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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:
- There exists $A \ge 0$ such that, for every $y \ge 0$ , $\displaystyle \sum_{p \le y} f(p) \log p \le Ay$ .
- There exists $B \ge 0$ such that $\displaystyle \sum_p \sum_{k \ge 2} \frac{f(p^k)\log(p^k)}{p^k} \le B$ .
Then for all $x>1$ , $\displaystyle \sum_{n \le x} f(n) \le (A+B+1) \frac{x}{\log x} \sum_{n \le x} \frac{f(n)}{n}$ .
Proof.
Dividing the inequality $\displaystyle \log x \sum_{n \le x} f(n) \le (A+B+1)x\sum_{n \le x} \frac{f(n)}{n}$ by $\log x$ yields the desired result. 
The theorem has an obvious corollary:
Corollary If $f$ satisfies the conditions of the theorem, then for all $x>1$ , $\displaystyle \sum_{n \le x} f(n)=O\left(\frac{x}{\log x} \sum_{n \le x} \frac{f(n)}{n}\right)$ .
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