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Old 2020-02-23, 23:58   #3
Dr Sardonicus
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Feb 2017

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Originally Posted by wildrabbitt View Post

the following is something I've been reading.

He started from the power series

\(\sum_{n=1}^\infty \big(\frac{n}{q}\big)x^n=\frac{1}{1-x^q}\sum_{m=1}^{q-1}\big(\frac{m}{q}\big)x^m=\frac{xf(x)}{1-x^q}\)

say, and by putting this in the formula

\(\Gamma(s)n^{-1}=\int_0^1 x^{n-1}(\log x^{-1})^{s-1} \mathrm{d}x\)

he obtained

\(\Gamma(s)L(s)=-\int_0^1\frac{f(x)}{x^q-1}(\log x^{-1})^{s-1}\mathrm{d}x\)

I'm stuck because I can't see how he put what he put in the formula.

Can anyone explain it step by step?
I took the liberty of TEXing your equations. That makes them a whole lot easier to read, for me at least. You can probably drop the big parens.

Is that supposed to be (log(x)-1^(s-1)? That would be log(x)1-s, yes?
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