Short sums of certain arithmetic functions

Abstract

uniformly for 1 6 q 6 y. This result has turned out to be very useful in a wide range of applications. A closer inspection of its proof reveals, in the case q = 1, that : (a) g needs only be sub-multiplicative, i.e. g(mn) 6 g(m)g(n) for (m,n) = 1 with g(1) = 1 ; (b) the constant implicit in the ⌧ sign depends only on A, B and ↵ ; (c) given ↵, condition (ii) above need only hold for a particular = (↵). Shiu’s result has been generalised by Nair [5] to sub-multiplicative functions of polynomial values in a short interval. In this paper, we weaken the property of sub-multiplicativity significantly to appreciably widen the range of application of such a result. Consider, for any fixed k 2 N, the class Mk(A,B, ) of non-negative arithmetic functions F (n1, . . . , nk) such that

Locations

• Acta Mathematica - View - PDF