Let $S$ be the space/statespace/system

(maybe with parts/states/aspects $s_0, s_1, s_2, s_3, s_4,\dots$)

$S \equiv 1 + T \equiv 1 - t$

or similar … here $1$ is the neutral/constant thing in the theory and $T$ resp. $t$ is what's really interesting about $S$.

E.g. you have a scattering matrix and $1$ is the free propagation and $T$ is the interaction.

$\dfrac{1}{S}$ … flipped encoding, switches low and far behavior, represents the weight of $X$.

$Q(t):=\dfrac{1}{1-t}=\sum_{n=0}^\infty t^n$

It starts out as $Q(t)=1+t+{\mathcal O}(t^2) \approx 1-T$, but it diverges once $t$ reaches $1$.

(And I observe $S\,Q(t)=2\,Q(t)-1$.)

$\log(S)=\log(1+T)=\sum_{n=0}^\infty\frac{(-1)^n}{n}T^n$

… logarithmic encoding, alternating+declining coefficients give very good convergence.

We want to understand this, in a broad sense, as tamed version of the original: $\log(T) < T$.

But, for $Q(t)$ interpreted in a field, its proper singularity isn't tamed by $\log$:

$\log(Q(t))=\log\left(\dfrac{1}{1-t}\right)=-\log(1-t)=\sum_{n=0}^\infty\frac{1}{n}t^n$

still diverges at $\lim{t\to 1}$.

$\zeta_S$ … Some gluing together of data of $S$.

Sometimes zetas are somewhat obscured using $\exp$'s chained with $\log$'s, in the spirit of above.

$\Pi$ is a gluing together of some aspects of a system.

It's a convolution (in the literal and the metaphorical sense) of structure (e.g addition for polynomials and multiplication for Dirichlet-like objects such as the Riemann zeta function)

For primes $p$, set $t_z=p^{-z}$ and define

$\zeta_\text{Riemann}(z):=\prod Q(t_z)=\prod_\text{primes p}\frac{1}{1-p^{-z}}$.

See Polylogarithm:

$\log\left(\dfrac{1}{1-t}\right)={\mathrm{Li}}_1(t)$

$\zeta_\text{Riemann}(z)={\mathrm{Li}}_s(1)$

Wikipedia: Local zeta-function, Weil conjectures # Statement of the Weil conjectures

StackExchange: What is a zeta function? (Great answer!)

Riemann zeta function