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Old 2020-02-23, 17:31   #1
Jul 2014

3·149 Posts
Default something I don't understand to do with Dirichlets theorem


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?

Last fiddled with by wildrabbitt on 2020-02-23 at 17:32
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