A Turan--Kubilius type inequality on shifted products

International audienceIn 1934 Turan proved that if f(n) is an additive arithmetic function satisfying certain conditions, then for almost all m <= n the value of f(m) is "near" the expectation Sigma(p <= n) f(p)/p. Later Kubilius sharpened this result by proving that the conditions in Turan's theorem can be relaxed, and still the same conclusion holds. In an earlier paper we studied whether this result has a sum set analogue, i.e., if f (n) is an additive arithmetic function and A, B are "large" subsets of {1,2,...,n}, then for almost all a is an element of A, b is an element of B, the value of f (a + b) is "near" the expectation? We proved such a result under an assumption which is slightly milder than Turan's condition, but is not needed in Kubilius estimate. In this paper we prove the multiplicative analogue of this theorem by proving a similar result with ab + 1 in place of a + b