My math professor once said, “I’ve know the existence of these math problems for many years. And I assure you, there are a lot easier ways to make a million dollars”

grigori perelman doesn't agree.

Even if the prize was a billion dollars, no one could solve it

@kruth 666 If you were smart (which you would be) and capitalised on it, you could make a hell of a lot more than a million dollars.

@kruth 666 "have you saved the entirety of mankind recently, including the possible future universe? if so, click here for your chance to win a new Kia."

So he was a millionaire then?

Can I just appreciate how well the animation is? Literally, WOW.

Do you know where california is?

Ikr, as a motion artist myself my jaw was dropped throughout the video

No, you cant

Figuratively, WOW.

Great production

I really appreciate that you explain the more “basic” things (e.g. what a log function is). It makes the video feel welcoming to people who aren’t necessary very good at math (like me, lol)

yeah but then other parts of it they just brush over like it's nothing

Log function is just a reverse function of exponential function.

@emigoldber i dont even think it was intended but it is pretty good

nice pun

"The proof of the Riemann hypothesis is trivial and left to the reader as an exercise" - Bernhard Riemann

@Nen Master5 whats the epic sci-channel?

@Nen Master5 Owen is great, but please don't go advertise unrelated creators in the comments. it's considered spam.

and he died before he could write it down... so sad, so sad

@Nen Master5 can we just have the epic sci channel instead?

@Emilio Prato its a satirical parody

Thank you, Quanta Magazine. My understanding of the Riemann Hypothesis went from 0% to 15%. Great job (I mean it).

15%? sheesh, i guess ur a bit off by about +14.999997%

Whoever does these animations, massive props to you. These are literally the best math illustrations I've ever seen.

I@Hans Meiser y

@John from tennessee i

@Hans Meiser 3Green1Red

Agreed - I'm an uneducated outsider - but my daughter loves math , even though she failed classes in high school because her strong synesthesia caused her to fail classes in high school because she's make arithmetic mistakes and couldn't finish the answers on time. She's always looking for ways to show me why she finds these ideas so "lyrical and lovely" ( her phrase") and this gives me some idea of what she enjoys

It says at the end Guan-Huei Wu Clay Shonkwiler

This is like becoming an astronaut, discovering a previously unknown planet, finding a river on that planet, and at the bottom of the river is the perfectly fitting other half to a broken rock you found in a river on Earth as a kid. The Universe sees the look on your face and laughs silently.

@Sasha Botalova LMAO

garbage analogy and nonsensical as well :)

@Ramaraksha He never mentioned a magic man.

When pure mathematics comes with lucid explanations, and the two are complemented by a perfect vanilla icing of aesthetic graphics. A million thanks for this amazing presentation.

Brilliant explanation. This makes me love math even more. There is so much beauty and mystery in mathematical patterns.

For the first time in my 46 years, I have truly understood what the Riemann Hypothesis actually is. Thank you!

Literally if my math teacher had just said “logarithms are to exponents what division is to multiplication,” I would have had much less trouble with them. Thanks dude

​@InfluxDecline It depends on the underlying function, i.e. if 2^3=8 is the application of the function f(x)=x^3 to the argument x=2 then its inverse function (in the R domain) is g(x)=cbrt(x). If it is the application of the function f(x)=2^x to the argument x=3 then its inverse function (in the R+ domain) is g(x)=log2t(x).

@InfluxDecline In Multiplication you have a * b = b * a so there is only one inverse operation. In exponentiation, a ^ b is not equal to b ^ a - we need two inverse operations which are roots and logarithms.

@refresh I suggest you never read Ayn Rand, or read her and then read Kierkegaard or Descartes, or Plato...any philosopher who will rinse the nasty idiocy of Rand from your mind forever.

