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

Discussion in 'Math Research' started by Harald Helfgott, Aug 30, 2009.

  1. 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.]
     
    Harald Helfgott, Aug 30, 2009
    #1
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  2. Harald Helfgott

    Olivier Guest

    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
     
    Olivier, Aug 31, 2009
    #2
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