On the Modified Selberg Integral

الأصل · EN

We give a kind of approximate majorant principle result for the modified Selberg integral, say (N,h), of essentially bounded f: → (i.e., bounded by arbitrary small powers); i.e., we get an upper bound, in terms of the modified Selberg integral of a related function F (with |f μ|≪ F μ, in the supports intersection), getting a square-root cancellation for the error-terms. Here (N,h) is the mean-square (in N<x≤ 2N) of the averaged short sum of, say, f:=g, minus its expected value; i.e., 1 h∑ₘ≤ ₕ∑₀≤ |ₙ₋ₓ|<ₘf(n)-Mf(x,h), with expected value Mf(x,h) (say, ≈ h∑d≤ ₓg(d)/d); so, this mean-square weights, on average, the f-values in (almost all, i.e. all, but o(N) possible exceptions) the short intervals [x-h,x+h], with mild restrictions on h (say, h→ ∞ and h=o(N), when N→ ∞).