Personally I dont understand that description at all. I have used logs a lot in my research. I found that just using them, and seeing how they straighten out a regression line was enough. Also, they're useful for the normal distribution, because no matter how far you go in terms of Standard deviations below the mean a negative log (e.g. log -1) is always a positive number, whereas with "raw" numbers, you rapidly get into negative numbers. We can be said to live in a logarthmic universe.

@rhombicuboctahedron You could say “What is the log to the base n of sqrt(n),” and the answer would be 1/2. The log here is the power 1/2.

Really well done!! I am not a mathematician but it feels my brain was struck by lightning. Great feeling.

Well done! Great animations go a very long way to illuminating the discussion which is as relatively simple and clear as possible. Thank you.

I loved this video and the math explanations. I could like it 10 times if possible. Great explanation. That is an example of how math should be taught. I am an engineer and at university I had a few good professors, but no professor was as good as this video.

Thanks for making this video, even though its only touching the surface it was really informative and easy to understand this problem and even harder to solve.

WHO WANTS TO BE A MILLIONAIRE?! Mathematicians: "No thanks..."

They do not use proper tools to solve the problem.

@Jack Schwartz the issue is convergence-divergence -- infinitely many 'infinite equations' can be solved.

@ONXsDITTU WAT no. still an outstanding problem. cripes.

Lets see if I get deleted again

@H B I pointed out that it's unsolvable due to the fact that the answer is an infinite equation thus unsolvable, and they deleted it. SMH Take Care All

Thanks. Wonderful ☺️. When you're working with waves and need figures as we require, a reduction of complexity makes subjects understandable as you've so aptly done.

Brilliant explanation. This video will help add many different layman’s perspectives on this problem, if only for a few moments, and hopefully help crack it.

Im so glad I found this video when it was released. Rewatching this now made me appreciate the language of math so much more now that I'm taking a math degree.

I can't believe I understood this. I've heard about this for years, but this is the first explanation I've seen that makes sense. Great video.

I have watched countless videos about the Riemann Hypothesis, the Riemann's Zeta function, etc. And this is only one that actually explains the connecction between this function and the distribution of prime numbers. The harmonics part has never been explained to me before. Well done, now I can finally truly understand why this is such a big deal for mathematicians. Well done!

I was going to say the same!

You are on the right track ..

Odd harmonics create square waves, I learned this in audio engineering. So cool to know the mathematical basis for music.

@dodo meme you're going to have a FIELD day when you find out about quantum mechanics

These videos are so well done and are so much better then the way I had to learn some of these concepts. It used to be just a mess of fuzzy equations without context or explaining. I guess I was stuck with poor teachers...

I remember reading the first few pages of Prof. Dr. Kumar Eswaran almost a year ago and can still remember his shot being tremendously clear. I still hope he didnt provide the proof, but it seems it was genius.

This is the best video explanation of the Riemann Hypothesis. Thank you for taking the time and effort to produce it.

Brilliant video, thank you so much for sharing. Maybe the person or machine that manages to solve this problem might have to think in a superior or different dimension.

Reimann, gauss, euler and all other guys did all this stuff without matplotlib😳 I can't even imagine the extent of their hardwork and dedication

@imCurveee that's a myth, computers aren't making people dumb, it's just the generalized access to internet that simply brought a lot of dumb people into public forums.

Back then it was art 🖼

Corn

they simply loved the numbers more than current generation love pokimane, sweet anita and so on.

Yeah because they did actual math and not numerical BS. :P

Very well produced and outstanding narration. Well done!

Thanks for making someone who'd thought since 6th grade that he could live without learning more Maths that it can still be learned and had fun with whilst in the learning process.

Loved the presentation. You made it easier to understand. Thankyou sir

This is great content, very well presented.

Proving the Riemann Hypothesis is probably one of the hardest ways to make a million dollars.

it's easier to open a casino in the North Pole.

Nevermind! It´s fun!

I KNOW.

If not the hardest…

Yet know it's been proved it's wrong

I love the way he says he'll guide you through this in plain language. Incomprehensible to me non math person.

