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Let $f \colon X \to X$ be a topological dynamical system. Write $|\operatorname{Fix}(f^n)|$ for the number of fixed points of the $n$-fold composition of $f$. Then the Ruelle zeta function is defined as $$\zeta(s) = \exp \bigg( \sum_{n \geq 1} |\operatorname{Fix}(f^n)| \frac{s^n}{n}\bigg).$$ Contrast this with the Riemann zeta function, defined via the formula $$\zeta(s) = \sum_{n \geq 1} \frac{1}{n^{-s}}.$$ From my understanding, we call something a zeta function if it resembles the Riemann zeta function. But these two functions look absolutely nothing alike. How can I see the similarity? I presume I need to make some kind of clever substitution to compare them, but I don't see how.

NumbersAreNice
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Perhaps you can better see a connection from the product formulae for the two zeta functions.

For the Riemann zeta function, we have

$$\zeta(s) = \prod_{p\text{ prime}}\frac{1}{1-p^{-s}}$$

and for the Ruelle zeta function, we have

$$\zeta(s) = \prod_{\gamma \text{ periodic orbit}}\frac{1}{1-s^{|\gamma|}}$$

where $|\gamma|$ is the length of the orbit $\gamma$ (the prime period of its constituent points).

At least these formulae are much similar in form than the summation formulae, of course there is still a difference in that, for one, the variable is an exponent appearing in the denominator of the multiplicative terms, and in the other it just appears as the base of a power.

Dan Rust
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    Based on this, if I were to try to find an analogue of the Riemann zeta function in dynamical systems, I would replace $s^{|\gamma|}$ with $|\gamma|^{-s}$. Why don't we do this? (I could also reverse the question, and ask why the Riemann zeta function doesn't have $s^{-p}$ rather than $p^{-s}$, but that is arguably far less sensible.) Is there anything deeper that makes Riemann's and Ruelle's zeta functions both zeta functions, whatever that may truly mean? – NumbersAreNice Nov 12 '19 at 16:07
  • It's a good question, and for a suitable answer, I would refer to this great answer https://math.stackexchange.com/questions/429616/what-is-a-zeta-function. The main point is that zeta functions encode something about basic constituent pieces of the object you're studying. – Dan Rust Nov 12 '19 at 16:51
  • Thank you. I am not a number theorists, so I have no feeling for the significance or true meaning of zeta functions. The answer there sounds like it could be very much of help. – NumbersAreNice Nov 12 '19 at 16:54
  • I'm also not a number theorist so I can't add much more than the excellent answer there! – Dan Rust Nov 12 '19 at 16:55