# 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 $c\geq 0$ and $0\leq\lambda<2$ such that, for every prime $p$ and every positive integer $k$, $f(p^{k})\leq c\lambda^{k}$.

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. 1.

There exists $A\geq 0$ such that, for every $y\geq 0$, $\displaystyle\sum_{p\leq y}f(p)\log p\leq Ay$.

2. 2.

There exists $B\geq 0$ such that $\displaystyle\sum_{p}\sum_{k\geq 2}\frac{f(p^{k})\log(p^{k})}{p^{k}}\leq B$.

###### Proof.

Let $f$ the hypotheses of the lemma.

Let $y\geq 0$. Thus,

 $\displaystyle\sum_{p\leq y}f(p)\log p$ $\displaystyle\leq c\lambda\sum_{p\leq y}\log p$ $\displaystyle\leq c\lambda y\log 4$ by this theorem (http://planetmath.org/UpperBoundOnVarthetan).

Also:

 $\displaystyle\sum_{p}\sum_{k\geq 2}\frac{f(p^{k})\log(p^{k})}{p^{k}}$ $\displaystyle\leq\sum_{p}\sum_{k\geq 2}\frac{c\lambda^{k}\cdot k\log p}{p^{k}}$ $\displaystyle\leq c\sum_{p}\log p\sum_{k\geq 2}k\left(\frac{\lambda}{p}\right)% ^{k}$ $\displaystyle\leq c\sum_{p}\log p\cdot\frac{2\left(\frac{\lambda}{p}\right)^{2% }-\left(\frac{\lambda}{p}\right)^{3}}{\left(1-\frac{\lambda}{p}\right)^{2}}$ $\displaystyle\leq\frac{2c}{\left(1-\frac{\lambda}{2}\right)^{2}}\sum_{p}\log p% \left(\frac{\lambda}{p}\right)^{2}$ $\displaystyle\leq\frac{2c\lambda^{2}}{\left(1-\frac{\lambda}{2}\right)^{2}}% \sum_{p}\frac{\log p}{p^{2}}$ $\displaystyle\leq\frac{2c\lambda^{2}\zeta\left(\frac{3}{2}\right)}{\left(1-% \frac{\lambda}{2}\right)^{2}}$, where $\zeta$ denotes the Riemann zeta function

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

Title Wirsing condition WirsingCondition 2013-03-22 16:08:45 2013-03-22 16:08:45 Wkbj79 (1863) Wkbj79 (1863) 13 Wkbj79 (1863) Definition msc 11N37 DisplaystyleSum_nLeXYomeganO_yxlogXy1ForYGe0