T Nation

Riemann Zeta Function

[quote]Dr. Pangloss wrote:
I watched the video again.

I got so mad I had to lay down.[/quote]

I didn’t even watch it. Saw Riemann in the title and just said screw it.

[quote]SkyzykS wrote:

[quote]Dr. Pangloss wrote:
I watched the video again.

I got so mad I had to lay down.[/quote]

I didn’t even watch it. Saw Riemann in the title and just said screw it.
[/quote]

If you think Riemann’s works are bad, try reading some of von Neumann’s.

I too am glad that Dr. Matt is here.

[quote]Dr.Matt581 wrote:

[quote]spar4tee wrote:

[quote]csulli wrote:
Dr. Matt is the coolest[/quote]

He’s so cool that patients go see him when they’re healthy and stay out of his sight when they’re ill because they know they’re not good enough for him. He’s so cool that he has patients despite not being a MD. He’s so cool that he’s a MD anyway. He’s cool that numerical constructs are his patients.[/quote]

LOL, I would never have chosen MD as a profession. I never wanted to take up a trade, but if I had decided on a trade I would have accepted the commission I was offered by the Russian Army (officer training was required for most university students in the Soviet Union and continued for the first years of the Russian Federation, and I excelled at my military studies but my grandfather would have been really disappointed if I joined the military). Academia is where I belong and I could not imagine myself as happy as I am now in any other career.[/quote]

I can’t see myself doing anything non-entrepreneurial in the long term. I never really wanted to go the doctor/lawyer route either.

[quote]Dr.Matt581 wrote:

[quote]Dr. Pangloss wrote:
I still don’t believe it. It’s make no intuitive sense whatsoever.[/quote]

This is not an intuitive process, and this article is very misleading. This sum is indeed accurate, but it is not the only value that can be assigned to the sum of natural numbers. Under the standard axioms applied to the field of real numbers, this sum is indeed divergent, which is what would be intuitive to most people since the field axioms as applied to real numbers are what are taught in primary, secondary, and post-secondary classes up until complex numbers and other things become a factor. This means that it approaches some kind of infinite value, or the limit in some other way does not exist. In this case it is one form of an infinitely positive value.

The problem is that people are taught these topics without it ever being explained to them that this is not the only way to do things mathematically. Things like addition and multiplication (which when applied to real numbers using the standard field axioms are the exact same thing) are taught as pure, unalterable facts to people and by people who do not understand that the reason that these are true are because we are following arbitrary rules made up by mathematicians for a specific purpose and they can be altered or thrown out altogether under certain circumstances. For example, the field axioms applied to real numbers and the operations that arise from them do not apply at all to imaginary numbers (which, despite the name, are very much real. Now, when we apply the field axioms to other, more complicated groups and fields and such we get new definitions of operations like addition and multiplication and we can define things like the Riemann Zeta function, or Ramanujan summation which we can apply to sum the set of natural numbers and get -1/12 and other values like 1/4 (which is also a valid convergent value for the sum of natural numbers under some conditions) both of these are extremely useful to physicist like myself.

[/quote]

But isn’t the error of the film clip fairly clear?
The function of arithmetic addition is not the same as estimating (a statistical) average. All the other arithmetic manipulations which follow therefore are in error.

The proof of the zeta function using limit theorem is not so mysterious


and notice (1) that the zeta function is not the addition series represented in the clip, and (2) the specific value of -1/12 applies to a unique operand, -1.

[quote]DrSkeptix wrote:

But isn’t the error of the film clip fairly clear?
The function of arithmetic addition is not the same as estimating (a statistical) average. All the other arithmetic manipulations which follow therefore are in error. [/quote]

I don’t know, I didn’t watch the video but what you described is a common error that many people make when trying to derive and prove this summation (it is actually pretty similar to the first attempts by Ramanujan to derive this until he found a valid one). Among other things it does require violating several additive laws. The proper derivation and proof are really quite complicated but very elegant and beautiful.

No offense, but I will not read any proofs or explanations of anything on wikipedia, even though they may be valid. However, I will say that there is a way to prove the summation from the article using the Riemann Zeta function, but one must use the field of complex numbers, not real, to do so and it takes a lot of manipulation. First, the sum of natural numbers does not lend itself to a direct input to the Riemann Zeta function (or rather, you can but it reduces back to the original divergent summation anyway so it is not useful in this case). We must first express the original summation by it’s corresponding Dirichlet series, which converges to the Zeta function when the real part of s is greater then one. Then, through the use of analytic continuation we can use zeta-regularization to arrive at the sum converging to z(-1) = -1/12. There are, however, much easier ways to derive and prove this convergence.

