\sum_{n<=N} y^{omega(n)}

Let y>1. Let f(N;y) = \sum_{n<=N} y^{omega(n)}.

It is easy to show by elementary means that f(N;y) << N (log N)^{y-1}.

It is also true that f(N;y) \sim c N (log N)^{y-1} for some constant c
(perhaps c=1?). I vaguely remember how to do this (contour integration
of a fractional power of z(s) around s=1), but does anybody have a
reference? Furthermore - is there a (reasonable) elementary way to do
this?

H A Helfgott


[Mod note: the OP informs me that omega(n) is the number of prime
divisors of n.]

 

Hi,

Tenenbaum, Théorie analytique et probabiliste des nombres,
chapter II.6, Theorem 2, when y < 2.
Tenenbaum is anyway one of the main reference on the Selberg
Delange method.

For an elementary treatment, use Levin-Fainleib
(Which gives you access to \sum_{n<=N} y^{omega(n)}/n)
and a similar treatment due to Wirsing
gives you \sum_{n<=N} y^{omega(n)}.

Annexe of my future-famous-book, Theorem 21.2
It relies on Theorem 21.1 which is fully explicit,
though the constants there are most probably not very
efficient.

Wirsing, {E.} 1961.
Das asymptotische {V}erhalten von {S}ummen {\"u}ber multiplikative
{F}unktionen.
{\em Math. Ann.}, {\bf 143}, 75--102.


Best,
Amities,
Olivier