Top quality content. Extremely clear. Thanks.

Thank you so much! If I had these videos during my mathematics studies it would really be more fun. And surely result in more passion.

These visualizations of the Complex Plane helped me understand complex numbers REALLY EASILY.

Riemann was a complete genius. It's good that this video correctly credit him and Gauss for laying the foundations for SR. I've only seen them credited once before. Most people seem to think that Einstein came up with it out of thin air and it's been stated we'd have had to wait 80 years if Einstein hadn't done it. Apart from Gauss and Rheimann disproving this, Hilbert was working on his own form of SR in parallel with Einstein.

Riemann's hypothesis continues to amaze me. He sure found something amazing

Very didactic. Easily explained. Congratulations! I loved this curiosity.

Thank you for explaining it so clearly!

You just made mathematics fun, I understood only half of it but the video was great, glad I discovered your channel! :)

Math is fun tho. It's bad teachers that make it unfunny.

@ArghyaFromNE Imagine that has a proof...

@Gabriel La Rosa that's s good first step tho

Understanding only a half of the Riemman Hypothesis is not the way to demonstrate it...

So true, people need more of videos like that, to explain complex issues in a simpler and more streamlined way, to get more brains on solving real problems

Golden content and visual explanation. Thank you very much :)

Your explanation of the Riemann hypothesis is wonderful!

University professors should be coerced to watch this and actually present this topic with sense of clarity seen here. Thank you for doing what literally a briefcase worth of tuition fees couldn't.

Absolutely loved it! Thanks professor.

This explained things very intuitively. Thanks for sharing.

Look at 13:21. At each prime p, the value of the function does not increase by log(p), as stated in the video. If that were the case, each step would be larger than the one before. Instead, the steps increase sometimes by large amounts, sometimes by small ones. For example, the step labeled "log(p)" at 13:21 is at the prime 13, i.e., the size of the step is log(13), but the step at 17, which should be log(17), is smaller. It should be bigger. What am I missing?

I feel the puzzle has more to do with frequency like that of synthesis in musical synthesizers utilizing different filters(analog) or digital(no filter) and how noise can be represented A pulsar can even be represented as a pulse(control voltage) Fast Radio Bursts hold priority in the realm of spirits… And Really the secret answer is: Oscilloscope

Function Hypothesis: Twos Thirds (inverted Fourths), fifths, half an octave (6), 7 the magic number… 11 the interloper… Sexagesimal and roots or functions using pi … etc… Wolf Rayet Star Longevity of a White dwarf/ Magnetar/ Pulsar/Neutron Star Erratic Red Dwarf star Highly Metallic Star (remnants of large rich 2nd/3rd generation stars) Quark Star Boson Star (Dark Matter star) (Stars surrounded by a Hydrogen and Helium bubble (early universe) Primordial Black Hole Interlocked Universe Ringularity vs. gyroscope

@Kappa Lock Ah. That makes sense. Thanks. I'm going to have to re-watch the whole video now, because it seems like I missed something important.

In the video he said that we also gonna jump by logp when we encounter p squared, p^3, p^4, etc... So the small jump you talked about was at 16 which is 2^4, so that is log2, and that's why it's smaller

I hope they answer, I’m gonna comment to attract attention ⚠️

That was a great presentation. I saw it for the first time just moments ago. I would like some feedback if possible? Riemann asserted, in his paper/presentation, that whilst the real part of the roots of the Zeta Function, so far as he could tell, appeared to be equal to 1/2, that the validity of this unproven observation was inconsequential to his papers conclusion that the prime counting function tends to Li(x). If Riemann was correct, are we trying to prove or disprove an empty and meaningless hypothesis (1+1=2) or was Riemann unaware of the importance a critical (pardon the punn) point in his analysis? .

