On partial derivatives of some summatory functions

Let f be a real arithmetic function and let g:[1,∞[→ R be a smooth function. We describe two emblematic instances in which saddle-point estimates may be used to evaluate the frequency, on the set of integers n x, of the event (n) g(n)\ from those relevant to the event (n) y\. The first example revisits Dickman's historical contribution to the theory of friable integers. The second is concerned with the distribution of the squarefree kernel of an integer.