@\slimits@@@nxyω(n)=Oy(x(logx)y-1) for y0

Within this entry, ω refers to the number of distinct prime factors function, 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 1.

For y0, nxyω(n)=Oy(x(logx)y-1).


Since yω(pk)=y for all p and k, the real-valued nonnegative multiplicative functionMathworldPlanetmath yω(n) the Wirsing condition with c=y and λ=1. Thus:

nxyω(n) =Oy(xlogxnxyω(n)n)

Title @\slimits@@@nxyω(n)=Oy(x(logx)y-1) for y0
Canonical name displaystylesumnleXYomeganOyxlogXy1ForYge0
Date of creation 2013-03-22 16:11:22
Last modified on 2013-03-22 16:11:22
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 9
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 11N37
Related topic AsymptoticEstimate
Related topic DisplaystyleYOmeganOleftFracxlogXy12YRightFor1LeY2
Related topic WirsingCondition