@\slimits@@@nx(τ(n))a=Oa(x(logx)2a-1) for a0


Within this entry, τ refers to the divisor functionDlmfDlmfMathworldPlanetmath, refers to the floor function, log refers to the natural logarithmMathworldPlanetmathPlanetmath, p refers to a prime, and k and n refer to positive integers.

Theorem.

For a0, nx(τ(n))a=Oa(x(logx)2a-1).

The Oa indicates that the constant implied by the definition of O depends on a. (See Landau notationMathworldPlanetmathPlanetmath for more details.)

Proof.

Let a0. Since (τ)a=idaτ, id is completely multiplicative, and τ is multiplicative, (τ)a is multiplicative. (See composition of multiplicative functions for more details.)

For any y0,

py(τ(p))alogp =py2alogp
=2apylogp
2aylog4 by this theorem (http://planetmath.org/UpperBoundOnVarthetan).

Also,

pk2(τ(pk))apklog(pk) =pk2(k+1)apkklogp
plogpk2(k+1)a+1pk
plogpp2k2(2k)a+1pk-2
2a+1p1p32k2ka+12k-2
2a+3ζ(32)k2ka+12k, where ζ denotes the Riemann zeta functionDlmfDlmfMathworldPlanetmath.

Since

limk|((k+1)a+12k+1)(ka+12k)|=limk|(k+1)a+12kka+12k+1|=limk(12)(k+1k)a+1=12(limkk+1k)a+1=12,

k2ka+12k converges by the ratio testMathworldPlanetmath. Thus, by this theorem (http://planetmath.org/AsymptoticEstimatesForRealValuedNonnegativeMultiplicativeFunctions), nx(τ(n))a=Oa(xlogxnx(τ(n))an). Therefore,

nx(τ(n))a =Oa(xlogxnx(τ(n))an)
=Oa(xlogxpx(1+k=1logxlogp(τ(pk))apk))
=Oa(xlogx(exp(pxk=1logxlogp(k+1)apk)))
=Oa(xlogx(exp(pxk=1logxlogp(2k)apk)))
=Oa(xlogx(exp(2apxk=1logxlogpkapk)))
=Oa(xlogx(exp(2a(loglogx+Oa(1)))))
=Oa(xlogx(exp(log(logx)2a)))
=Oa(xlogx(logx)2a)
=Oa(x(logx)2a-1).

Title @\slimits@@@nx(τ(n))a=Oa(x(logx)2a-1) for a0
Canonical name displaystylesumnleXtaunaOaxlogX2a1ForAge0
Date of creation 2013-03-22 16:09:53
Last modified on 2013-03-22 16:09:53
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 15
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 11N37
Related topic AsymptoticEstimatesForRealValuedNonnegativeMultiplicativeFunctions
Related topic DisplaystyleYOmeganOleftFracxlogXy12YRightFor1LeY2