2018/03/23 by Krishnaswami Alladi, Alladi, Krishnaswami, Todd J. Molnar +1
Mathematics · #Analytic Number Theory Research
paper · pdf · doi:10.48550/arxiv.1803.08964
We obtain uniform estimates for Nk(x,y), the number of positive integers\nn up to x for which \ωy(n)=k, where \ωy(n) is the number of\ndistinct prime factors of n which are <y. The motivation for this problem\nis an observation due to the first author in 2015 that for certain ranges of\ny, the asymptotic behavior of Nk(x,y) is different from the classical\nsituation concerning Nk(x,x) studied by Sathe and Selberg. We demonstrate\nthis variation of the classical theme; to estimate Nk(x,y) we study the sum\nSz(x,y)=\∑n\≤ xz\ωy(n) for Re(z)>0 by the Buchstab-de Bruijn\nmethod. We also utilize a certain recent result of Tenenbaum to complete our\nasymptotic analysis.\n