---
date: 2021-10-29 02:02
modification date: Friday 29th October 2021 02:02:42
title: Riemann Hypothesis
aliases: [Riemann Hypothesis, RH, GRH]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #NT/analytic
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

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# Riemann Hypothesis

# GRH

Statement: for $L(\chi, s)$ an arbitrary [L function](Unsorted/L%20function.md), its meromorphic continuation to $\CC$ has zeros only along the line $\Re(s) = {1\over 2}$.

Obviously wildly open! Some interesting conditional results:

- Compare to [Dirichlet's theorem on primes in arithmetic progressions](Unsorted/Dirichlet's%20theorem%20on%20primes%20in%20arithmetic%20progressions.md). Fix an [arithmetic progressions](arithmetic%20progressions) $A = A(a, d) = \ts{a, a+d, a+2d,\cdots}$ for $a,d$ coprime and define $\pi_A(x) \da \ts{p\in A \st p\leq x, p\text{ prime }}$; then GRH implies an asymptotic estimate in $x$:
$$
\pi_A(x) \sim \phi(d)\inv\int_2^x {1\over \log t}\dt + \bigo\qty{x^{{1\over 2} + \eps}}
$$
- GRH implies a weak form of [Goldbach](Goldbach).

## EGRH

More generally, the [Dedekind zeta function](Unsorted/Dedekind%20zeta%20function.md) $\zeta_K(s)$ of a [number field](Unsorted/number%20field.md) $K$, the conjecture is that if $s$ is a zero with $\Re(s) \in (0, 1)$, then $\Re(s) = {1\over 2}$.

Conditional results:

- An effective version of [Chebotarev density](Unsorted/Chebotarev%20density.md).

## Computing zeta(2)

Compute $\zeta(2)$:  consider $f(z) \da \pi \cot(\pi z)$, whence $f(z) = f(z+1)$, has poles at every $z\in \ZZ$ with residue 1, and is bounded at $i\infty$.
 
 Then:
 
 ![Pasted image 20211029020500.png](Pasted%20image%2020211029020500.png)
 
 ![Pasted image 20211029020511.png](Pasted%20image%2020211029020511.png)
 
 ## Notes
 
 ![Pasted image 20211029020530.png](Pasted%20image%2020211029020530.png)

# Primes

![](attachments/Pasted%20image%2020220504204919.png)

See [Connes operator](Connes%20operator):
![](attachments/Pasted%20image%2020220504205139.png)