@\slimits@@@nxyΩ(n)=O(x(logx)y-12-y) for 1y<2

Within this entry, Ω refers to the number of (nondistinct) prime factorsMathworldPlanetmath functionMathworldPlanetmath (http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction), μ refers to the Möbius function, log refers to the natural logarithmMathworldPlanetmathPlanetmath, p refers to a prime, and d, k, m, and n refer to positive integers.


For 1y<2, nxyΩ(n)=O(x(logx)y-12-y).


Let g be a function such that yΩ=1*g. Then g is multiplicative and g=μ*yΩ. Thus:

nxyΩ(n) =dxmxdg(d) by the convolution method
=O(x2-y(celoglogx)y-1) for some c>0

Note that a result for y=2 (and therefore for y2), such as nx2Ω(n)=O(xlogx), is unobtainable, as evidenced by this theorem (http://planetmath.org/DisplaystyleXlog2xOleftsum_nLeX2OmeganRight). On the other hand, the asymptotic estimates nx2ω(n)=O(xlogx) and nxτ(n)=O(xlogx) are true.

Title @\slimits@@@nxyΩ(n)=O(x(logx)y-12-y) for 1y<2
Canonical name displaystylesumnleXYOmeganOleftfracxlogXy12yrightFor1leY2
Date of creation 2013-03-22 16:09:15
Last modified on 2013-03-22 16:09:15
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 18
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 11N37
Related topic AsymptoticEstimate
Related topic ConvolutionMethod
Related topic DisplaystyleXlog2xOleftsum_nLeX2OmeganRight
Related topic DisplaystyleSum_nLeXYomeganO_yxlogXy1ForYGe0
Related topic DisplaystyleSum_nLeXTaunaO_axlogX2a1ForAGe0
Related topic 2omeganLeTaunLe2Omegan