asymptotic estimates for real-valued nonnegative multiplicative functions

Note that, within this entry, $p$ always refers to a prime, $k$, $m$, and $n$ always refer to positive integers, and $\log$ always refers to the natural logarithm.

Theorem.

Let $f$ be a real-valued nonnegative multiplicative function such that the two following conditions are satisfied:

1. There exists $A\ge 0$ such that, for every $y\ge 0$, ${\displaystyle \sum _{p\le y}}f(p)\log p\le Ay$.

2. There exists $B\ge 0$ such that ${\displaystyle \sum _p}{\displaystyle \sum _{k\ge 2}}{\displaystyle \frac{f(p^k)\log (p^k)}{p^k}}\le B$.

Then for all $x\mathrm{>}\mathrm{1}$, ${\displaystyle \mathrm{\sum }_{n\mathrm{\le }x}}f\mathit{}\mathrm{(}n\mathrm{)}\mathrm{\le }\mathrm{(}A\mathrm{+}B\mathrm{+}\mathrm{1}\mathrm{)}\mathit{}{\displaystyle \frac{x}{\log \mathit{}x}}\mathit{}{\displaystyle \mathrm{\sum }_{n\mathrm{\le }x}}{\displaystyle \frac{f\mathit{}\mathrm{(}n\mathrm{)}}{n}}$.

Proof.

Dividing the inequality $\log x{\displaystyle \sum _{n\le x}}f(n)\le (A+B+1)x{\displaystyle \sum _{n\le x}}{\displaystyle \frac{f(n)}{n}}$ by $\log x$ yields the desired result.∎

The theorem has an obvious corollary:

Corollary.

If $f$ the conditions of the theorem, then for all $x\mathrm{>}\mathrm{1}$, ${\displaystyle \mathrm{\sum }_{n\mathrm{\le }x}}f\mathit{}\mathrm{(}n\mathrm{)}\mathrm{=}O\mathit{}\mathrm{\left(}{\displaystyle \frac{x}{\log \mathit{}x}}\mathit{}{\displaystyle \mathrm{\sum }_{n\mathrm{\le }x}}{\displaystyle \frac{f\mathit{}\mathrm{(}n\mathrm{)}}{n}}\mathrm{\right)}$.