Which Dirichlet Series is obtained as the Mellin transformation of
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Originally Posted by EinStone Which Dirichlet Series is obtained as the Mellin transformation of Now let which means . We then get Therefore This Dirichlet series can then be written in terms of . Note: We require for the original sum to converge. It also converges absolutely for and that's why we can move the integral sign into the summation.
Last edited by chiph588@; April 3rd 2010 at 09:55 AM.
Ok thanks, what can you say about ? Deduce that
Originally Posted by EinStone Ok thanks, what can you say about ? Deduce that Just use the equation .
Last edited by chiph588@; April 3rd 2010 at 11:36 AM.
Originally Posted by chiph588@ Just use the equation . Here's a proof of this.
Ok good, can you still evaluate ?
Originally Posted by EinStone Ok good, can you still evaluate ? What do you mean?
OK I will ask from beginning. I learned that the Mellin transformation transforms Power series in Dirichlet Series. If and then . Now in my case Then we should have . How do I get your result from here?
Originally Posted by EinStone OK I will ask from beginning. I learned that the Mellin transformation transforms Power series in Dirichlet Series. If and then . Now in my case Then we should have . How do I get your result from here? The definition of the Mellin transformation is We have , so . Now simplify a bit to get . Now let which means . We then get Therefore So now, let and , where . Note , thus . So as you can see (hopefully), we do get your identity for this case, namely .
Last edited by chiph588@; April 3rd 2010 at 05:55 PM.
ah I found my mistake. My Power series had coefficient and not just . Thanks for the post .
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