The moral of story is never fully assume “that’s it”.

[quote]Dr.Matt581 wrote:

[quote]DrSkeptix wrote:

But isn’t the error of the film clip fairly clear?
The function of arithmetic addition is not the same as estimating (a statistical) average. All the other arithmetic manipulations which follow therefore are in error. [/quote]

I don’t know, I didn’t watch the video but what you described is a common error that many people make when trying to derive and prove this summation (it is actually pretty similar to the first attempts by Ramanujan to derive this until he found a valid one). Among other things it does require violating several additive laws. The proper derivation and proof are really quite complicated but very elegant and beautiful.

No offense, but I will not read any proofs or explanations of anything on wikipedia, even though they may be valid. However, I will say that there is a way to prove the summation from the article using the Reimann Zeta function, but one must use the field of complex numbers, not real, to do so and it takes a lot of manipulation. First, the sum of natural numbers does not lend itself to a direct input to the Reimann Zeta function (or rather, you can but it reduces back to the original divergent summantion anyway so it is not useful in this case). We must first express the original summation by it’s corresponding Dirichlet series, which converges to the Zeta function when the real part of s is greater then one. Then, through the use of analytic continuation we can use zeta-regularization to arrive at the sum converging to z(-1) = -1/12. There are, however, much easier ways to derive and prove this convergence.
[/quote]

Ok. After walking the dog, I recant.
(We agree on z(-1)= -1/12)

Now then, without reference to “averaging,” one can prove that the series S = 1-1+1… equals 1/2.
Add S to S with the “frame shift.” Then 2S = 1 and S = 1/2
(No reference to limits there.)

Not so fast, said my dog. Consider a little set theory. Within S are an infinite number of sequences of 1 -1+1… of n length. It is easy to prove that for any sequence of n length = 1, a sequence of n+1 length = 0. The converse is also true. But no sequence of finite n length will equal 1/2.

Aha! That word “finite.” The infinite sequence = 1/2, even if any particular finite sequence = either 1 or 0.

Which leads me to a question in nuclear magnetic resonance…

[quote]DrSkeptix wrote:

[quote]Dr.Matt581 wrote:

[quote]DrSkeptix wrote:

But isn’t the error of the film clip fairly clear?
The function of arithmetic addition is not the same as estimating (a statistical) average. All the other arithmetic manipulations which follow therefore are in error. [/quote]

I don’t know, I didn’t watch the video but what you described is a common error that many people make when trying to derive and prove this summation (it is actually pretty similar to the first attempts by Ramanujan to derive this until he found a valid one). Among other things it does require violating several additive laws. The proper derivation and proof are really quite complicated but very elegant and beautiful.

No offense, but I will not read any proofs or explanations of anything on wikipedia, even though they may be valid. However, I will say that there is a way to prove the summation from the article using the Reimann Zeta function, but one must use the field of complex numbers, not real, to do so and it takes a lot of manipulation. First, the sum of natural numbers does not lend itself to a direct input to the Reimann Zeta function (or rather, you can but it reduces back to the original divergent summantion anyway so it is not useful in this case). We must first express the original summation by it’s corresponding Dirichlet series, which converges to the Zeta function when the real part of s is greater then one. Then, through the use of analytic continuation we can use zeta-regularization to arrive at the sum converging to z(-1) = -1/12. There are, however, much easier ways to derive and prove this convergence.
[/quote]

Ok. After walking the dog, I recant.
(We agree on z(-1)= -1/12)

Now then, without reference to “averaging,” one can prove that the series S = 1-1+1… equals 1/2.
Add S to S with the “frame shift.” Then 2S = 1 and S = 1/2
(No reference to limits there.)

Not so fast, said my dog. Consider a little set theory. Within S are an infinite number of sequences of 1 -1+1… of n length. It is easy to prove that for any sequence of n length = 1, a sequence of n+1 = 0. The converse is also true. But no sequence of finite n length will equal 1/2.

Aha! That word “finite.” The infinite sequence = 1/2, even if any particular finite sequence = either 1 or 0.[/quote]

You might want to try again. S = 1 - 1 + 1. S + S = (1 - 1 + 1) + (1 - 1 + 1) = 2, so you get 2S = 2 and S = 1. I am not sure what you mean by the “frame shift” can you specify what operation you are trying to perform?

