How the Riemann zeta function reveals the hidden distribution of prime numbers
What does the Riemann zeta function have to do with the distribution of primes?
I explore how Euler connected the infinite sum of prime reciprocals to the Riemann zeta function using the Euler product formula. By bridging number theory and calculus, I demonstrate that the divergence of the harmonic series proves there are infinitely many primes, revealing a profound link between simple arithmetic and complex analysis.
The Euler product formula is a way of encoding unique prime factorization in calculus, serving as a vital bridge between number theory and the study of infinite series.
- bradrn
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...
- lioeters
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.
- briandw
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.
- dist-epoch
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).
- bwfan123
Love the writing style, and I wish all math was written this way. These articles are accessible to anyone who has had some high school math background, and paints a historical backdrop on how the sausage was made (with links to the original papers). Some of these ideas took millenia to develop, and when presented in neat capsule, students are cheated of the mystery and struggle that went into the progression and development of the ideas. I would argue that all math education has to be re-invented to follow this style, ie, uncover the historical progression, wonder and mystery.