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 logarithmMathworldPlanetmathPlanetmath.

Let f be a real-valued nonnegative multiplicative functionMathworldPlanetmath. The Wirsing condition is that there exist c,λ with c0 and 0λ<2 such that, for every prime p and every positive integer k, f(pk)cλk.

The Wirsing condition is important because of the following 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 A0 such that, for every y0, pyf(p)logpAy.

  2. 2.

    There exists B0 such that pk2f(pk)log(pk)pkB.


Let f the hypotheses of the lemma.

Let y0. Thus,

pyf(p)logp cλpylogp
cλylog4 by this theorem (http://planetmath.org/UpperBoundOnVarthetan).


pk2f(pk)log(pk)pk pk2cλkklogppk
2cλ2ζ(32)(1-λ2)2, where ζ denotes the Riemann zeta functionDlmfDlmfMathworldPlanetmath

Hence, A=cλlog4 and B=2cλ2ζ(32)(1-λ2)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