Decay of Mean Values of Multiplicative Functions

Abstract

AbstractFor given multiplicative function f , with |f(n)| ≤ 1 for all n, we are interested in how fast its mean value (1/x) Σn≤xf(n) converges. Halász showed that this depends on the minimum M (over y ∈ ℝ) of Σp≤x (1 – Re(f(p)p–iy )/p, and subsequent authors gave the upper bound ⪡ (1 + M)e–M. For many applications it is necessary to have explicit constants in this and various related bounds, and we provide these via our own variant of the Halász-Montgomery lemma (in fact the constant we give is best possible up to a factor of 10). We also develop a new type of hybrid bound in terms of the location of the absolute value of y that minimizes the sum above. As one application we give bounds for the least representatives of the cosets of the k-th powers mod p.