My musings about Riemann Hypothesis

Youtube math videos are what I tend to watch both to wake up in the morning, and wind down enough to sleep. This includes:

If I’d had 3blue1brown’s π characters to help me in my undergraduate physics course, I would have easily better a letter grade higher. Unless, you know, I had decided I really needed to contribute to his python animation libraries, in which case, my grades might been even lower…

I watch them on the Roku TV, interspersed with

And I’ve regulary gotten up early to watch the FreeRide World Tour:

And I guess I’m now getting 80% of my (international) news via:

I’m now up to episode 44 of Zeta Explained, and I just watched a second version of the Prime Counting function history from PhysicsExplainedVideos. Some of the intermediate videos, in the 20 to 30 number do not always explain what the motivation for that explanation. Trust that it will come back in a later video. In this business of explaining things first, Mr.Zeta is like many Neurodivergents :-) Like the Zeta Explained #32: Möbius Inversion, Lindelof, and The Diophantine Approximation video.

This series is a huge effort, and very appreciative. Mr. Zeta is sometimes repetitive on some things, and while he appologizes if the viewer hasn’t done any complex analysis (I have, that was a reason I repeated third year physics), it’s not really clear to me that anyone who hasn’t done some (and forgotten a lot) is going to get it.

Everytime he says, “And if you believe me..”, and I hear Max Smart each time: https://www.youtube.com/watch?v=uWIHDZz0WW8, https://www.youtube.com/watch?v=bA-UPrK_x-k … “A boy scout with rabbies…”

Videos 39/40/41 with the Riemann paper on the left leaves the content kinda small, particulary with me on the couch, watching on TV…

Mr.Zeta’s voice sounds identical to an IETF colleague of mine. Uncanny.

Now that I’ve shared my watch list, and high-level thoughts… on to my very detailed ponders.

Riemann was able to map the Euler product version of the Zeta Function (which clearly involves primes!) into this J(x) weighted prime counting function. To recover the non-weighed π(x), one needs to subtract off the effects of powers of primes up to x^(1/2). This implies that one has to know what the primes are up to that number. I’m pretty sure this is built into the definition of the Möbius Inversion.

If not for this, I think, given some large number G, to find the next prime one could otherwise calculate π(G), then π(G+2), π(G+4), … (well, the divisibility test for three is easy (even in binary), and there are a few other easy to eliminate intervals))… and if π(G+n) > π(G), then there was a prime between G and G+n.

I think, if one was starting from 1 that calculating up to π(G) is okay, because along the way, one has to calculate π(G^{1/2}), so if one is able to remember all the primes up to this huge number, then doing the corrections is “easy”.

I was thinking at first that eliminating the weighted jumps of J(x) might be easy as they less than one, only as the PhysicsExplainedVideos shows, J(100) = 28…, while π(100)=25, so there are at some places where the number might cross an integer due to the sum of the weights. But, between J(100) and J(102), it would have to go up by at least one, since 101 is a prime. So maybe this could be feasible?

Related is that calculating J(x) requires knowledge of some number of non-trivial zeros of the zeta function. They are believed to be irrational, so there is no closed form. And there are an infinite number of them. Calculating those numbers might be more difficult (time and memory complexity) than just finding big primes by Sieve method. I don’t know… I always pondered that Sieve test back in the 1980s, when that was the go-to test for integer math on microprocessors.

So, the first related question is: what is the error on J(x) as a function of how many non-trivial zeroes are used, and how precisely those non-trivial zeros are known? How expensive are the zeros to calculate? (I think the voyager golden record contains the fine-structure of hydrogen, I would have added some non-trivial zeros…)

I spent a lot of brain cells trying to understand why the critical line is at 0.5 and not another number. I see the 1/2 pop out in episode 20, Hadamard’s Factorization, … nope,not there… Oh, yes, in the Euler-Maclaurin Summation. It’s right there as a stand-alone real term. The rest of the terms could be complex, and zeros would be when the other terms work out to be -1/2.

In the end, the symmetry of ζ(1-s) = ζ(s), implies the mirror is at s=1/2. I think that the offset of the pole to s=1, is also reflected in the difference between Γ(n) = (n-1)! I was pondering why, if ζ(1) is a pole, why isn’t ζ(0) also a pole since s=0, ζ(1-s) = ζ(s), as ζ(1-0) = ζ(0)? I think it’s because, when extending ζ(s) to values s<1, using Riemann’s Xi Function, which includes Γ(s/2) as well, which also has a pole at s=0, and I kinda feel like maybe that helps. (Episode 15, 17 and end of 42)

(EDIT: well, watch Zeta Explained #46: Zeta at Zero. https://www.youtube.com/watch?v=1tBeocmw8bE Yes. (1/2)2Γ(s/2) => Γ(1+s/2))

So my next question is: what would happen to the J(x) prime counting function is one of the non-trivia zeroes was not on the critical line? I think that means, steping forward to episode 44, Primes as Waves… minute 8:20, ρ = 0.5 +- iα, that if α = a + ib, then ρ = 0.5 + i(a+ib) = 0.5-b + ia. That is, if α is complex , then ρ is not on the critical line.

This would change the expansion of Li(), and the denominator x^(0.5) would now be x^(0.5-b). I think that changes the damping factor, but not the oscillation?

Well, while I used to play a VLSI designer on TV, I don’t do that anymore, and I’ve never been a number theorist, not even on TV.