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Let $\omega(k)$ be the prime omega function, it counts how many distinct prime factors k has.

The dirichlet series for $\omega(k)$ can be written as,$$\sum_{k=1}^\infty\frac{\omega(k)}{k^s}=\prod_{p}\frac{1}{1-p^{-s}}*\sum_{p}\frac{1}{p^s}=\zeta(s)*P(s)$$ I know I cant re-write $$\sum_{k=0}^\infty\frac{\omega(ak+b)}{(ak+b)^s}=\prod_{p\equiv\text{b mod a} }\frac{1}{1-p^{-s}}*\sum_{p\equiv\text{b mod a}}\frac{1}{p^s}$$ But can I re-write it, as somthing similar?

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You should learn about Dirichlet characters. There are $\phi(q)$ (completely multiplicative) Dirichlet characters $\chi$ modulo $q$, and summing over all of them one gets the following nice relationship, for any $a$ that is relatively prime to $q$: $$ \frac1{\phi(q)} \sum_{\chi\pmod q} \overline{\chi(a)} \chi(n) = \begin{cases}1, &\text{if } n\equiv a\pmod q, \\0, &\text{if } n\not\equiv a\pmod q. \end{cases} $$ Therefore (for $1\le a\le q$ and $(a,q)=1$) \begin{align*} \sum_{k=0}^\infty \frac{\omega(qk+a)}{(qk+a)^s} &= \sum_{n=1}^\infty \frac{\omega(n)}{n^s} \frac1{\phi(q)} \sum_{\chi\pmod q} \overline{\chi(a)} \chi(n) \\ &= \frac1{\phi(q)} \sum_{\chi\pmod q} \overline{\chi(a)} \sum_{n=1}^\infty \frac{\omega(n)\chi(n)}{n^s} \\ &= \frac1{\phi(q)} \sum_{\chi\pmod q} \overline{\chi(a)} \prod_p \frac1{1-\chi(p)p^{-s}} \sum_p \frac{\chi(p)}{p^s} \\ &= \frac1{\phi(q)} \sum_{\chi\pmod q} \overline{\chi(a)} L(s,\chi) \sum_p \frac{\chi(p)}{p^s}, \end{align*} where $L(s,\chi)$ is a Dirichlet $L$-function.

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    Whats a dirichlet L function, what are those lines above those characters?2012-12-29
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    What is meant by the summation over x ( mod q) what does that mean?2012-12-29
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    The lines over the characters are complex conjugates. My suggestion is that you read up on Dirichet $L$-functions and Dirichlet characters (on the web, or from a book on analytic number theory). They are extremely pertinent to questions like these about arithmetic progressions. Dirichlet $L$-functions can be expressed in terms of Hurwitz zeta functions, but they are much more convenient (they have Euler products, for example).2012-12-29
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    Greg I have a good understanding of high school calculus, trigonometry, algebra, and I am familear with many topics in number theory, and have conducted my own deep studdies with additive function. Could you give me a recomendation to a good book on analytic number theory using only the skills I have now?2012-12-29
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    I would recommend Apostol's "Introduction to Analytic Number Theory".2012-12-29
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    Does it use techniques beyond those that I have mentioned? Also out of curriousity is Eric Naslund your graduate student? He also asked for a proof of the statement regaurding the fractionalpart sum on the vonmangoldt function, I am willing to give you a proof of this right now if your not to buisy, I realized it wasn't as deep as I thought it was. In the proof I use fourier series and a little complex analysis, although I have never read anything really formal on the subject. So you might have to brush it up a little if thats no problem. Also can I post the proof on this page?2012-12-29
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Define a generalization of the prime zeta function, where the primes range over the congruene classes: $$P_{a,b}(s)=\sum_{p\equiv b \text{ mod a}}\frac{1}{p^s}$$ Also define the indicator function $$\delta_{a,b}(k)=\text{1 if k mod a = b},\text{ 0 if k mod a}\ne b$$ Then for b>0,$$\sum_{k=0}^\infty\frac{\omega(ak+b)}{(ak+b)^s}=\frac{1}{a^s}\sum_{j=1}^a\sum_{k=1}^a\delta_{a,b}(kj)P_{a,k}(s)\zeta(s,\frac{j}{a})$$ Where $\zeta(s,q)$ is the hurrwitz zeta function.

This seems very straight forward,

I don't understand what it is you did with your "dirichlet characters"