The Riemann Hypothesis, explained

I always wanted to extend my tutorial[1][2][3] with a 4th part, to make the connection between Euler Gamma function and Riemann Zeta Function.

I will link to your article, it's really well written.

[1] https://mourafiq.com/2015/08/30/extensions-of-the-factorial-...

[2] https://mourafiq.com/2015/09/09/extensions-of-the-factorial-...

[3] https://mourafiq.com/2015/10/09/extensions-of-the-factorial-...

I did get hung up on the complex graph showing the first few non-trivial zeros. It's a very busy graph! I ended up realizing that you graphed the contours Re(zeta(s))=0 and Im(zeta(s))=0.

E.g. "Riemann hypothesis likely remains unsolved despite claimed proof" https://www.newscientist.com/article/2180504-riemann-hypothe...

> 24 September 2018

> Michael Atiyah claims to have found a proof for the Riemann hypothesis

> One of the most famous unsolved problems in mathematics likely remains unsolved. At a hotly-anticipated talk at the Heidelberg Laureate Forum today, retired mathematician Michael Atiyah delivered what he claimed was a proof of the Riemann hypothesis, a challenge that has eluded his peers for nearly 160 years.

Because you define J(x) as https://cdn-images-1.medium.com/max/2000/1*EjlcuLa3Z_wnmYREs... so it shouldn't really approximate pi(x) that well.

Where the OP wrote

> Those numbers you can’t divide into other numbers, except when you divide them by themselves or 1?

would have read better with

"Those numbers you can't factor into ..."

I.e., it is easy to take 2 and "divide it into 6" but can't factor 2 into a product of other numbers except itself and 1.

Here is the Euler formula for middle school.

We start with an infinite product of power series for all primes:

(1 + 2 + 2^2 + ... + 2^k + ...) *

(1 + 3 + 3^2 + ... + 3^k + ...) * ...

(1 + p + p^2 + ... + p^k + ...) * ...

Let's open the parentheses in an orderly manner without running into infinity too prematurely (i.e. taking only a finite number of non-unit terms in each product):

1 + 2 + 3 + 2^2 + 5 + 2 * 3 + 7 + 2^3 + 3^2 + 2 * 5 + 11 + 2^2 * 3 + ...

Now, if we apply any completely multiplicative function f(ab) = f(a)f(b) to the terms we get

(1 + f(2) + f(2^2) + ... + f(2^k) + ...) *

(1 + f(3) + f(3^2) + ... + f(3^k) + ...) * ...

(1 + f(p) + f(p^2) + ... + f(p^k) + ...) * ... =

1 + f(2) + f(3) + f(4) + f(5) + f(6) + ...

When f(n) = n^(-s) we get the zeta function on the right and an infinite product of converging sums (s > 1) on the left.

The converging sums S = 1 + p^(-s) + p^(-2s) + ... are of course 1/(1 - p^(-s)) as S = 1 + S * p^(-s).

Voilà,

  1 + 2^(-s) + 3^(-s) + ... = 1/(1 - 2^(-s)) * 1/(1 - 3^(-s)) * ... * 1/(1 - p^(-s)) ...

SUM m(n) * n^(-s) = (1 - 2^(-s))(1 - 3^(-s))(1 - 5^(-s))...(1 - p^(-s))...

(just open the parentheses on the right, that's what middle-schoolers do).

And what happens if we multiply our power series S by 1 - p^(-s)? They cancel each other, so the whole product is 1.

The "astonishing connection" of the Euler formula turns out to be the fundamental theorem of arithmetic in disguise.

The true mystery can be the viewed from a vector angle, i.e. view the zeta value as $\zeta(c) = V(c) \bullet V(0)$, where $V(x) = (1^{-x}, 2^{-x}, 3^{-x}, ...)$ for x a complex variable.

A more thorough treatment is @ https://read.barnesandnoble.com/book/egg-n-our-face-3#1

--- Zeta In Terms Of Vectors for RH

Anyone with a little more in-depth knowledge able to comment on this supposed proof?