[quote]DrSkeptix wrote:
Which leads me to a question in nuclear magnetic resonance…[/quote]

By all means, ask away, but it is way past my bedtime so my answer will have to wait until tomorrow.

[quote]Dr.Matt581 wrote:

[quote]DrSkeptix wrote:

[quote]Dr.Matt581 wrote:

[quote]DrSkeptix wrote:

But isn’t the error of the film clip fairly clear?
The function of arithmetic addition is not the same as estimating (a statistical) average. All the other arithmetic manipulations which follow therefore are in error. [/quote]

I don’t know, I didn’t watch the video but what you described is a common error that many people make when trying to derive and prove this summation (it is actually pretty similar to the first attempts by Ramanujan to derive this until he found a valid one). Among other things it does require violating several additive laws. The proper derivation and proof are really quite complicated but very elegant and beautiful.

No offense, but I will not read any proofs or explanations of anything on wikipedia, even though they may be valid. However, I will say that there is a way to prove the summation from the article using the Reimann Zeta function, but one must use the field of complex numbers, not real, to do so and it takes a lot of manipulation. First, the sum of natural numbers does not lend itself to a direct input to the Reimann Zeta function (or rather, you can but it reduces back to the original divergent summantion anyway so it is not useful in this case). We must first express the original summation by it’s corresponding Dirichlet series, which converges to the Zeta function when the real part of s is greater then one. Then, through the use of analytic continuation we can use zeta-regularization to arrive at the sum converging to z(-1) = -1/12. There are, however, much easier ways to derive and prove this convergence.
[/quote]

Ok. After walking the dog, I recant.
(We agree on z(-1)= -1/12)

Now then, without reference to “averaging,” one can prove that the series S = 1-1+1… equals 1/2.
Add S to S with the “frame shift.” Then 2S = 1 and S = 1/2
(No reference to limits there.)

Not so fast, said my dog. Consider a little set theory. Within S are an infinite number of sequences of 1 -1+1… of n length. It is easy to prove that for any sequence of n length = 1, a sequence of n+1 = 0. The converse is also true. But no sequence of finite n length will equal 1/2.

Aha! That word “finite.” The infinite sequence = 1/2, even if any particular finite sequence = either 1 or 0.[/quote]

You might want to try again. S = 1 - 1 + 1. S + S = (1 - 1 + 1) + (1 - 1 + 1) = 2, so you get 2S = 2 and S = 1. I am not sure what you mean by the “frame shift” can you specify what operation you are trying to perform?[/quote]

S = 1 - 1 + 1 - 1 etc
S =( ) 1 - 1 + 1 etc.*


2S = 1, and
S = 1/2

which of course works for only sequences S of infinite length.


*edited to indicate a sequence of infinite length with a “frame shift.”

[quote]DrSkeptix wrote:

S = 1 - 1 + 1 - 1 �¢?�¦
S = 1 - 1 + 1�¢?�¦.


2S = 1, and
S = 1/2

which of course works for only sequences S of infinite length.[/quote]

I do not know if you are trying to use some kind of operator on S to transform it or not, if you are pleas tell me what it is and I can tell you whether it is valid or not, but the way you have it written you have two different sets.

S = 1 - 1 + 1 - 1 and
S’ = 1 - 1 + 1

They are not the same, so the sum of the two would be (S + S’) = 2, not 2S = 2.

Edit: now I really am going to bed. My wife hates it when I stay up this late.

[quote]Dr.Matt581 wrote:

[quote]DrSkeptix wrote:
Which leads me to a question in nuclear magnetic resonance…[/quote]

By all means, ask away, but it is way past my bedtime so my answer will have to wait until tomorrow.[/quote]

Oh! I would so like to learn how to even ask the question. And please, forgive in advance, my naive ignorance.
I will leave it to you: "Polar atoms are in a perfectly alternating high Gauss field. They either are spinning “up” or “down” in it. + or -. 1 or -1. In what state of the magnetic field are they “1/2”?

[quote]Dr.Matt581 wrote:

[quote]DrSkeptix wrote:

S = 1 - 1 + 1 - 1 �??�?�¢?�??�?�¦
S = 1 - 1 + 1�??�?�¢?�??�?�¦.


2S = 1, and
S = 1/2

which of course works for only sequences S of infinite length.[/quote]

I do not know if you are trying to use some kind of operator on S to transform it or not, if you are pleas tell me what it is and I can tell you whether it is valid or not, but the way you have it written you have two different sets.

