Study of weighted average or sum of a multiplicative function

Question

In the study of a multiplicative function $f$, sometimes the weighted average of the function $\sum_{1\leq n\leq x} (1-\frac{n}{x}) f(n)$ is studied instead of the sum $\sum_{1\leq n\leq x} f(n)$. Why do we want to study the weighted average instead of the sum in some cases? Is it because sometimes it's impossible to get an asymptotic formula for $\sum_{1\leq n\leq x} f(n)$?

Answer 1

One moral is that studying averages of terms $a(n)$ is easier than studying individual terms. Thus we might hope to understand $$ A(X) = \sum_{1 \leq n \leq X} a(n) $$ even if $a(n)$ is hard to figure out. And similarly, we might hope to understand $$ \widetilde{A}(X) = \sum_{1 \leq n \leq X} A(n) $$ even if $A(n)$ individually is hard to study. In this last expression, note that $a(1)$ is counted a total of $X$ times (once from $A(1), once from $A(2)$, etc.), $a(2)$ is counted $X - 1$ times (once from $A(2), once from $A(3)$, etc), and so on. In fact $$ \widetilde{A}(X) = \sum_{1 \leq n \leq X} (X - n + 1) a(n). $$ Up to a small relabeling of $X$ (and dividing by $X$), this is essentially the same as $$ \sum_{1 \leq n \leq X} (1 - n/X) a(n)$$ as in the OP.

This is one reason why this might work out better.

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More generally, there's no reason to stop there. Why not repeatedly average, a lot? This gives one motivation for the so-called Cesaro weights (or Riesz weights). Each additional summation gives a little bit of extra hope.

Taking many of these weighted summations and combining them together is the basic idea behind a general method in analytic number theory called "Landau's method". It's very powerful! For example, you can see this directly in the first displayed equation and description in section 2.2 of Uniform bounds in lattice point counting and partial sums of zeta functions (https://arxiv.org/abs/1710.02190).

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More generally still, a big idea when studying multiplicative functions is that properties of the functions are exposed in the analytic properties of the associated Dirichlet series. The basic summatory function $A(X)$ is retrievable from the Dirichlet series $\sum a(n) n^{-s}$ by Perron's formula, $$ A(X) = \frac{1}{2 \pi i} \int_{2 - i\infty}^{2 + i\infty} \sum_{n \geq 1} \frac{a(n)}{n^s} X^s \frac{ds}{s}. $$ But it's usually analytically easier to insert functions with nicer decay properties on the right. The Riesz means come from studying $$ \frac{1}{2\pi i} \int_{2 - i\infty}^{2 + i\infty} \sum_{n \geq 1} \frac{a(n)}{n^s} \frac{\Gamma(s)}{\Gamma(s + k + 1)} X^{s + k} ds. $$ (Both here and above I use $2$ to mean a number in the region of absolute convergence of the Dirichlet series, and don't consider that further). The point is that the gamma functions give more obvious decay, which might make studying the integrand easier.

We can instead insert any meromorphic weight function we want. Different choices give different perspectives, but the fundamental idea is that nicer analytic behavior can lead to problems that are easier to understand.