---
date: 2022-12-30 17:06
modification date: Friday 30th December 2022 17:06:18
title: "zeta functions"
aliases: [zeta functions]
---

Last modified date: NaN

---

- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [[motivic measure]]

---

# zeta functions

![2023-01-09-11](attachments/2023-01-09-11.png)
![2023-01-09-12](attachments/2023-01-09-12.png)

![](attachments/2023-02-10-zta1.png)

## Euler Product Expansion

- Euler products for scheme-theoretic zetas appearing in the [Weil Conjectures](Unsorted/Weil%20Conjectures.md):
	- $$\begin{align}\log\zeta_X(t) &= \sum_{r\geq 1}\size X(\FF_{q^r}) \, {t^r\over r} \\&= \sum_{x\in \abs{X}} \qty{\sum_{r\geq 1} {t^{r\cdot \deg(x)} \over r}} \\ &= -\sum_{x\in \abs{X}}\log(1-t^{\deg(x)}) \\ \implies \zeta_X(t) &= \prod_{x\in \abs{X}} \qty{1-t^{\deg x}} \inv\quad\underset{t\da q^{-s}}\leadsto\quad \prod_{x\in \abs X}(1-\Norm(x)^{-s})\inv\end{align}$$
	- Here $\Norm(x) = \size\qty{\OO_{X, x}/\mfm_x}$ is the size of the finite residue field and $\abs{X}$ is the set of closed points. Note each $x$ yields $\deg(x)$ many $L\dash$rational points for $L\da \FF_q^{\deg(x)}$.
	- $X\da \spec \ZZ$ recovers Riemann zeta.

# Motivic

![](attachments/2023-02-10-mmotiviczeta.png)