New equation of prime-zeta? part 1

consider the arithmetic function

Aa(n)={1,n is prime0,otherwiseAa(n)=\begin{cases} 1 &, \text{n is prime}\\ 0&,\text{otherwise}\end{cases}

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

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

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

so: ya(n)={k(1)k,n=p1p2p3...pk(1)k,n=p1p2p3...pk1pk20,otherwiseya(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 ω(n)\omega(n) is the nmber of prime factors of n and μ(n)\mu(n) is the möbius function


part-2

Note by Aareyan Manzoor
3 years, 8 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

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

[example link](https://brilliant.org)example link
> 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 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

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

Aareyan Manzoor - 3 years, 8 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...