Multiplicative functions of polynomial values in short intervals

Abstract

Multiplicative functions of polynomial values in short intervalsby Mohan Nair (Glasgow)1. Introduction.Let d(n) denote the divisor function and let P (n) be an irreducible polynomial of degree g with integer coefficients.In 1952, Erdős [2] showed that there exist constants c 1 and c 2 , which may depend on P , such thatfor x ≥ 2. This result was generalized by Delmer [1] who showed that for any l ∈ N,with s = 2 l -1 and where the c i may depend, in addition, on l.The lower bound in Erdős's result is fairly straightforward but the upper bound is a combination of characteristically ingenious ideas.In 1971, Wolke [6] clarified these ideas and together with several important contributions of his own, showed that any suitable sum of the form n≤x f (a n ) can be similarly bounded.Here f is any non-negative multiplicative function with f (p l ) ≤ c 1 l c 2 , c 1 , c 2 constants, and {a n } is a sequence of natural numbers with a structure amenable to the sieve method.Applying his results to f (n) = d(n) and a n = |P (n)|, he recovered Erdős's result and, indeed, gave several other interesting applications.The very generality of Wolke's results meant that the bounds obtained lacked uniformity with respect to any particular class of sequences {a n }.In 1980, Shiu [5] obtained such a uniformity for the class of arithmetic progressions, i.e. for linear polynomials and refined the Erdős-Wolke method, in this particular case, to include a larger class of multiplicative functions as well as to obtain a short-interval result.He considered the class of non-negative multiplicative functions f which satisfy the weaker conditions f (p l ) ≤ A l 0 , f (n) ≤ A 1 (ε)n ε for any ε > 0 and constants A 0 , A 1 .

Locations

• Acta Arithmetica - View - PDF