Finding the nontrivial zeros of the Riemann Zeta Function using Desmos

In this video I discuss extending the Riemann Zeta Function using the Dirichlet Eta Function, and use this along with Desmos to find the first few nontrivial zeros.

All nontrivial zeros of the Riemann Zeta Function so far calculated by humans (there are about 10^13 of them) all have real part 1/2, which falls in line with the Riemann Hypothesis which says that all of these zeros have real part 1/2.

The Riemann Hypothesis is a currently unsolved math problem, and is attached to a $1,000,000 prize awarded by the Clay Mathematics Institute.

[Windprinc]

You made this abstract thing easy to understand. This is the first time I understood the terms “continuation” and “analytic continuation” and the graphs made things much more interesting 👍🏼

[philip]

You are so eloquent and chose great backing music. I also like how you overlayed yourself over the drawing board. Good job!

[txikitofandango]

Very cool and gives me ideas for how to evaluate other functions and solve other equations in desmos, with the right setup

[madhuragrawal]

Fascinating, clear, and very well presented! Thank you so much, I'm looking forward to more exciting videos from this channel

[HurricaneEmily]

This was a start for me. I still don’t know how to calculate the zeta function but at least I know more than I did before. I didn’t know how to raise a natural number to a complex power. Is there a way to calculate the zeta function without using an app? You mentioned integrals.

The feedback I would give on the video is that it would be much easier to see the math without your face in frame. Maybe you could introduce a video with your face and then just let the math take up the screen. I couldn’t see it very well in the first part and I couldn’t read the graph at all when you were using the app.

But very cool. Thank you for the hard work.

[piee]

Feels good to see someone use a tool to its full potential. Loved it!

[Cully]

Excellent presentation to a challenging subject. You make it seem all so natural. Followup graphics very illustrative.

[tim]

My goal is to understand the comments and the video, both equally great.

[EyalRotem-iw]

Thank you for the excellent explanation. I'm glad to discover your channel.

[hotmole]

Thank you for the wonderful explanation, it had a flow and I learned a ton

[WildEngineering]

damn bro hopefully you see you have a future on YouTube and upgrade your camera!
edit* the proof you finished at 4:18 is beautiful

[carlowood]

I am adding this comment after watching the first 7 plus something minutes. But it refers to the time stamp of 4:53. You show that the complex sum converges AT LEAST when Re(s) > 1, using the triangle inequality. But then you go on and say "we proberen that this ONLY converges when Re part is greater than one". Huh? That triangle inequality is FAR off from the actual value of the sum! I am convinced that the sum will perfectly *converge* for real parts of s less than 1, as long as the imaginary part non-zero. I trust that analytic continuation will also work for those cases, but you can't say that the sum will (always) diverge if Re(s) <= 1.

[owenpawling]

I did this a while ago in desmos. Very good!

[pietro]

Great video, thank you! I just wanted to say that the 'false intersection' at 18:13 it's not actually false, the eta function has zeros on the line x=1, the first one is at 2pi/ln2 which is a little bigger than 9, exactly where the intersection occurs, so it shouldn't go away when k is larger.

[DolphyWind]

Cool video! I am looking forward to study complex analysis by myself in the future!

[nightowl]

The two curves always seem to intersect at right angles when you have a riemann-zero. Is that a coincidence?

[tylosenpai]

Thanks, i can't find this "Analytic" calculation anywhere else that i can understand

[rva]

15:20 so both expressions need to be zero, but then in the graph intersections are searched, not zeros of each expression, then there I get lost!

[Wotsit]

@andrewdsotomayor

Re Transcript lines 19:03 to 19:39. I have always been curious about proving rigorously that the real and imaginary parts of the Zeta Function do in fact converge to zero at the exact same point and also on the line Re(s) = 0.5. Here you are showing that you get “really close”, but how can you be sure that if K was unimaginably large that the intersection doesn’t miss the line Re(s) = 0.5? I don’t think this is sufficiently rigorous unless it’s detailed somewhere else.

This is only a check on the first non-trivial zero and it seems that you need to go “infinitely far” along your series to nail this. Similar in a sense to the Riemann Hypothesis itself where you have to go “infinitely far” up the critical zone to check the location of all the zeros.

Grateful for clarification please.

Nonetheless this is an excellent visualisation of the issues. Thank you for this!

[Rhino-Flea]

well... purely from the desmos pic... is there a way to "get rid of" the curves that don't cross the y-axis and the ones that have no bounded x value so as to simplify this question?