The Riemann Hypothesis and even the much weaker statement that there are no non-trivial roots of the Riemann Zeta Function with real part 1, imply that π(x) ~ Li(x), which is one of the equivalent forms of the so called Prime Number Theorem. So Riemann wouldn't be correct if he had stated that the Riemann Hypothesis was inconsequential to the Prime Number Theorem. However, he never did. In his epochal 1859 paper he stated that his hypothesis seemed secondary to him for his further investigations which he presented in this paper. As a main result he established the close and obscure connection between the roots of the Riemann Zeta Function and the prime counting function and his argumentation did indeed not depend on the Riemann Hypothesis. The ideas and techniques introduced by Riemann led to a proof of the Prime Number Theorem in 1896 and the crucial argument in this proof was that there are no zeros of the Riemann Zeta Function with real part 1. You asked if mathematicians are trying to prove a meaningless statement because the Prime Number Theorem can be deduced by a weaker statement. Well, it is very hard to understand the importance of the Riemann Hypothesis. I will try to give you a short answer. If you have further questions just write me an email. You can find the adress in my channel info. After Riemann published his 9 pages work on number theory and the zeta function, all gaps and inaccuracies appearing in his paper could be fixed within the following decades. The only (besides a related statement) open question from Riemanns highly influential work is the Riemann Hypothesis and it remains open for over 150 years. This shows that there is a huge lack of understanding in the fundamental theory of the Riemann Zeta Function. The ideas of Riemann inspired many mathematicians and generalizations of the Riemann Zeta Function were found (for example the Dedekind Zeta Functions for algebraic number fields). The generalization to function fields lead to the so called Weil Conjectures which involve an analogue of the Riemann Hypothesis over finite fields. The proof of the Weil Conjectures required a whole reformulation of algebraic geometry which in turn involves a lot of intricate and heavy abstract machinery (done in wide parts by Grothendieck) and the proof of the Riemann Hypothesis over finite fields was done by Grothendiecks student Deligne. This proof is one of the great mathematical achievements of the last century and laid the foundation for modern number theory. The impact of Riemanns work is in this sense still seen today in modern number theory for example in more recent works as the proof of the Modularity Theorem (Fermat's Last Theorem) or the Langlands Program. So there is a huge influence and impact of Riemanns work and the Riemann Hypothesis is by no means a meaningless statement.

Amazing video 😮😍👏👏👏 Very easy to understand! More videos like this please

That shot of complex functions at ~ 9:40 was amazing

A very nice Video, very good explained and real entertaining 🙂 I love cool science content, thank you very much 🙃😊

To me, the modified prime counting function looks suspiciously similar to the "infinite product of infinite sums of primes power" that Euler proved to be the same as zeta function. I wonder if they represent the same thing.

This was a beautiful presentation. Thank you.

This is literally the best video on YouTube explaining why the Rieman hypothesis is related to the prime numbers and why proving it is so important. Other videos only briefly mentions that it's important because the 'prime number distribution is encoded in the function', like bruh that doesnt explain it enough. This video also beautifully shows how anaylitcal continuation works.

@Joh Embrey So let me ask you this: even if someone were to prove Riemann’s hypothesis what will it actually accomplish? In terms of tangible benefits for mankind??

A Solution for the RIEMANN ZETA FUNCTION is extremely valuable because It also point to Solutions for enhancing the Poincare conjecture, Hodge Invariance conjecture as it relates to PRIME NUMBERS and Doing Arithmetic past ZERO or Singularity as it is called in Analytic Geometry , and Algebraic Geometry, and it Directly points to the Prime factorization Algorithm , the Division algorithm, and the QUADRIATIC FORMULA This Solves many DIMENSIONS and RANK IN THE COMPLEX FUNCTION PLANE for MANIFOLD like The Kahler MANIFOLD ,CALIBU YAU MANIFOLD simeoustanesly and Points to Soulutions to the entire Millennium Prize Problems proposed by The Early 20th Century Philospher and Mathematician David HILBERT

@Titus Wong Keep in mind that this is one of the only attempts you can find on the internet at explaining the connection between the Riemann Hypothesis and prime numbers. I wrote a paper on the Riemann Hypothesis last month and when researching I could not find any explanation like the one at 12:40, even in books that were entirely about the Riemann Hypothesis.

