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Ooo😊😊😊poooo

6:55 😅😅😅😊😊😊😊😊 7:01 7:02

Po99oooo😅ooo😊99889ppp😊p😅oo99😊😊😅9😊😊9😊o😊9😊 12:03 oo😊o😊ooo😊oooo9ooo0😊ook o9😅op9ook 😊o99😊9p99popolice p😊😊9😅

Once again making people think its normal summation, but its not

Let's be clear about this. 1+2+3+... does not equal -1/12. The series is the result of a function definition that doesn't work at -1. However, it's true that there is another more complicated function definition that gives the same values where the first definition works, and also works at -1. It's that other function that has the value -1/12 at -1. A theoretical physicist tries to calculate something and gets the result 1+2+3+... . They guess that maybe they used the wrong maths, and maybe the right maths would give that other function so the answer is -1/12. If experiments then agree with this prediction the physicist becomes famous; if not they shrug and try a different way to calculate it. Edited : I typed +1 when I meant -1. Hey ho.

@@john_g_harris What 1+2+3+... equals, depends on your particular choice of how to assign values to infinite series. It's not possible to assign any finite value to it if you choose to adopt the standard definition but there are other definitions. The Ramanujan summation of 1+2+3+... does equal -1/12. Which particular definition is relevant to solving any particular problem can vary depending on the context in which the summation arises.

Classic..

@@john_g_harris Regularization of the Riemann zeta function at s = -3 is used in calculating the Casimir effect and more generally in quantum mechanics there's a fair amount of renormalization where techniques are used to get a finite sum from a divergent series to get actual results. The argument that the sum of 1+2+3+ ... does not equal -1/12 because it uses a different method of getting the result comes up a lot. While it's important to recognize that yes, it doesn't mean "equals" in the same way as other "equals", this exact sort of thing has happened before. By the rules of basic arithmetic, the sum of a rational number and another rational number is a rational number. But take all the nonnegative integers and sum the reciprocal of their factorials and you get the transcendental number e. However, getting to this result, or for that matter getting the result of any convergent infinite series requires a different technique than basic arithmetic. This is not a controversial result today because people are used to the concept of limits and zero, but in the time of Pythagoras or Archimedes, it would have been jus as controversial as summing the positive integers to -1/12. There's an apocryphal story that a member of the Cult of Pythagoras came up with a proof that the square root of 2 is irrational and that the Pythagoreans were so incensed with the result because it broke the rules that they believed in that they took him out to sea in a boat and returned without him. Archimedes came very close to inventing calculus but couldn't make the final conceptual leap because the Ancient Greeks did not believe zero. The idea of using limits to get a result and getting an irrational number from an infinite sum of rational numbers would have been quite controversial.

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🦀

yep lol

It's sort of like how π keeps showing up even when you don't see a circle anywhere near.

@@namelastname4077 You can spend all your time contemplating the miseries of life and inevitablility of death if you want - personally I prefer to spend mine getting excited about fun cool things

So you’re saying Matt is kind of a Parker Yitang Zhang

@@TimMaddux exactly

🤣🤣🤣

He *makes *breakthroughs

@@ophello Haha, thanks.

His age is a mathematical constant, rather than a variable.

idk man, hes aged a bit since his smosh days

He's asymptotically aging

He is probably a Youkai lol

A healthy even diet, with an odd snack here and there

I finally feel a little bit better seeing someone else feel the same.

That's not true, it always have been a mixture of both hard and easy topics. Take the last 6 videos, for example, I would argue 3 are very "simple"/"easy" ("Making a klein bottle", "a hairy problem" and "cow-culus")

I recommend the 3Blue1Brown video on the riemann zeta hypothesis for background here. It is visually beautiful.

“This is HEAVY, doc” -Marty McFly

Wow it’s Verlisify! The search for Siegel zeroes so hard they call it Verlisify. Verlisify isify whoo whoo

Same here! My thesis is in fluid dynamics but this is way more interesting to me

There are already connections. By the nature of the riemann zeroes generating the prime number theorem, you get twin prime conjecture somewhat easily.

What's your thesis on? Hope you are finding it interesting.

What's the source on that first bit? How critical is the mistake?

Remember that Wiles’ proof of Fermat’s Last Theorem had a mistake. Give it time and we’ll see.

This actually does happen on sites like Math Overflow

Hey, dude, check this out! This stuff is fire! Read it while on shrooms, it will blow your mind!

What r u talking about?

Yes you can ! Do it if you love math

If love it it's possible, if you still have doubt then watch David goggins Then if you still don't go after it you will regret it

@@gauravbharwan6377 You wanna be a mathematician too, bro?

R u kidding me? This is number theory. Ofc he's very talented. He was concidered the best back in the school

It truly is fascinating how long number theory reaches into other fields of mathematics in order to even begin to grasp the nature of primes

It's a mistake. ϗ is the ligature for the Greek word "kai" which means "and." It is similar to the ampersand "&" in English.

probably a mistake by whoever typeset the animation. the hand written letter is chi and as far as i can see that's the standard notation as well. interesting to see this letter tho; it's new to me.

@@heavenlyactsatheavycost7629 ϗ is a ligature for the Greek word "και," which means "and"! It's similar to how the ampersand (&) is a ligature "et," the Latin word for "and."

Padilla also wrote a script xi when he should have written zeta.

@@jaoswald Probably thinking of the completed zeta function.

Except Heath-Brown's theorem will almost certainly will never prove the twin prime conjecture, because the Riemann Hypothesis is widely believed to be true.

