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Thread: Dirichlet

  1. #1
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    Dirichlet

    I am sorry for being rude in my previous thread, I would like to apologize and resubmit my question with all the correct notation and hopefully present it more clearer. My text is Elementary Number Theory by Jones and Jones. I do not understand Dirichlet product and series well at all. I am sure the definitions and lemmas in my text would be enough to tackle these problems, but I cannot seem to grasp them. Any help would be greatly appreciated, and I thank you for your time.

    $\displaystyle k$ 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

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

    Suppose $\displaystyle h$ is completely multiplicative. Define $\displaystyle f^{*k}$ inductively by $\displaystyle f^{*1}$=$\displaystyle f$and $\displaystyle f^{*(k+1)}$=$\displaystyle f*f^{*k}$.

    Prove that:
    (a)$\displaystyle (fh)*(gh) = (f*g)h$
    (b)$\displaystyle (fh)*(gh)^{*k} = (f*g^{*k})h$
    (c)$\displaystyle (fh)*h = (f*u)h$
    (d)$\displaystyle (fh)*h^{*k} = (f*u^{*k})h$
    (e)$\displaystyle h^{*k} = u^{*k}h$
    (f) If $\displaystyle h$ is non-zero, then $\displaystyle (\mu^{*k}h) * h^{*k} =I$
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  2. #2
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    Quote Originally Posted by shadow_2145 View Post
    (a)$\displaystyle (fh)*(gh) = (f*g)h$
    How are you defining $\displaystyle fh$? Do you mean $\displaystyle f\circ h$? The reason why I am afraid to assume you mean $\displaystyle f\circ h$ is because $\displaystyle f,h$ are arithmetic functions, which can mean, $\displaystyle f,h:\mathbb{N}\mapsto \mathbb{C}$ and therefore composition would not make sense. Unless, here your arithmetic functions have their range as a subset of the naturals.
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  3. #3
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    Are you saying that everything would make sense if it was in the form of $\displaystyle f\circ h$? Or not?
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    Quote Originally Posted by shadow_2145 View Post
    Are you saying that everything would make sense if it was in the form of $\displaystyle f\circ h$? Or not?
    I am asking how do you define $\displaystyle fh$? I said it it reasonable to define $\displaystyle fh = f\circ h$ as longs as the ranges of $\displaystyle f,h$ are subsets of $\displaystyle \mathbb{N}$ otherwise the composition will not make sense. . So is it okay to assume that $\displaystyle f,g:\mathbb{N}\mapsto \mathbb{N}$?
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  5. #5
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    Yes, they are subsets of $\displaystyle \mathbb{N}$. It is ok to assume what you stated. $\displaystyle f,g:\mathbb{N}\mapsto \mathbb{N}$ is correct. Sorry!
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    Quote Originally Posted by shadow_2145 View Post
    (a)$\displaystyle (fh)*(gh) = (f*g)h$
    To show these functions are the same you need to show for any $\displaystyle n$ we have $\displaystyle [(f\circ h)*(g\circ h)](n) = [(f*g)\circ h](n)$.

    $\displaystyle [(f\circ h)*(g\circ h)](n) = f(h(n))*g(h(n)) = \sum_{d|n} f(h(d))g(h(n/d))$.
    $\displaystyle [(f*g)\circ h](n) = \sum_{d|n}f(h(n))g(h(n/d))$.

    Thus, they are the same. Try doing the other ones in the same way.
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  7. #7
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    Thank you! I will try, but (b) and (d) are also confusing me with the $\displaystyle ^{*k}$
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