but any advanced analytic
number theory text (e.g. [IwKo2004]) will cover this material.
Unfortunately, while the Fourier-pseudorandomness of Λ is well under-
stood, the linear forms and correlation conditions are essentially equivalent
to (and in fact slightly harder than) the original problem of obtaining asymp-
totics for linear patterns in primes, and so using Λ for the pseudorandom
measure would result in a circular argument. Furthermore, correlations such
as
En∈[N]Λ(n)Λ(n + 2)
(which essentially counts the number of twin primes up to N) are notoriously
difficult to compute. For instance, if one tries to expand the above sum using
(1.62), one ends up with expressions such as
d,d ≤N
μ(d)μ(d )
n≤N:d|n,d |n+2
log
n
d
log
n + 2
d
.
24The obius function μ is defined by setting μ(n) := (−1)k when n is the product of k
distinct primes for some k 0, and μ(n) = 0 otherwise.
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