Amazing Tarot Card Reading. Is Anandi Dhawan Dead/Alive ??

Amazing Tarot Card Reading. Is Anandi Dhawan Dead/Alive ??

THIS IS HOW YOU TEACH MATHEMATICS!!! MATHEMATICS IS ABSOLUTELY AMAZING, BUT SOME TEACHERS MAKE IT LOOK LIKE IT'S INCREDIBLY DIFFICULT SO MANY FIND IT BORING. HOPE MORE TEACHERS ARE LIKE THIS!!!

especially when the importance of such equations or numbers is explained to the students very simple example: differentiation and integration are the first two "complicated" math you encounter in high school, but teachers fail to explain their importance and that never made them interesting (to me).

I already knew this material. I enjoyed this video for the pedagogy, the way you explained it.

It was an exceptional visualization of what is the Riemann Hypothesis.

I was able to follow until 13:10, after that it was a difficult one for me to follow. An excellent video, perhaps I need to work more on understanding the building blocks of this problem a bit more

Thank you Professor Kontorovich! Amazing! Mathematics can be so Beautiful!

Brilliantly explained, thank-you

I've watched this video and don't distrust math as much. I appreciate how you talk about math as a tool and what makes this interesting, rather than make it a 'I know more than you' lecture 👍

I've always held the hypothesis that this would probably be solved with a proof by contradiction. Since it claims all lie on that one line, say there is a number off of that line and try to go from there. I've been out of school too long to mess with trying to tackle this.

I have discovered a truly marvellous proof of this, but it's much too large for this youtube comment to contain. Therefore it is left as an exercise to the reader.

Fave YouTube comment ever

@Macitron3000 kk

Yeah it’s a lot like like 👍🏽 and m

Where is your proof?

It’s so crazy to know the distribution of prime numbers is dependent upon imaginary numbers, things that would ordinarily seem to be completely unrelated.

@A Name You Can't Remember agree

If I know one thing about math, is that all of it is 100% related even if we don't know the relation yet.

Top notch quality content.... Thank you for this

Now just imagine if they had the computing power we have today.

This presentation is a real treat 🍰🍯🍹 I feel so grateful everytime I watch thisl 🙏

For those who saw Beautiful Mind, this was the puzzle Nash was working on at the end of the movie. There is a Dover book from Edwards, "Reimann's Zeta Function". 305 pp. The first 25 pages explain Reimann's original 8 page paper. The rest of the book tackles developments since 1859 (up to 1974). Edward's book is presented as a guide to the primary sources. If you saw "The Man Who Knew Infinity", Hardy and Ramanujan also did work related to the conjecture. Turing also worked on the problem, taking a computational approach. Just so you know the competition and how it relates to nerd culture. I get stuck just trying to draw a Greek Zeta.

Thanks, Greg! I just ordered this book on Amazon. :-)

Surgeon - Cool. I was just going to recommend Derbyshire's "Prime Obsession" that I find the most helpful intro for laymen & serious investigators.

Urm...what?

Trauma Surgeon There's another very good book, entitled "Prime Obsession" that alternates chapters on the theory with biographical chapters on Riemann. If you love math, it's a wonderful book. Highly recommended.

Heyy thanks I didn't know this book existed!

That was fantastic, even for a non-math nerd such as myself.

Wow. I sure wish I understood any of this. Eventually I will, has I've recently became fascinated with primes and this is too good to overlook.

Awesome explanation sir 🔥🔥👍

Great video, but it is very frustrating that most videos stop here. I'd like to learn how to evaluate the function within the critical strip, how was he able to prove this relationship etc.

