asymptotic estimates for real-valued nonnegative multiplicative functions

Note that, within this entry, p always refers to a prime, k, m, and n always refer to positive integers, and log always refers to the natural logarithmMathworldPlanetmathPlanetmath.


Let f be a real-valued nonnegative multiplicative functionMathworldPlanetmath such that the two following conditions are satisfied:

  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.

Then for all x>1, nxf(n)(A+B+1)xlogxnxf(n)n.


logxnxf(n)=nxf(n)(logx-logn+logn)=nxf(n)log(xn)+nxf(n)lognnxf(n)(xn)+nxf(n)pknlog(pk)xnxf(n)n+pkxlog(pk)nx and pknf(n)xnxf(n)n+pkxlog(pk)nx and pknf(pk)f(npk)xnxf(n)n+pkxlog(pk)mxpkf(pk)f(m)xnxf(n)n+pxf(p)logpmxpf(m)+pxk2f(pk)log(pk)xpkmxpkf(m)mxnxf(n)n+mxf(m)pxmf(p)logp+xmxf(m)mpxk2f(pk)log(pk)pkxnxf(n)n+mxf(m)(Axm)+xmxf(m)mBxnxf(n)n+Axnxf(n)n+Bxnxf(n)n(A+B+1)xnxf(n)n

Dividing the inequalityMathworldPlanetmath logxnxf(n)(A+B+1)xnxf(n)n by logx yields the desired result. ∎

The theorem has an obvious corollary:


If f the conditions of the theorem, then for all x>1, nxf(n)=O(xlogxnxf(n)n).

Title asymptotic estimates for real-valued nonnegative multiplicative functions
Canonical name AsymptoticEstimatesForRealvaluedNonnegativeMultiplicativeFunctions
Date of creation 2013-03-22 16:08:42
Last modified on 2013-03-22 16:08:42
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 11
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 11N37
Related topic AsymptoticEstimate
Related topic DisplaystyleSum_nLeXTaunaO_axlogX2a1ForAGe0