Investigating primes is nearly as old as mathematics itself and its reasonable to assume other ideas where discovered in the hopes of applying them to various problems incorporating prime numbers.
From a practical, applied, perspective, “understanding” primes, that is making their “hidden” structure a known “truth”, would either confirm or deny the Riemann hypothesis wherein many other conjectures that assume the hypothesis to be true would also be “truely” known.
Or from TFA:
> …In the 19th century, research on these kinds of statements led to the development of much of modern number theory. In the 20th century, it helped inspire one of the most ambitious mathematical efforts to date, the Langlands program. And in the 21st, work on these sorts of primes has continued to yield new techniques and insights.
> …Their[the article’s sunbjects’] proof, which was posted online (opens a new tab) in October, doesn’t just sharpen mathematicians’ understanding of the primes. It also makes use of a set of tools from a very different area of mathematics, suggesting that those tools are far more powerful than mathematicians imagined, and potentially ripe for applications elsewhere.
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.
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.
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.
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 ;-)
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.
Investigating primes is nearly as old as mathematics itself and its reasonable to assume other ideas where discovered in the hopes of applying them to various problems incorporating prime numbers.
From a practical, applied, perspective, “understanding” primes, that is making their “hidden” structure a known “truth”, would either confirm or deny the Riemann hypothesis wherein many other conjectures that assume the hypothesis to be true would also be “truely” known.
Or from TFA:
> …In the 19th century, research on these kinds of statements led to the development of much of modern number theory. In the 20th century, it helped inspire one of the most ambitious mathematical efforts to date, the Langlands program. And in the 21st, work on these sorts of primes has continued to yield new techniques and insights.
> …Their[the article’s sunbjects’] proof, which was posted online (opens a new tab) in October, doesn’t just sharpen mathematicians’ understanding of the primes. It also makes use of a set of tools from a very different area of mathematics, suggesting that those tools are far more powerful than mathematicians imagined, and potentially ripe for applications elsewhere.