Math Encounters - Primes and Zeros: A Million-Dollar Mystery

How can we quickly determine how many primes there are less than some huge number? The great mathematician Georg Friedrich Bernhard Riemann proposed a solution 160 years ago — but we still haven't been able to verify his hypothesis. Join mathematician Brian Conrey, founding Executive Director of the American Institute of Mathematics, as he shares some of the colorful history surrounding the world's greatest-ever math challenge.

Math Encounters is a public presentation series celebrating the spectacular world of mathematics, presented by the Simons Foundation and the National Museum of Mathematics.

For further information, call the National Museum of Mathematics at

[RSLT]

Dr Brian Great lecture! and thank you you very much for explaining the history and importance of R.H.

[goldenera]

Best video that explains such a complex phenomenon in a simple language - thanks a bunch

[alvinuli]

When you get a call from an unidentified number you can say "I will pick the call if and only if it is a prime number", what is hard, is to do so before the call finishes.

[MostlyIC]

WOW, lots of gems in this talk. I've seen the graph at 40:00 for real-part = 1/2 in several other youtube videos and have always wondered what it would look like at something not exactly 1/2 and here we have it !!!

[monkeymathematician]

The graph shown at 49:55 made my first artificial neuron fire. Thank you Professor 😁

[LydellAaron]

Little engaged smart dude at 30:53 made my day. I love the idea of parents bringing their children to an event like this to inspire interest.

49:35 higher you go, more zeros. Sinusoids with increasing frequency might have this behavior. I've seen lots of zeros on multiplied sinusoids (in desmos graphing calculator simulations), but the blowups, I visualize as a form of complex exponential combined with wave construction interferences.

That Mexican hat wavelet (aka Ricker wavelet aka Higgs wavefunction) is symmetric and capable of blowing up like this too, and then settling extremely quickly. It also has adjustable sharpness by adjusting sigma (I have been playing around with that special Wavelet in desmos graphing calculator).

You'd need somehow to combine those various qualities, which you can only do with wave functions. 1:07:44 a spectral interpretation as he puts it.

In my own observations on primes, prime frequencies never cross zero at the same point, within a 1Hz 2π period. Went and filed a patent pending to exploit this interesting property for quantum computation.

He suggests describing orthogonal gaussian matrices at 1:09:00 but after playing with the Ricker wavelet, that model looks more suitable, and it's not a random distribution. So a matrix of orthogonal Ricker wavelets is what I would play around with. 1:10:30 Ricker is unitary and hermitian if I understand it right.

1:17:31 I'm glad they cleared up that random matrix theory is not actually random distributions, because I was mentally stuck there too. There IS an order (a spectral order of resonances) involved here even though their distributions appears random.

[flnvohz]

Watching the presentation I was like struck by lightning at 59:08, as 5 is not the first prime "of interest", but 3! I never realized, that the Clarke transform is linked to the Riemann Hypothesis in this way!

[bazsnell]

Wonderful stuff! Thank you for making these available to the public throughout the world.

[bjorntorlarsson]

I understand the quartets (like 101, 103, 107, 109) as occuring when the more significant figures don't add up to a sum divisible by 3, together with the 1, 3, 7 and 9 that all prime numbers must have as their least significant figure. If the figures of a number adds up to a number divisible by 3, then that number itself is divisible by 3.

[enlongchiou]

From every row of Pascal triangle deduce 1-2-1=-2, 1-3-3+1=-4, 1-4-6+4-1=-6, all trivial zero of zeta function, have such pattern from(x-1)^n of Pascal triangle, flip -, + sign from 3rd term on.

[Maria-szfc]

here for the 7 from the web: To determine whether a number is divisible by 7, you have to remove the last digit of the number, double it, and then subtract it from the remaining number. If the remainder is zero or a multiple of 7, then the number is divisible by 7.

[Larry-fw]

Can the faulty arithmetic of 9/11 be examined here ? Three thousand missing bodies seemed to totally eliminate any credibility of a building collapses theory given that no other examples exist in world history of such a low rate of recovery. Twenty five years hasn’t given us another example either . Add to that the fact it happened twice on the same day to two identical buildings at the same location and it certainly looks impossible.

[flnvohz]

When identifying the squares vector, when do we stop? In the case of P=5 we stop at 4, P=7 at 9. Is it, that we stop when an identified square is hit again? P=5: 4, 9; P=7: 9, 16 ?

[flnvohz]

As I do not really understand the consequences of the vectors addition as shown at 59:08, I may ask: what if we substract all vectors first and then add the square vectors two times? As all vectors, starting from 0 end up in a complete way round, can't the target be reached by adding the square vectors only?

[DavidBrown-omcv]

"... the list of primes goes on forever ..." According to Edward Fredkin, infinities, infinitesimals, perfectly continuous functions, and local sources of randomness are figments of human imagination and do not occur in nature. I conjecture that the Riemann Hypothesis is true but unprovable in ZFC, but ZFC and Peano Arithmetic are contrary to empirical existence. There is a saying on Wall Street: Trees do not grow to the sky. Do an arbitrarily large number of positive integers occur in nature? Consider some conjectures: (1) There are three fundamental levels of physics: classical field theory, quantum field theory, & string theory, (2) There exist positive integers W, X, Y, & Z — each greater than 1 and less than 10, 000 — such that the amount of classical information is < W^X, the amount of quantum information is < Y^ (W^X), and the amount of stringy information is < Z^(Y^(W^X)), (3) String theory with Fredkin's finite nature hypothesis suggests = (3.9±.5) * 10^–5 . Is Professor MIlgrom of the Weizmann Institute the world's greatest living scientist? Google "pavel kroupa dark matter" & "riccardo scarpa mond arxiv",

[davidwilkie]

If another aspect of POV is to use Euler's implied symbolically connected inference in the projection-drawing picture-plane, on the Blackboard that is, how effectively the innate "whole message" of logarithmic condensation-coordination vanishing-into-no-thing Singularity-point positioning is exposed to view depends on the assembly of functional abstractions shown us by our Teachers. Eg if we are familiar with the entangled connection of Absolute Zero-infinity reference-framing containment positioning NOW of/by i-reflection Singularity-point, the expectation is that all potential positioning possibilities are contained, recognisable within the holography dimensionality of e-Pi-i 1-0-infinity sync-duration, represented by the Unit Circle of Infinity @.dt instantaneously.

But the Mathologer's and 3BLUE 1BROWN Students know how to speak the million dollar language, to qualify.

[akhilin]

it amounts to "statistical arithmetics" .

[goldenera]

@33:44 why have you chosen Cosine function? should this not be divided by sqrt(X)? I mean should it be Cos (Gamma N Log(X))/ sqrt (X) ?

[alastairbateman]

WOW I can't believe it! In Dec 2017 I did a YouTube video titled 'Fermat's Last Theorem : 125=126' and now I find it confirmed by Prof Brian Conrey and the GREAT Carl Freidrich Gauss no less who got 125 and 126 respectively for one of their sums of the same data at 21 minutes into the video.

[alain-michelmace]

at 42 min you say there is no zeros on the side Re(z)<0. I fear ir is not exactly right, there are trivial zeros, z=-2n... but apart from that i liked your speech. thanks