Applying Mellin Transformation

**EinStone** · April 3rd 2010, 07:40 AM

Which Dirichlet Series is obtained as the Mellin transformation of
$\frac{1}{e^t+ 1}= \sum_{n=0}^\infty (-1)^n (e^{-t})^{n+1}$

**chiph588@** · April 3rd 2010, 08:42 AM

Originally Posted by EinStone

Which Dirichlet Series is obtained as the Mellin transformation of
$\frac{1}{e^t+ 1}= \sum_{n=0}^\infty (-1)^n (e^{-t})^{n+1}$

$\left\{\mathcal{M}f\right\}(s) = \int_0^\infty \frac{x^{s-1}}{e^x+1}dx = \sum_{n=0}^\infty (-1)^n\int_0^\infty x^{s-1}e^{-x(n+1)} dx$

Now let $t=x(n+1)$ which means $dt=(n+1)dx$ .

We then get $\left\{\mathcal{M}f\right\}(s) = \sum_{n=0}^\infty \frac{(-1)^n}{(n+1)^s}\int_0^\infty t^{s-1}e^{-t} dt$

Therefore $\left\{\mathcal{M}f\right\}(s) = \Gamma(s)\sum_{n=0}^\infty \frac{(-1)^n}{(n+1)^s} = \Gamma(s)\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s}$

This Dirichlet series can then be written in terms of $\zeta(s)$ .

Note: We require $x>0$ for the original sum to converge. It also converges absolutely for $x>0$ and that's why we can move the integral sign into the summation.

**EinStone** · April 3rd 2010, 09:07 AM

Ok thanks, what can you say about $\frac{\left\{\mathcal{M}f\right\}(s)}{\zeta(s)}$ ? Deduce that

$\zeta(s) = \frac{1}{\Gamma(s)(1-2^{1-s})} \int_0^s \frac{t^{s-1}}{e^t+1}$

**chiph588@** · April 3rd 2010, 09:48 AM

Originally Posted by EinStone

Ok thanks, what can you say about $\frac{\left\{\mathcal{M}f\right\}(s)}{\zeta(s)}$ ? Deduce that

$\zeta(s) = \frac{1}{\Gamma(s)(1-2^{1-s})} \int_0^s \frac{t^{s-1}}{e^t+1}$

Just use the equation $\zeta(s)(1-{2^{1-s}})= \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s}$ .

**chiph588@** · April 3rd 2010, 10:37 AM

Originally Posted by chiph588@

Just use the equation $\zeta(s)(1-{2^{1-s}})= \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s}$ .

Here's a proof of this.

**EinStone** · April 3rd 2010, 11:41 AM

Ok good, can you still evaluate $<br /> \frac{\left\{\mathcal{M}f\right\}(s)}{\zeta(s)}<br />$ ?

**chiph588@** · April 3rd 2010, 11:46 AM

Originally Posted by EinStone

Ok good, can you still evaluate $<br /> \frac{\left\{\mathcal{M}f\right\}(s)}{\zeta(s)}<br />$ ?

What do you mean?

**EinStone** · April 3rd 2010, 02:26 PM

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 $F(s) = \sum_{n=0}^\infty (-1)^n (e^{-s})^{n+1}$

Then we should have $f(s) = \frac{1}{\Gamma(s)} \int_0^\infty \sum_{n=0}^\infty ((-1)^n (e^{-e^{-t}})^{n+1}) * t^{s-1}dt$ . How do I get your result from here?

**chiph588@** · April 3rd 2010, 03:05 PM

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 $F(s) = \sum_{n=0}^\infty (-1)^n (e^{-s})^{n+1}$

Then we should have $f(s) = \frac{1}{\Gamma(s)} \int_0^\infty \sum_{n=0}^\infty ((-1)^n (e^{-e^{-t}})^{n+1}) * t^{s-1}dt$ . How do I get your result from here?

The definition of the Mellin transformation is $\left\{\mathcal{M}f\right\}(s) = \int_0^\infty x^{s-1}f(x)dx$

We have $f(x) = \sum_{n=0}^\infty (-1)^n e^{-x(n+1)}$ , so $\left\{\mathcal{M}f\right\}(s) = \int_0^\infty x^{s-1}\sum_{n=0}^\infty (-1)^n e^{-x(n+1)}dx$ .

Now simplify a bit to get $\left\{\mathcal{M}f\right\}(s) = \sum_{n=0}^\infty (-1)^n \int_0^\infty x^{s-1}e^{-x(n+1)}dx$ .

Now let $t=x(n+1)$ which means $dt=(n+1)dx$ .

We then get $\left\{\mathcal{M}f\right\}(s) = \sum_{n=0}^\infty \frac{(-1)^n}{(n+1)^s}\int_0^\infty t^{s-1}e^{-t} dt$

Therefore $\left\{\mathcal{M}f\right\}(s) = \Gamma(s)\sum_{n=0}^\infty \frac{(-1)^n}{(n+1)^s} = \Gamma(s)\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s}$

So now, let $g(s) = \sum_{n=0}^\infty \frac{a_n}{n^s}$ and $G(z) = \sum_{n=1}^\infty a_n z^n$ , where $a_n=(-1)^{n+1}$ .
Note $G(z) = \sum_{n=0}^\infty a_{n+1} z^{n+1}$ , thus $G(e^{-t}) = \sum_{n=0}^\infty a_{n+1} (e^{-t})^{n+1}$ .

So as you can see (hopefully), we do get your identity for this case, namely $g(s) = \frac{1}{\Gamma(s)}\int_0^\infty G(e^{-t})t^{s-1}dt$ .

**EinStone** · April 4th 2010, 02:22 AM

ah I found my mistake. My Power series had coefficient $e^{-t}$ and not just $t$ . Thanks for the post

.

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