This is a really nice explanation -- it flowed really well, and I liked the sieve approach to proving the product-over-primes formula (the other standard way relies on a clever trick, and being clever is fun-but-not-good-for-exposition).
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.
If you like this topic and didn’t read all the way to the bottom, the author’s recommendation of “Prime Obsession” by John Derbyshire is spot on. It’s a fantastic book.
> 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.
I think they're mislabelled and misdescribed a bit, and what those graphs actually show is what you get if you (1) compute an approximation to J using only a limited number of Riemann zeros, and then (2) use the inversion formula to compute what pi(x) would be if #1 had given you the exact value of J. So the y-axis shouldn't be labelled "J(x)", it should be labelled something more like "pi(x)".
Caution: I haven't actually done the computation and checked.
[EDITED to add:] Now I have, or at least one of them, for the "first 35 roots" one. I took "first 35 roots" to mean "first 35 with positive imaginary part, plus their complex conjugates". I got a graph that was much wigglier, and a much closer match to pi(x) than the one in the article. So then I thought maybe it was meant to be 35 roots in total -- though you really do want to take those conjugates in pairs, so the odd number is strange. Anyway, I tried with the first 17 pairs: still much too wiggly. With the first five pairs of zeros, I get a good (but not perfect) match for the graph in the article.
It is easy to take prime number 2 (from the famous treatise A Short Table of Even Primes) and "divide it into" 6. And it is easy to take 6 and factor it into primes 2 and 3.
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.
It's the very same thing that struck me right away. The English is off, even though I understood perfectly well what the author intended to communicate. Then it clicked, he's Norwegian, and most assuredly English is a second language.
> [The Euler Product Formula] states that the sum of the zeta function is equal to the product of the reciprocal of one minus the reciprocal of primes to the power s. This astonishing connection laid the foundation for modern prime number theory...
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):
It may boil down to the way we view the problem: whether to focus on the zeta values or the structure of the function.
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.
Book 'Prime Obsession' by John Derbyshire is also very good detailing the history and basic math behind Riemann Hypothesis. It is very accessible, well written and captivating.
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-...