Beyond the LSD method for the partial sums of multiplicative functions

The Landau-Selberg-Delange method gives an asymptotic formula for the partial sums of a multiplicative function f whose prime values are $$ on average. In the literature, the average is usually taken to be $$ with a very strong error term, leading to an asymptotic formula for the partial sums with a very strong error term. In practice, the average at the prime values may only be known with a fairly weak error term, and so we explore here how good an estimate this will imply for the partial sums of f, developing new techniques to do so.