By L. Badescu, D. Popescu

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7]. 1 for the solution). Yet, in the cases which naturally occur in sieve theory (classical and otherwise), there is an extensive literature available,5 and we can select for our applications very strong results of various types. Even if we select a sieve support L other than the traditional one of squarefree integers L, as we will do at some point (and as Zywina also did) with L = m | m is squarefree and g(m) L where g is some other multiplicative function ‘close to m’on average, bounds for f (m) g(m) M are also known (we will use a very general result of Lau and Wu [88]).

D. in the case of integers) of m and n, and write m = m d = m ∪ d, n = n d = n ∪ d (disjoint unions). According to the multiplicative deﬁnition of Bm and Bn , we can write ϕ = ϕm ⊗ ϕd , ϕ = ϕn ⊗ ϕ d for some unique basis elements ϕm ∈ Bm , ϕd , ϕd ∈ Bd and ϕn ∈ Bn . m’ of m and n. 9) (which, usually, is not a basis element in B[m,n] ). 11 Let m, n, ϕ and ϕ be as before. We have [ϕ, ϕ ](ρ[m,n] (y)) = ϕ(ρm (y))ϕ (ρn (y)) for all y ∈ Y , hence W (ϕ, ϕ ) = [ϕ, ϕ ](ρ[m,n] (Fx ))dμ(x). X Now we can hope to split the integral according to the value of y = ρ[m,n] (Fx ) in Y[m,n] , and evaluate it by ﬁrst summing the main term in an equidistribution statement.

Note that this means that for any conjugacy-invariant subset ⊂ G , union of a set of conjugacy classes such that ⊂ d −1 (p(α)) = Y , we have ν( )= | | . |Gg | We turn to the question of ﬁnding a suitable orthonormal basis of L2 (Y , ν ). This is provided by the following general lemma, which applies equally to H = G and to H = Gm for m ∈ S( ). 2 Let H be a ﬁnite group, H g a normal subgroup with abelian quotient = H /H g . Let α ∈ and let Y be the set of conjugacy classes of H with image α in .