Allow me to give context to the question, which appears in the box at the bottom.
A very general hope from the theory of Dirichlet series is to try to extract information about the coefficients of a Dirichlet series like $$ D(s) = \sum_{n \geq 1} \frac{a(n)}{n^s}.$$ Frequently, we're interested in partial sums like $$ A_0(X) := \sum_{n \leq X} a(n),$$ which we might get at by performing an inverse Mellin transform like $$ A_0(X) = \frac{1}{2\pi i} \int_{(\sigma)} D(s) X^s \frac{ds}{s}$$ for $\sigma$ in the realm of absolute convergence of the Dirichlet series. One might proceed by shifting the line of integration left, picking up residues, etc. But in practice this can be hard, since convergence issues can rear their angry heads.
Sometimes, we can instead get information about $$A_d(X) := \frac{1}{d! X^d} \sum_{n \leq X} a(n) (X - n)^d$$ through the transform $$ A_d(X) = \frac{1}{2\pi i} \int_{(\sigma)} D(s) X^s \frac{ds}{s(s+1)\cdots(s+d)},$$ which might converge better.
My question is very classical:
- What can we say about the nonweighted partial sums $A_0$ from knowledge of the behaviour of the weighted partial sums $A_d$, perhaps for all $d > d_0$ for some $d_0 > 0$?
- Can we say more if $a(n) \geq 0$ for all $n$?
This sounds like the sort of thing that would have been understood really well by classical analytic number theorists, or mathematicians like Cauchy, Hardy, or Littlewood. But I haven't dealt too much with these weighted partial sums yet, nor do I know of places where they've been applied classically and successfully before.