Approximating by a local model

Abstract

It is high time we show in a somewhat general setting how to approximate a given weighted sequence by a local model. Let us start with such a sequence (f(n)) _n together with an additional function ψ∞ (which will take care of the size constraints), for which we assume the following bound:

$$\sum\limits_{q \leqslant D} {\mathop {m{\text{ax}}}\limits_{a\,\bmod q} \left| {\sum\limits_{n \equiv a\left[ q \right]} {f\left( n \right){\psi _\infty }\left( n \right) - {f_q}\left( a \right)X/q} } \right| \leqslant E}$$

((17.1))

for some parameters D, E, X and (f _q ) _q . The Bombieri-Vinogradov Theorem falls within this framework with ψ∞ being the characteristic function of real numbers ≤ N and E = N/(Log N)^A, together with D = √N/(Log N)^B for some B = B(A); then f(n) = Λ(n) and f _q = q𝟙u _q /ϕ(q), and finally X = N. Note that the function f _q that appears is precisely the one we used as a local model for the primes. The parameter X is here for homogeneity and could be dispensed with, simply by incorporating it in f _q . However, in usual applications, X will be here to treat the dependence on the size, i.e. the contribution of the infinite place, while f _q will be independent of it and only accounts for the effect of the finite places. We shall need some properties of these f _q ’s, namely:

$$\left. {\forall d} \right|q,\forall a\bmod d,\quad J_{\tilde d}^{\tilde q}{f_q} = {f_d}.$$

((17.2))

This equation may look unpalatable, but here is an equivalent formulation:

$$\forall d\left| {q,\quad {f_d}\left( a \right)/d = \sum\limits_{\mathop {b\;\bmod q}\limits_{b \equiv a\left[ q \right]} } {{f_q}\left( b \right)/q} } \right.$$

((17.3))

where it is maybe easier to consider f _q /q as one function (the density, as in (13.1) and (13.2)).