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 $c,\lambda \in ℝ$ with $c\ge 0$ and 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 $f$ the Wirsing condition, then it automatically the conditions in this theorem (http://planetmath.org/AsymptoticEstimatesForRealValuedNonnegativeMultiplicativeFunctions). Those conditions are:

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

Proof.

Let $f$ the hypotheses of the lemma.

Let $y\ge 0$. Thus,

Also:

Hence, $A=c\lambda \log 4$ and $B={\displaystyle \frac{2c{\lambda }^2\zeta \left(\frac{3}{2}\right)}{{\left(1-\frac{\lambda }{2}\right)}^2}}$.∎