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Wirsing condition
Note that, within this entry, $p$ always refers to a prime, $k$ always refers to a positive integer, and $\log$ always refers to the natural logarithm.
Let $f$ be a real-valued nonnegative multiplicative function. The Wirsing condition is that there exist $c, \lambda \in \mathbb{R}$ with $c \ge 0$ and $0 \le \lambda <2$ such that, for every prime $p$ and every positive integer $k$ , $f(p^k) \le c \lambda^k$ .
The Wirsing condition is important because of the following lemma:
Lemma If a real-valued nonnegative multiplicative function $f$ satisfies the Wirsing condition, then it automatically satisfies the conditions in this theorem. Those conditions are:
- 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 $\displaystyle \sum_p \sum_{k \ge 2} \frac{f(p^k)\log(p^k)}{p^k} \le B$ .
Proof. Let $f$ satisfy the hypotheses of the lemma.
Let $y \ge 0$ . Thus,
| $\displaystyle \sum_{p \le y} f(p)\log p$ | $\displaystyle \le c\lambda \sum_{p \le y} \log p$ |
| $\displaystyle \le c\lambda y \log 4$ by this theorem. |
Also:
| $\displaystyle \sum_p \sum_{k \ge 2} \frac{f(p^k)\log(p^k)}{p^k}$ | $\displaystyle \le \sum_p \sum_{k \ge 2} \frac{c\lambda^k \cdot k\log p}{p^k}$ |
| $\displaystyle \le c\sum_p \log p \sum_{k \ge 2} k\left( \frac{\lambda}{p} \right)^k$ | |
| $\displaystyle \le c\sum_p \log p \cdot \frac{2\left( \frac{\lambda}{p} \right)^2-\left( \frac{\lambda}{p} \right)^3}{\left( 1-\frac{\lambda}{p} \right)^2}$ | |
| $\displaystyle \le \frac{2c}{\left( 1-\frac{\lambda}{2} \right)^2} \sum_p \log p \left( \frac{\lambda}{p} \right)^2$ | |
| $\displaystyle \le \frac{2c\lambda^2}{\left( 1-\frac{\lambda}{2} \right)^2} \sum_p \frac{\log p}{p^2}$ | |
| $\displaystyle \le \frac{2c\lambda^2\zeta\left( \frac{3}{2} \right)}{\left( 1-\frac{\lambda}{2} \right)^2}$ , where $\zeta$ denotes the Riemann zeta function |
Hence, $A=c\lambda \log 4$ and $\displaystyle B=\frac{2c\lambda^2\zeta\left( \frac{3}{2} \right)}{\left( 1-\frac{\lambda}{2} \right)^2}$ . 
Wirsing condition is owned by Warren Buck.
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