Keep pumping out content like this. Love the level of detail & creativity in these videos.

@Mikhail Fedorov That’s called passive learning

Me too. It makes me feel like I'm doing something with my life even though I'm slouching back and passively consuming someone else's hard work.

Brilliant - totally fascinating - thanx for putting this together. Really appreciated. Cheers 🥂

I've tried explaining to people that imaginary numbers are indeed real but no one ever understands. I'm glad to see some agreement out in the wild.

For anybody interested in learning more about the history of this search, I highly recommend the book The Music Of The Primes by Marcus du Sautoy!

What about the non-trivial 1s, or 0.5s, or 0.0001s? In other words, do any OTHER solutions of the function also lie on specific parts of the graph? What about the furthest values between each two zeroes? Is there a pattern to them? How about instead of using (i^2 = -1) for the imaginary plane, try i^2 = -pi, or -e. Has all this sort of stuff been tried?

If this reaches to you, please create a video for each millennium problem with the clarity of this video. A great introductory video to dummies like us. I never knew Riemann hypothesis had a harmonic side.

@Ruinenlust P vs NP is by far the easiest to explain. "Is it faster to find the solution to a problem, or test a plausible answer to see if it's right?"

@GonnaEndYa wtf are you talking about

@Ruinenlust Question: Want scientific Recommendations? Youtubers to check out?

Ya no one told me there was harmonics...

What's the friggin problem.. Is that life though? Like the inverter that makes the quantums into reality. Is the universe a taco bell cinnamon twist donut rolled into a croissant?

"There is only one way to be sure of it and it's the same way the ancient Greeks did their math: rigorous, absolute mathematical proof" - chills!

There's one other way. Finding the one prime that breaks the hypothesis. Hence the computer search.

That was quite beautiful. Far better explained that my complex analysis professor. The traditional way we teach and learn is obsolete. And no, I don't mean to disrespect teachers, I'm a high school maths teacher myself.

I love how in daily life one piece of evidence is enough to form conclusions but in math a computer can check the non-trivial zetas trillllllllllions of times and they're still like, "ahhhhh idk it's still too early to tell"

Really good when it's explained very slowly, lost me as soon as it started glossed with the zeta function. Specifically 7:08 lost me completely.

I think you deserve $1 million just for explaining this hypothesis in a clear and understandable language. Well done!

@MM wut

Lol

@pxlqq > Hi. Thanks for noticing. Yet thats a bit cryptic. Care to expand your comment?

@MM among us

Unfortunately, despite the rhetoric, most maths pros, like Riemann himself, really don't want know why R's zeta formula functions as it does, nor why RH remained unsolved for more than 157 years. Also, like Riemann, nor do they want to learn or do anything other than what they are doing inside The Box of the current paradigm of their fave maths niche. If that were untrue Riemann could have solved RH--IFF he could've gotten out of his tumnel-vision syndrome (& outa The Box). Also, if the culture of current maths was not allergic to superior theory & metatheory of maths & logic it would be easy to get my proofs reviewed, published & verified. As is, that's almost impossible. Sigh...seems a shame to let 21 years of good work and next-gen maths go to waste. Oh, well...humanity is clearly stuck with a culture of cowardice, conceit & corruption. So, i guess we're doomed. So, nothing matters. Rite?

Great video! Well-explained, beautiful graphics and a dive into history - brilliant!

Math has never really been my thing, but this was fascinating.

This presentation confirms I did the right thing by bailing out of high school 12th grade (senior) math, called trigonometry (I think). I could see that I was headed for a guaranteed grade of F (not good for someone headed to college). I could see that the subject matter would be totally useless to me in my life going forward. My HS guidance counselor agreed and mercifully transferred me into a "Speech and Drama" class, whre my first assignment was to memorize Shakespeare's "All The World's a Stage speech" and deliver it in front of the class, which scared the hell out of me (a stutterer). One of my greatest triumphs was that I performed the speech flawlessly and earned an "A" grade in the class.

