Ever since a derivation of the prime number theorem using combinatorial arguments occurred to me, I have wondered whether there might be a direct relation between analytic combinatorics and analytic number theory. After further investigation, I realised that such a fundamental relation exists via the Mellin transform.

If we define:

$$f(x) = \frac{1}{e^x-1}$$

the Mellin transform of $$f$$ yields:

$$\mathcal{M} f(s) = \int_{0}^\infty x^{s-1} \frac{e^{-x}}{1-e^{-x}} dx = \int_{0}^\infty x^{s-1} \sum_{n=1}^\infty e^{-nx} dx = \sum_{n=1}^\infty \int_{0}^\infty x^{s-1} e^{-nx} dx$$

Now, let’s note that given:

$$\int_{0}^\infty x^{s-1} e^{-nx} dx$$

if we set $$u:=nx$$,

$$\int_{0}^\infty \big(\frac{u}{n}\big)^s \cdot e^{-u} \frac{du}{u} = \frac{1}{n^s} \int_{0}^\infty u^{s-1} e^{-u} du$$

$$\implies \mathcal{M}f(s) = \zeta(s) \cdot \Gamma(s)$$

I think that this relation between the Riemann zeta function and the Euler gamma function might explain why the Riemann zeta function shows up with great regularity in combinatorial settings, particularly in statistical mechanics and quantum field theory.