S = 1 - 1 + 1 - 1 and
S’ = 1 - 1 + 1

They are not the same, so the sum of the two would be (S + S’) = 2, not 2S = 2.

Edit: now I really am going to bed. My wife hates it when I stay up this late.[/quote]

The forum hieroglyphics could not record a space or an ellipsis.
I edited the original.
The “frame shift” is the maneuver of adding the infinite sequence, but displaced by one digit for visual clarity.

Good night, Professor.

Two docs now. Nice.

[quote]DrSkeptix wrote:

[quote]Dr.Matt581 wrote:

[quote]DrSkeptix wrote:

S = 1 - 1 + 1 - 1 �??�??�?�¢?�??�??�?�¦
S = 1 - 1 + 1�??�??�?�¢?�??�??�?�¦.


2S = 1, and
S = 1/2

which of course works for only sequences S of infinite length.[/quote]

I do not know if you are trying to use some kind of operator on S to transform it or not, if you are pleas tell me what it is and I can tell you whether it is valid or not, but the way you have it written you have two different sets.

S = 1 - 1 + 1 - 1 and
S’ = 1 - 1 + 1

They are not the same, so the sum of the two would be (S + S’) = 2, not 2S = 2.

Edit: now I really am going to bed. My wife hates it when I stay up this late.[/quote]

The forum hieroglyphics could not record a space or an ellipsis.
I edited the original.
The “frame shift” is the maneuver of adding the infinite sequence, but displaced by one digit for visual clarity.

Good night, Professor.
[/quote]

Yes, this is the part that confused me the most and needed to understand how this “frame shift” works. It would complete my day :slight_smile:

Great thread and illuminating posts.

Reminds me of the basics of “transfinite math” I learned way back when; ordinal numbers and cardinal numbers and how diff “sizes” of the infinite seemed straightfwd. For example:

The sum of all natural numbers (1, 2, 3, …) is the first cardinal: aleph-zero, aleph-nought, or aleph-null
The sum of all real numbers (1, 1.1, 1.11, … 2.345, … 3, … ) was the next cardinal:

edit - symbols won’t show up properly

As you guys know I suck at math but when I read the article I thought that maybe that this is just one of those weird quirks were infinity exists just not in our knowledge of mathematical numeric symbols? Still confused I am mathematically challenged forgive me ahha. I remember learning that music can be broken down into numeric values as well but not perfectly, is it along the same lines that our number system needs more numbers or it is not possible to reach whole numbers, or are we just obsessed with whole numbers? I am trying to understand sorry for the confusion I think once I comprehend what math does I will learn it better.

[quote]thethirdruffian wrote:

[quote]Dr.Matt581 wrote:

[quote]Dr. Pangloss wrote:
I still don’t believe it. It’s make no intuitive sense whatsoever.[/quote]

This is not an intuitive process, and this article is very misleading. This sum is indeed accurate, but it is not the only value that can be assigned to the sum of natural numbers. Under the standard axioms applied to the field of real numbers, this sum is indeed divergent, which is what would be intuitive to most people since the field axioms as applied to real numbers are what are taught in primary, secondary, and post-secondary classes up until complex numbers and other things become a factor. This means that it approaches some kind of infinite value, or the limit in some other way does not exist. In this case it is one form of an infinitely positive value.

The problem is that people are taught these topics without it ever being explained to them that this is not the only way to do things mathematically. Things like addition and multiplication (which when applied to real numbers using the standard field axioms are the exact same thing) are taught as pure, unalterable facts to people and by people who do not understand that the reason that these are true are because we are following arbitrary rules made up by mathematicians for a specific purpose and they can be altered or thrown out altogether under certain circumstances. For example, the field axioms applied to real numbers and the operations that arise from them do not apply at all to imaginary numbers (which, despite the name, are very much real. Now, when we apply the field axioms to other, more complicated groups and fields and such we get new definitions of operations like addition and multiplication and we can define things like the Riemann Zeta function, or Ramanujan summation which we can apply to sum the set of natural numbers and get -1/12 and other values like 1/4 (which is also a valid convergent value for the sum of natural numbers under some conditions) both of these are extremely useful to physicist like myself.

[/quote]

And this is also why we engineers don’t let physicists actually design mechanical objects.[/quote]

Lol you are a douche. Losers who don’t what else to do go in engineering.