Decay of Mean Values of Multiplicative Functions

For given multiplicative function f, with |f(n)| <= 1 for all n, we are interested in how fast its mean value (1/x)(sigma)f(n) converges. Halasz showed that this depends on the minimum M (over y (element of) (R) of (sigma)( 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 Halasz-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.