alt Hacker NewsWhat does the Riemann zeta function have to do with the distribution of primes?
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Personally I’ve found this article a much more approachable overview to the question in the title: https://golem.ph.utexas.edu/category/2019/09/the_riemann_hyp...
I’ve seen the connection between the Riemann zeta function and primes talked about but never an accessible explanation. Although I must say that the beginning was fairly easy to follow, I got overwhelmed about a quarter into it. Feels like I need a week of study to really comprehend the entire article.
Good article, but there is one step in the reasoning that rubs me the wrong way:
> This equation might seem a little hard to solve, but at this point you might notice something funny: $equation$
> So, if F′(x)log(x)=1,F′(x)log(x)=1, then
> Thus, our mystery function F(x)F(x) obeys F′(x)log(x)=1.F′(x)log(x)=1. From here you deduce F′(x)=1/log(x),F′(x)=1/log(x), so by integrating you get
But it does not seems that F needs to have that trait, just that 1 works in that instance. It's sufficient but not necessary. How can you tell that it is that simple solution which is the right one?
The Riemann Hypothesis is a weird thing. It is about as important as Fundamental Theorem of Algebra or Fundamental Theorem of Calculus in number theory. Yet we are nowhere near to proving it. And yet it is so powerful that we are trying to prove things assuming RH is correct.
Stupid question on toilet: if prime numbers are 1s and other integers are 0s. What do we get from this “digital” landscape?
There is a whole YouTube channel just about this subject. Tens of hours in total, in 30-50 min episodes handling little chunks of this matter (Zeta/Riemann/primes)
Easy to follow without requiring advanced math, great visualizations.
https://www.youtube.com/@ZetaExplained/videos
However needing tens of hours of video to explain what the Riemann Hypothesis is, without glossing over details, tells you something about it's difficulty (as a statement).
And as if all of that were not head-exploding enough, I'm still searching for an implementation of the zeta function for complex arguments...
Although the author turned out to be a fairly despicable person, Prime Obsession is an absolutely wonderful book on this subject.
What is the relationship between Riemann zeta function and Zipf's law? And between words and primes?
I love that this is a site solely dedicated to the subject of interesting Diophantine equations, written by two mathematics PhD students. Going deep on a narrow focus, I respect that approach.
The article itself feels a bit long without a satisfying payoff at end. But then again, the journey is entertaining even for a non-specialist, and the lack of a strong conclusion is due to the unsolved problem of the Riemann hypothesis. The open-ended question is probably irresistible for some personality types that can't stand the suspense and demand a resolution.
Perhaps a way to strengthen the ending is to explain why the question of the distribution of prime numbers is worth solving, and what are the larger implications.