# Thread: Two more questions (cont.)

1. ## 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.

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

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

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

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

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

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

(c) $\displaystyle p_k * \lambda_k = I$

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

Any help/hints/suggestions would be greatly appreciated. Thanks!

2. Again $\displaystyle \mu ^{\left[ k \right]} \left( n \right) = \left\{ \begin{gathered} \mu \left( n \right){\text{ if }}n = m^k {\text{ for some }}m \in \mathbb{Z}^ + \hfill \\ 0{\text{ otherwise}} \hfill \\ \end{gathered} \right.$

(b) We have$\displaystyle \sum\limits_{\left. d \right|n} {\mu ^{\left[ k \right]} \left( n \right)} = p_k \left( n \right)$

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

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

Thus: $\displaystyle \sum\limits_{n = 1}^\infty {\frac{{p_k \left( n \right)}} {{n^s }}} = \frac{{\zeta \left( s \right)}} {{\zeta \left( {s \cdot k} \right)}}$

(c) Hint: Multiply the DSGF of $\displaystyle p_k \left( n \right){\text{ and }}\lambda _k \left( n \right)$
(a) Consider the different cases
(d) Use part (a) and the definition of $\displaystyle {\mu \left( n \right)}$ (see whether our function is multiplicative) or use the series in (b) and remember/prove that $\displaystyle \sum\limits_{n = 1}^\infty {\frac{{\left| {\mu \left( n \right)} \right|}} {{n^s }}} = \frac{{\zeta \left( s \right)}} {{\zeta \left( {2 \cdot s} \right)}}$