Siegel primes would essentially guarantee very large fluctuations in the sequence of prime numbers, so much so that primes would inevitably need to be close together every so often. However, fluctuations of the primes appear to be FAR smaller than even the Riemann Hypothesis guarantees, so this method will almost certainly not prove the Twin Prime conjecture.

But if I understand correctly, Heath-Brown's theorem states that if there are no Siegel Primes then the Twin Prime Conjecture is false. And they said it's widely believed that these zeroes don't exist. So doesn't that mean that it's also believed the Twin Prime Conjecture is false?

@@VoodoosMaster I think the inexistence of Siegel Zeros doesn't prevent the Twin Prime conjecture to be true. 08:05 The statement says that one of them has to be true, meaning that at least one of them is true if the other is false (but maybe both are true, hahaha). However, both being false is not possible according to the theorem.

@@jagatiello6900 Ohhh got it, thank you. Then it's not as exciting as I imagined lol

nice one but of course it's plugged into the keyboard

ϗ is the ligature for the Greek word "καί" which means "and." It is similar to the ampersand "&" in English, which is a ligature for "et," the Latin word for "and."

death knell*

“death nail” made me laugh… Mutant offspring of “death knell” and “nail in the coffin” 😂

@@oldvlognewtricks Thanks for that! That will teach me to be more careful when using expressions! Well, English is my second language... I've corrected the error because it distracted from the point I try to make...

I think it would only disprove the generalised one, since we would know there's some generalised zeta function that has a non-trivial zero off the line, but it doesn't show that there's a non-trivial zero off the line on the original zeta function

If it is a Siegel zero for one Dirichlet character it doesn't mean automatically it is one for another.

In addition, the Riemann zeta function doesn't have real zeros inside the critical strip, so all of its non-trivial zeros are complex (i.e. not purely real). See e.g. Titchmarsh book on the RZF, p.30. Chapter 2, Section 12. Although the RZF can't have Siegel zeros, this doesn't imply a thing about the original RH either, for there still could be off the line complex zeros somewhere inside the critical strip.

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Yes - and technically speaking, the concept of "a Siegel zero" is not well-defined (you can always choose a small enough constant c so that any given zero is more than c/logD away from 1). You need an infinite collection of zeros that converge to 1 very rapidly in order to call the whole set a *collection* of siegel zeros.

Did you know the following fact about Riemann and primes: Riemann's hands each had a prime number of fingers!

@@u.v.s.5583 Using the term "digits" would have been more correct and a double entendre. Missed opportunity.

It's impossible to have one Siegel zero- you just lower the constant until it isn't a Siegel zero. You need the infinite family to eliminate all possible constants.

@@btf_flotsam478 I guess this boils down on the definition of a Siegel zero. How do you define it? Is the Siegel zero defined in terms of multiple L-functions (meaning it's a zero for all L(s,\chi_q) for any \chi_q) or is it defined for one single Dirichlet L-function? I thought any exceptional zero that we can find for one specific Dirichlet L-function was called a Siegel zero and then we look at the collection of these zeros (for all Dirichlet characters) to formulate Heath-Browns theorem. Or do you call these just zeros and then define the Siegel zero to be one zero for all L(s,\chi_q)?

It's fairly simple. Instead of the Riemann zeta function Sum (1/n^s) we look at the functions Sum (f(n)/n^s) for some additional function f(n) called Dirichlet character. If one chooses f(n):=1 then we get the Riemann zeta function.

No, it is so called in honor of the famous mathematician Bernhard Generalized Riemann (1967-1975)

And, of course, his work supports the generalised Riemann Hypothesis anyway.

No wrong

As far as I understood, the Riemann-hypothesis is a special case of the generalized one. If you disprove the generalized one, you disprove every one of its special cases, so the Riemann-hypothesis is then false.

How many watermelons did Matt have?

It should correctly be: - "if you find infinite Siegel zeros (one for each Dirichlet character chi mod q), then the twin prim conjceture is true." - "if you find a Siegel zero for the principal Dirichlet character, then you have found a zero for the Riemann zeta function above the 1/2 line (since there is an identity) and thus disproved the RH"

"Trivial" in this phrase probably means something like "a PhD student in number theory can prove this as an exercise", not a common usage of that word.

Instead of D it should be log(q) where q is the modulus of the Dirichlet character and c is just a constant. Both can be derived as upper bounds. If I remember correctly the theorem is due to Gronwall and Titchmarsh. It's a theorem about real and complex Dirichlet characters for L-functions.

That is correct If the GRH is true, the RH is true But the converse is not the case.

A Siegel zero disproves a different statement, the generalized Riemann hypothesis (GRH), not the actual Riemann hypothesis (RH). The GHR is a statement for a class of functions (so called Dirichlet L-functions).

The generalized Riemann hypothesis states that all nontrivial zeros of the generalized Riemann zeta function have a real part of 1/2. The real part of Siegel zeros is within a certain distance of 1, instead of 1/2; if they exist, they would be nontrivial zeros whose real parts are not 1/2, which would contradict the GRH (and thus disprove it).

Because the Riemann hypothesis says that all (non-trivial) zeros lies on the 1/2 vertical line of the graph. If you find a non-trivial zero outside the 1/2 vertical line, then the Riemann hypothesis is false by definition, and a siegel zero is precisely a non-trivial zero outside the 1/2 vertical line (specifically, one very close to 1 as the video explains).

No, probably both are true. What can't happen is that they're both false, that there are only finitely many twin primes and the GRH fails.