Integral Representation of the Zeta Function: $\zeta(s)=\frac1{\Gamma(s)}\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$

Question

How does one get from this $$\zeta(s)=\sum_{k=1}^{\infty}\frac1{k^s}$$ to the integral representation $$\zeta(s)=\frac1{\Gamma(s)}\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$$ of the Riemann Zeta function?

I can see that it can be rewritten as

$$\Gamma(s)\zeta(s)=\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$$ and the Gamma function as an integral yields

$$\zeta(s)\int_{0}^\infty \frac{x^{s-1}}{e^x}dx=\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$$

But this approach does not work as the right integral does not converge. So how does one go from the summation to the integral representation?

Answer 1

Recall that for $t>1$, $$\frac{1}{t-1}=\sum_{n=1}^\infty t^{-n}$$

Then substitute $t=e^{x}$ for $x>0$.

Then substituting $x=\frac{v}{n}$ in the $n$th term of the integral, you get:

$$\int_0^{\infty} x^{s-1}e^{-nx}\,dx=\frac{1}{n^s}\int_0^\infty v^{s-1}e^{-v}dv = \frac{1}{n^s}\Gamma(s)$$

Answer 2

$\newcommand{\dd}{\mathrm{d}}$

The Mellin transform of a function $f$ is given by

\begin{align} \left\{\mathscr{M}f\right\}(s)=\phi(s)=\int_0^{\infty}x^{s-1}f(x)\dd x \end{align}

For the function

\begin{align} f(x)=\frac{1}{e^x - 1} \end{align}

we have

\begin{align} \phi(s)&=\int_0^{\infty}\frac{x^{s-1}}{e^x - 1}\dd x \\ &=\sum_{n \geq1}\int_{0}^{\infty}x^{s-1}e^{-nx}\dd x \end{align}

Making the substitution $u=nx$ gives

\begin{align} \phi(s)&=\sum_{n \geq1}\frac{1}{n^s}\int_{0}^{\infty}u^{s-1}e^{-u}\dd u \\ &=\zeta(s)\Gamma(s) \end{align}

Equating this to the above integral gives the desired result:

\begin{align} \zeta(s)\Gamma(s)&=\int_{0}^{\infty}\frac{x^{s-1}}{e^x - 1}\dd x \\ \zeta(s)&=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{x^{s-1}}{e^x - 1}\dd x \end{align}

Answer 3

$$\mathscr{L}\{t^k\}=\int_0^\infty t^ke^{-st} = \dfrac{k!}{s^{k+1}}$$ is the well-known laplace transform of $t^{k}$

then like above in the first answer, sum over $s$ and swap integration and summation