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Math Help - Two more questions (cont.)

  1. #1
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    Two more questions (cont.)

    This is the 2nd part to my questions:

    I don't know if the following definition is needed, but I will include it.

    k is a positive integer and f,g and h are arithmetic functions. \hat{f} denotes the Dirichlet series for f, so

    \hat{f}(s) = \sum_{n=1}^{\infty}{f(n)/n^s}


    (2) Define p_k= u*\mu^{[k]}. Show that

    (a) If k>1 then for all primes p and all e \in N:

    p_k{(p^e)}= \left\{\begin{array}{cc}1,&\mbox{ if }e<k\\0, & \mbox{ if } e \geq k \end{array}\right.

    (b) \widehat{p_k}{(s)}= \frac{\zeta{(s)}}{\zeta{(ks)}}

    (c) p_k * \lambda_k = I

    (d) p_2 = |\mu| = \mu^2.

    Any help/hints/suggestions would be greatly appreciated. Thanks!
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  2. #2
    Super Member PaulRS's Avatar
    Joined
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    Again <br />
\mu ^{\left[ k \right]} \left( n \right) = \left\{ \begin{gathered}<br />
  \mu \left( n \right){\text{ if }}n = m^k {\text{ for some }}m \in \mathbb{Z}^ +   \hfill \\<br />
  0{\text{ otherwise}} \hfill \\ <br />
\end{gathered}  \right.<br />

    (b) We have <br />
\sum\limits_{\left. d \right|n} {\mu ^{\left[ k \right]} \left( n \right)}  = p_k \left( n \right)<br />

    Thus: <br />
\left( {\sum\limits_{n = 1}^\infty  {\frac{{\mu ^{\left[ k \right]} \left( n \right)}}<br />
{{n^s }}} } \right) \cdot \left( {\sum\limits_{n = 1}^\infty  {\frac{1}<br />
{{n^s }}} } \right) = \sum\limits_{n = 1}^\infty  {\frac{{\sum\limits_{\left. d \right|n} {\mu ^{\left[ k \right]} \left( n \right)} }}<br />
{{n^s }}}  = \sum\limits_{n = 1}^\infty  {\frac{{p_k \left( n \right)}}<br />
{{n^s }}} <br /> <br />

    But as you know by http://www.mathhelpforum.com/math-he...ped-again.htmlwe have <br />
\sum\limits_{n = 1}^\infty  {\frac{{\mu ^{\left[ k \right]} \left( n \right)}}<br />
{{n^s }}}  = \sum\limits_{n = 1}^\infty  {\frac{{\mu \left( n \right)}}<br />
{{n^{s \cdot k} }}}  = \frac{1}<br />
{{\zeta \left( {s \cdot k} \right)}}<br />

    Thus: <br />
\sum\limits_{n = 1}^\infty  {\frac{{p_k \left( n \right)}}<br />
{{n^s }}}  = \frac{{\zeta \left( s \right)}}<br />
{{\zeta \left( {s \cdot k} \right)}}<br /> <br />

    (c) Hint: Multiply the DSGF of <br />
p_k \left( n \right){\text{ and }}\lambda _k \left( n \right)<br />
    (a) Consider the different cases
    (d) Use part (a) and the definition of <br />
{\mu \left( n \right)}<br />
(see whether our function is multiplicative) or use the series in (b) and remember/prove that <br />
\sum\limits_{n = 1}^\infty  {\frac{{\left| {\mu \left( n \right)} \right|}}<br />
{{n^s }}}  = \frac{{\zeta \left( s \right)}}<br />
{{\zeta \left( {2 \cdot s} \right)}}
    Last edited by PaulRS; April 9th 2008 at 05:27 AM.
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