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,

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

Also:

${\displaystyle \sum _p}{\displaystyle \sum _{k\ge 2}}{\displaystyle \frac{f(p^k)\log (p^k)}{p^k}}$	$\le {\displaystyle \sum _p}{\displaystyle \sum _{k\ge 2}}{\displaystyle \frac{c{\lambda }^k\cdot k\log p}{p^k}}$
	$\le c{\displaystyle \sum _p}\log p{\displaystyle \sum _{k\ge 2}}k{\left({\displaystyle \frac{\lambda }{p}}\right)}^k$
	$\le c{\displaystyle \sum _p}\log p\cdot {\displaystyle \frac{2{\left(\frac{\lambda }{p}\right)}^2-{\left(\frac{\lambda }{p}\right)}^3}{{\left(1-\frac{\lambda }{p}\right)}^2}}$
	$\le {\displaystyle \frac{2c}{{\left(1-\frac{\lambda }{2}\right)}^2}}{\displaystyle \sum _p}\log p{\left({\displaystyle \frac{\lambda }{p}}\right)}^2$
	$\le {\displaystyle \frac{2c{\lambda }^2}{{\left(1-\frac{\lambda }{2}\right)}^2}}{\displaystyle \sum _p}{\displaystyle \frac{\log p}{p^2}}$
	$\le {\displaystyle \frac{2c{\lambda }^2\zeta \left(\frac{3}{2}\right)}{{\left(1-\frac{\lambda }{2}\right)}^2}}$, where $\zeta$ denotes the Riemann zeta function

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

Title	Wirsing condition
Canonical name	WirsingCondition
Date of creation	2013-03-22 16:08:45
Last modified on	2013-03-22 16:08:45
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	13
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 11N37
Related topic	DisplaystyleSum_nLeXYomeganO_yxlogXy1ForYGe0