# New equation of prime-zeta? part 1

consider the arithmetic function

$$Aa(n)=\begin{cases} 1 &, \text{n is prime}\\ 0&,\text{otherwise}\end{cases}$$

Read dirichlet convolution and dirichlet series . we have $ya=\mu*Aa=\sum_{p|n} \mu\left(\dfrac{n}{p}\right)$ This is because $$Aa$$ disappears over non-primes. Consider the case

1. n is square free.then $$\mu\left(\dfrac{n}{p}\right)=-\mu(n)$$ and when we add over all prime factors we get $$-\omega(n)\mu(n)$$.

2. Not squarefree but only one prime factor is squared. then it's möbius will be zero everywhere except the repeated prime factor. so we get $$(-1)^{\omega(n)}$$.

3. more than one prime factor is squared. then it will simply be zero.

so: $ya(n)=\begin{cases} -k(-1)^k&, n=p_1p_2p_3...p_k\\ (-1)^k &,n=p_1p_2p_3...p_{k-1}p_k^2\\ 0 &,\text{otherwise}\end{cases}$ I will continue this in part 2. Reshare if you enjoyed this.

btw $$\omega(n)$$ is the nmber of prime factors of n and $$\mu(n)$$ is the möbius function

part-2

Note by Aareyan Manzoor
2 years, 9 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

• bulleted
• list

1. numbered
2. list

1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

> This is a quote
This is a quote
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

I am writing part two now, where i will show the connection to prime zeta.

- 2 years, 9 months ago