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It's not clear to me what could be undecideable about the original proposition; either the continuation matches at the points in question, or it doesn't, and if it doesn't, how is it a continuation?
Enlightenment welcome.
Earlier quoted context omitted.
Man HR is harsh as hell. While yes, if a proof of RH is found it will probably use advanced tools, ever finding a proof with elementary tools is not totally out of the question. Regarding the Prime Number Theorem, which can be seen as a baby version of RH, experts in the field such as Hardy have been quoted as saying elementary proofs would not exist. Then 50 years after Hadamard and Poussin found the original proofs…
I've taken math up to calculus and linear algebra and enjoy watching 3B1B proofs. Is that enough to follow the elementary Prime Number Theorem proof? I've been interested in understanding that for a while. Any good links?
It was easy for me to follow along.
Zagier described the one that Ono gave him as "a cute problem about the asymptotic behavior of certain polynomials involving Euler's partition function, which is an old love of mine and of Ken's—and of about pretty much any classical number theorist."
"I found the problem intractable and I didn't really expect Don to get anywhere with it," Ono recalls. "But he thought the challenge was super fun and soon he had crafted a solution."
That sounds like a potentially useful problem-solving trick - pull a small but key problem out of your bigger problem, and then nerd-snipe your friends into solving it.
Four weeks pass, but nothing happens. Half a year later, the devil shows up again – in a rather gloomy mood.
"I’m sorry", he says. "I couldn’t prove the Riemann hypothesis either. But" – and his face lightens up – "I think I found a really interesting lemma…"
Neat. I mean, I don't understand the mathspeak since I stopped taking math after calculus in college, but it's neat to see fun approaches turn into viable tactics for finding a solution. By coincidence I toyed with the problem a bit this past week. Pretty difficult to tackle with a "basic" math toolbox (calc, trig, algebra, and some discrete math/number theory that's taught in many CS undergrad curricula). Much like…
Oh really? :-D
- reframe the real part of a complex input s as 1/2 + k for -1/2 - come up with a formula to sum the series. I don't have much math know-how in this regard, except for some tricks I learned in discrete math/algos class so I quickly ran out of ideas. Pairing sum terms together (like you can do to prove the sum of telescoping series, or of natural no's from 0 to N) doesn't seem to work.
- The original sum formula looks like sum(n=1, infinity, ((-1)^(n+1))/(n^s)) where s is the complex input. I tried to turn it into a continuous function by using -cos(pi*n) for the numerator and then use a definite integral, but the result is nasty before you even plug in the integral bounds. There's an imaginary term as the second argument for the gamma function, which I don't understand how to handle.
- show two different real values, r1 and r2, of zeroes at r1 + ci and r2 + di for some imaginary parts c and d. Then show the ratio r1/r2 = 1, and since we know r1 = 1/2 works, we know 1/2 is the only possible real part that works. Much like the summing stuff above, I couldn't figure out how to simplify things enough to even get an equation in terms of either r1 or r2.
- show that two zeroes on either side of the line s = 1/2 + iy cannot exist, or at least something like if one exists then the other must not exist, which contradicts the notion that we'd have two zeroes on either side of the line, where one has a real part between 0 and 1/2 (exclusive) and the other is in (1/2, 1). I don't have a good way of actually attacking this idea though.
There are some other ideas I tried, but the main problem is I can't simplify the sum. The sum itself plus the fact there's exponents in every denominator makes it especially hard to simplify algebraically, or at least I don't know how to do it.
Neat. I mean, I don't understand the mathspeak since I stopped taking math after calculus in college, but it's neat to see fun approaches turn into viable tactics for finding a solution. By coincidence I toyed with the problem a bit this past week. Pretty difficult to tackle with a "basic" math toolbox (calc, trig, algebra, and some discrete math/number theory that's taught in many CS undergrad curricula). Much like…
Man HR is harsh as hell. While yes, if a proof of RH is found it will probably use advanced tools, ever finding a proof with elementary tools is not totally out of the question. Regarding the Prime Number Theorem, which can be seen as a baby version of RH, experts in the field such as Hardy have been quoted as saying elementary proofs would not exist. Then 50 years after Hadamard and Poussin found the original proofs…
The idea for the paper was sparked two years ago by a "toy problem" that Ono presented as a "gift" to entertain Zagier during the lead-up to a math conference celebrating his 65th birthday. A toy problem is a scaled-down version of a bigger, more complicated problem that mathematicians are trying to solve. Zagier described the one that Ono gave him as "a cute problem about the asymptotic behavior of certain polynomia…
Both methods have yielded a lot of progress in mathematics. For example, solving a cut down version of Fermat’s Last Theorem got us the theory of ideals in algebra. Relatedly, Wiles work on the modularity theorem ultimately led to the actual proof of FLT.
I've seen a paper[1] that claims RH, as stated, is undecideable, and then proposes to prove that an analytic continuation of it has the property required. The idea seems to be that you can choose your continuation to have or not have the property. It's not clear to me what could be undecideable about the original proposition; either the continuation matches at the points in question, or it doesn't, and if it doesn't,…
However, the paper is clearly a serious attempt by someone inside academia who works in quantum computing, which is a related and heavily mathematical field.
You may have just given me a few hours of weekend reading material. :)
Earlier quoted context omitted.
Man HR is harsh as hell. While yes, if a proof of RH is found it will probably use advanced tools, ever finding a proof with elementary tools is not totally out of the question. Regarding the Prime Number Theorem, which can be seen as a baby version of RH, experts in the field such as Hardy have been quoted as saying elementary proofs would not exist. Then 50 years after Hadamard and Poussin found the original proofs…
I agree, I'm not saying a simple proof is ever out of the question either. But my failed attempt makes me wonder how much of a creative burst a typical person would have to have in order to come up with a simple solution in this case.
Earlier quoted context omitted.
Man HR is harsh as hell. While yes, if a proof of RH is found it will probably use advanced tools, ever finding a proof with elementary tools is not totally out of the question. Regarding the Prime Number Theorem, which can be seen as a baby version of RH, experts in the field such as Hardy have been quoted as saying elementary proofs would not exist. Then 50 years after Hadamard and Poussin found the original proofs…
I've taken math up to calculus and linear algebra and enjoy watching 3B1B proofs. Is that enough to follow the elementary Prime Number Theorem proof? I've been interested in understanding that for a while. Any good links?
Here’s one version of the proof: https://www.math.lsu.edu/~mahlburg/teaching/handouts/2014-72...
See also:
https://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute....
https://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute....
And, a related, very simple and elementary proof of Chebyschev’s theorem that there is always a prime number between n and 2n: https://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulat...