Beyond the LSD method for the partial sums of multiplicative functions

The Landau-Selberg-Delange (LSD) method gives an asymptotic formula for the partial sums of a multiplicative function $f$ whose prime values are $α$ on average. In the literature, the average is usually taken to be $α$ 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.

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