20200223, 17:31  #1 
Jul 2014
3·149 Posts 
something I don't understand to do with Dirichlets theorem
Hi,
the following is something I've been reading. He started from the power series \(\sum_{n=1}^\infty \big(\frac{n}{q}\big)x^n=\frac{1}{1x^q}\sum_{m=1}^{q1}\big(\frac{m}{q}\big)x^m=\frac{xf(x)}{1x^q}\) say, and by putting this in the formula \(\Gamma(s)n^{1}=\int_0^1 x^{n1}(\log x^{1})^{s1} \mathrm{d}x\) he obtained \(\Gamma(s)L(s)=\int_0^1\frac{f(x)}{x^q1}(\log x^{1})^{s1}\mathrm{d}x\) I'm stuck because I can't see how he put what he put in the formula. Can anyone explain it step by step? Last fiddled with by wildrabbitt on 20200223 at 17:32 
20200223, 20:17  #2  
Nov 2003
2^{2}×5×373 Posts 
Quote:


20200223, 23:58  #3  
Feb 2017
Nowhere
2^{5}×3^{3}×5 Posts 
Quote:
Is that supposed to be (log(x)^{1}^(s1)? That would be log(x)^{1s}, yes? 

20200224, 07:52  #4 
Jul 2014
3·149 Posts 
I just wrote what was written in the book.
I'm pretty sure it's supposed to be \(\log \frac{1}{x} \) otherwise it would have been written the way you wrote. 
20200224, 09:51  #5 
Jul 2014
3×149 Posts 
clarification
I aught to mention that
\( L(s)=\sum_{n=1}^{\infty}\big(\frac{n}{q}\big)n^{s}\) for which \(\big(\frac{n}{q}\big)\) is the Legendre symbol as is it is the first post. Don't know what to do if my latex is not parsing correctly. It comes out right when I preview the post. I can give more info if necessary. Last fiddled with by wildrabbitt on 20200224 at 09:52 
20200224, 12:49  #6  
Feb 2017
Nowhere
2^{5}×3^{3}×5 Posts 
Quote:
Your second equation is wrong. It should be (It is n^{s} on the left side, not n^{1}) 

20200224, 20:40  #7 
Jul 2014
3×149 Posts 
Thanks.
Are you substituting \(e^{u}\) for \(x\) ? What are you doin with \(u\)? Also, in case there's an obvious remedy to my latex not parsing properly, here's an example of what I wrote in the first post without the \\( and \\). \sum_{n=1}^\infty \big(\frac{n}{q}\big)x^n=\frac{1}{1x^q}\sum_{m=1}^{q1}\big(\frac{m}{q}\big)x^m=\frac{xf(x)}{1x^q} Last fiddled with by wildrabbitt on 20200224 at 20:44 Reason: needed to access page source 
20200225, 08:21  #8  
Dec 2012
The Netherlands
2^{4}·101 Posts 
Quote:
\[\sum_{n=1}^\infty \big(\frac{n}{q}\big)x^n=\frac{1}{1x^q}\sum_{m=1}^{q1}\big(\frac{m}{q}\big)x^m=\frac{xf(x)}{1x^q}\] 

20200225, 13:32  #9  
Feb 2017
Nowhere
1000011100000_{2} Posts 
Quote:
If you don't know how to make substitutions in integrals, it's high time you learn. Unfortunately, I'm probably not the one to teach you. 

20200225, 17:01  #10 
Jul 2014
3×149 Posts 
I didn't really get the answer I was hoping for so I thought I'd just ask the first things that it occurred to me to ask.
I do know how to make substitutions in integrals I just wasn't sure if that's what you were doing and it looked like you were making two substitutions. Thanks though. So you're substiting for x. I don't know what you're doing with u. 
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