The Distribution of Round Numbers

For positive x and positive integer y , let π( x , y ) denote the number of natural numbers less than x which have exactly y distinct prime factors. We show that the asymptotic formula for π( x , y ) given by Selberg for y ⩽ C log log x is in fact valid out to y = o {(log log x) 2 /(log log log x ) 2 }, and that for larger y , the old estimate must be multiplied by exp(− ½ y (log log log x ) 2 /(log log x) 2 ).. So modified, the estimate remains asymptotically correct at least to y =(log log x ) 2 /((log log log x ) 3/2 ; log log log log x ).