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The Riemann hypothesis makes me feel dumb - not just because I can’t solve it, no great shame in that - I genuinely get lost in amazement and wonderment by the

The Riemann hypothesis makes me feel dumb - not just because I can’t solve it, no great shame in that - I genuinely get lost in amazement and wonderment by the mind that develops a function, graphs it, and gleams some insight into numbers. Something about it I find humbling and makes me think about the archetype of mathematicians that lose their minds to numbers.

ykonstant 2y ago
It is mesmerizing, but do note it was not a single mind that produced this insight. It was centuries of work. It involved, among many others: 1. Newton and the Bernoulli family developing the theory of infinite series and connecting them to discrete sequences, 2. Wallis developing the first notions of infinite products and demonstrating the first non-trivial convergence of such, 3. Euler solving the Basel problem and linking the zeta function to the prime numbers (giving a new proof of the infinitude of primes), 4. Gauss and Eisenstein further using Euler's ideas and their own unique algebraic insights to understand primes in arithmetic progressions, and finally 5. Riemann taking the zeta function, putting it in the complex plane, revealing the unifying theme connecting the previous discoveries and making his own fundamentally new discoveries with the explicit formula. And of course the development only accelerated from that point on.
- airstrike 2y ago
  Thank you for this. I've favorited this comment so that I can read on each of these to sate my curiosity. Now I'm off to search for accessible resources for these topics for those of us non-mathematicians ;-)
  - ubnvfft 2y ago
    I think once you understand how to apply analytic continuation to the problem its relation to primes is much more apparent; even without a full understanding of the history. https://en.m.wikipedia.org/wiki/Analytic_continuation https://en.m.wikipedia.org/wiki/Analytic_continuation
- throwawaycities 2y ago
  That’s exactly how I begin to put it into context and rationalize this kind of work - he was a mathematician so this the kind of thing he worked on, and he was also working on a body of maths and knowledge. It’s much like physics and the great physics experiments throughout history for me, some of them I’d like to think I may have been able to develop, but others I just marvel at the ingeniousness of the experiments. Realistically in a vacuum I doubt I’d have even identified/defined prime numbers.

ykonstant 2y ago

It is mesmerizing, but do note it was not a single mind that produced this insight. It was centuries of work. It involved, among many others: 1. Newton and the Bernoulli family developing the theory of infinite series and connecting them to discrete sequences, 2. Wallis developing the first notions of infinite products and demonstrating the first non-trivial convergence of such, 3. Euler solving the Basel problem and linking the zeta function to the prime numbers (giving a new proof of the infinitude of primes), 4. Gauss and Eisenstein further using Euler's ideas and their own unique algebraic insights to understand primes in arithmetic progressions, and finally 5. Riemann taking the zeta function, putting it in the complex plane, revealing the unifying theme connecting the previous discoveries and making his own fundamentally new discoveries with the explicit formula. And of course the development only accelerated from that point on.

airstrike 2y ago
Thank you for this. I've favorited this comment so that I can read on each of these to sate my curiosity. Now I'm off to search for accessible resources for these topics for those of us non-mathematicians ;-)
- ubnvfft 2y ago
  I think once you understand how to apply analytic continuation to the problem its relation to primes is much more apparent; even without a full understanding of the history. https://en.m.wikipedia.org/wiki/Analytic_continuation https://en.m.wikipedia.org/wiki/Analytic_continuation
throwawaycities 2y ago
That’s exactly how I begin to put it into context and rationalize this kind of work - he was a mathematician so this the kind of thing he worked on, and he was also working on a body of maths and knowledge. It’s much like physics and the great physics experiments throughout history for me, some of them I’d like to think I may have been able to develop, but others I just marvel at the ingeniousness of the experiments. Realistically in a vacuum I doubt I’d have even identified/defined prime numbers.