Abstract
This paper is concerned with the quantityN(x, m), the number of positive integersn, 1⩽n⩽x, for whichΩ(n)=m, whereΩ:(n) denotes the total number of prime factors (counted with multiplicities) ofn. The main purpose of this article is to present three powerful analytic methods, due, respectively, to Selberg, van der Waerden, and the author, the combination of which allows one to completely characterize the asymptotic behavior of the quantityN(x, m) as the second parameter varies through all its possible values, namely 1⩽m⩽(log x)/log 2. These methods constitute, in a certain sense, a compact and effective set of analytical tools and apply also to the distribution function associated withn(x, m). All these methods are quite general and applicable to many other arithmetic functions.
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