Decay of mean values of multiplicative functions

For given multiplicative function f, with \f(n)\ less than or equal to 1 for all n, we are interested in how fast its mean value (1/x) Sigma(nless than or equal tox), f(n) converges. Halasz showed that this depends on the minimum M (over y is an element of R) of Sigma(pless than or equal tox)(1 - Re(f(p)p(-iy)))/p, and subsequent authors gave the upper bound much less than (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.