You realize this kind of research is done by 0.00..00001% of the population made up of the very best mathematical minds? You would never be expected to work on or know any of this.

About 50years ago an abstract mathematician tried to explain this to me, now (finally) I can see what he was getting at! Thanks.

I know very little about mathematics yet I was able to keep up with this video till the end. That's a rare talent you've got there, explaining such advanced concepts in plain English. Thank you!

Yeah it sounded nice

That is the talent only the TRUE professors posess. Feynman and sagan were like this.

Thank you for that, I sorta got lost at the harmonics part and had to rewatch a couple of times, but I think I understood it somewhat

A brilliant explanation. 99.99...% of mathematicians could not have done it better.

What a beautiful presentation. Thank you. I've subscribed.

I can imagine one way this problem may be solved but there is an issue What if there was a problem that you could prove works if and only if the reimann hypothesis is true, but then you prove it is true without relying on the reimann hypothesis, instead using a fact already proven, like e is transcendental or something (obviously something more complex than that but this is just an example) The thing is... what would this problem be? It is possible since Euler and Euclid both proved there are infinite prime numbers in different ways, with different logic behind it, but the issue is what would it be here?

@Nathan because if the reimann hypothesis is false and has to be true in order for our arbitrary theorem, whatever it may be, to be true, but then we prove our theorem true by means other than the reimann hypothesis, then there's a contradiction and it must be true in the end.

How can something be "if and only if" and then be proven without that contingency? All you'd be doing is an attempt at 2 semi-related, unproven, axioms.

Can we also have a video about why it's so difficult to prove, or rather why it's been so difficult for mathematicians to find the proof thus far?

@Lone Starr That's my understanding as well. However, what I don't understand is that your statement itself seems self contradictory. Basically, undecidable -> true -> you just decided it, it's true -> not undecidable? Unless "undecidable" means no Turing machine that tries to find counterexample halts?

@Patrick Withee any references of that?

Amazing Tarot Card Reading. Is Anandi Dhawan Dead/Alive ??

Amazing Tarot Card Reading. Is Anandi Dhawan Dead/Alive ??

Cuz its math.

as a math major... this still blows my mind regularly... the more i've dove into the subtle world of Zeta functions

Thanks, that is the best explanation of this I have heard.

1:54 For the buildings with multiple floors, are the colors of the numbers just random or did you have a certain way of coloring them based on stuff about the numbers?

Solid explanation! 😍👍

hold my beer, I got one A in math in high school, I got this

@Achyuth Thouta Gauss, Euler were physicists too. Secondly, almost all physicists are very good at mathematics. Paul Dirac, Werner Heisenberg, James Clerk Maxwell is one of the greatest mathematicians. Secondly, theoretical physics requires one to have an excellent mathematical understanding. Secondly, Alber God Einstein was one of the greatest mathematicians of his time. And finally, Newton wasn't "good" at both subject. At his time, he was most of both subjects.

"hOld My bEEr, i gOt onE A in maTh in hIgH school, i gOt THis." Wow, u can do algebra!

@Lambor_ghini He will drink vodka trust me

Yeah,u got this lol

@Gaurav Singh xd

In your example of convergent counting where you add 1, then 1/2, then 1/4 and say its convergent as 2 is the limmit doesn't make sense to me when I consider that there's infinite decimal places, so although it'll never reach 2, you can also never stop adding half of what you previously did. Rather than convergent or divergent, could that be considered parallel?

To think that those mathematicians came up with all of this centuries ago with just a pen and paper is mondblowing

What would happen to the harmonics that approximate Gauss log prime counting (GLPC) function if it is found that a non trivial zero lies outside the critical line? Would that zero would distort the approximation badly? or you would need infinite non trivial zeros outside the critical line for the approximation of the GPLC to break down