Euler's Product for the Zeta function via Boxes II | Math Foundations 241 | N J Wildberger

We continue exploring elementary connections between Box Arithmetic and Number theory, centered on Euler's product formula for the Zeta function. The Fundamental Identity of Arithmetic that we introduced in our last video plays a big role. But now we introduce some interesting variants: when we replace the numbers occurring in this Fundamental Identity with suitable powers of them, including also negative powers, and then apply the Sum operator as before.

What makes this works is a crucial multiplicative property of taking powers.

We make contact with another of Euler's famous formulas, for the sum of the reciprocal squares of natural numbers.

In this talk, we necessarily extend our usual finite careful mathematics with unbounded analogs. But that means that we need to be mindful of the exact meaning of the statements we make. Our general orientation is that the meanings of expression can be reduced to finite expressions by using the natural ordering of the Natural numbers to apply truncation consistently.

Video Contents :
00:00 Introduction
2:42 Powers And The Caret Operation
5:25 Applying f To The FIA
10:45 A Variant ( replacing with cubes)
13:25 Another Variant (replacing with reciprocals)
17:51 Suggests An Identity
19:51 Squared Reciprocals
21:39 Examples
24:32 A calculation

***********************

Here are the Insights into Mathematics Playlists:

Here are the Wild Egg Maths Playlists (some available only to Members!)

************************

[tomholroyd]

I love the distinction between invention and discovery. "Brought into being" as opposed to, "it's 2".

[santerisatama]

This is really really interesting.

On the second watch, now both 240 and 241, I started to get a little peak into FIA having some confluent connection with the deeply fascinating simplicity property of Stern-Brocot type constructs that numerically generate coprime fractions in order of magnitude.

I first heard of the simplicity property from Wildberger, of course. His presentations of SB-tree are also really worth a watch! :)

The simplicity of coprime fraction a/b is 1/ab. When summing together (in standard field arithmetics summation) all new mediants of a new generation sum up to 1/1 in the reduced form. If we put the generations on same row instead of the tree form of only new mediants, the sums of the rows become 1, 2, 3 etc.

What are the non-reduced forms of those rows?!

[santerisatama]

Fascinating!

What is the link between fractions of the type 1/1, 2/2, 3/3 etc. and shifting the exponent between nested boxes?

Coprime fractions a/b -> a/(b-1) seems a very fundamental relation. Also, perhaps we shouldn't think of 1/1, 2/2, 3/3 etc. totally identical with each other. The tautochrone property of cycloid can be given a simple anatomy by interpreting the non-reduced fractions as different positions on the cycloid arc, and the reduced form 1/1 as the same duration from each position... to where when? Between the boxes?

[ohault]

There is an interesting link here between this video and your previous one about limits. If we forget about PI from Euler and think geometrically about a circle, we could perhaps see two types of convergence, one using sums and another one using products where primes would play a role of bridging them together (not too slow, not too fast)

[markuzi]

We see the series with Power=1 does not converge. However we see the series with power=2 does converge. So the deeper question - Is there a value of power(index) between 1 and 2 that is the limiting value for convergence of the Euler series/product.

[antosha]

Hi Dr. Wildberger. Thank you so much for showing this construction. It is quite interesting! I assume that you are more comfortable with infinite boxes / multisets representing the sum of an infinite number of things than "traditional" infinite sums since the former can be written as a collection with an indication that its elements go on infinitely, while traditional infinite sums directly invoke an operation being done infinitely many times. I'm curious if you feel that this is a meaningful distinction between the two perspectives. One could argue that box arithmetic is just another way of conveniently "disguising" the fact that we've constructed an object which invokes an operation being done infinitely many times (using box addition an infinite number of times on finite boxes to construct an infinite box), similarly to how you claim modern mathematics has constructed an entire language to disguise its weak logical foundations.

[chadwainholness]

I believe MF 240 is missing/not public?

[santerisatama]

Trying to understand more of the context. Fundamental theorem of arithmetic is said to be a corollary of "Euclid's lemma:"

Euclid's lemma — If a prime p divides the product ab of two integers a and b, then p must divide at least one of those integers a or b.

Here's the original, translation from the Heath edition.

"BOOK VII, PROPOSITION 30.

If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.

For let the two numbers A, B by multiplying one another make C, and let any prime number D measure C; I say that D measures one of the numbers A, B.

For let it not measure A.

Now D is prime; therefore A, D are prime to one another. [VII. 29]

And, as many times as D measures C, so many units let there be in E.

Since then D measures C according to the units in E, therefore D by multiplying E has made C. [VII. Def. 15]

Further, A by multiplying B has also made C; therefore the product of D, E is equal to the product of A, B.

Therefore, as D is to A, so is B to E. [VII. 19]

But D, A are prime to one another, primes are also least, [VII. 21] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore D measures B.

Similarly we can also show that, if D do not measure B, it will measure A.

Therefore D measures one of the numbers A, B. Q. E. D."

What do we mean by division? Euclid means something different than what is currently taught in schools. Euclid means by "measures" continuous partitions of pure geometry constructions, for which equivalence relation 'neither more nor less' holds. Euclid's definition of mereology (aka inclusion relations") comes from the common notion 5 "The whole is greater than the part", ie. mereology is defined as nested inequivalence relations.

Proof of proposition 30 first evokes the previous proposition 29, which defines coprimes:

"PROPOSITION 29.
Any prime number is prime to any number which it does not measure.

Let A be a prime number, and let it not measure B; I say that B, A are prime to one another.

For, if B, A are not prime to one another, some number will measure them.

Let C measure them.

Since C measures B, and A does not measure B, therefore C is not the same with A.

Now, since C measures B, A, therefore it also measures A which is prime, though it is not the same with it: which is impossible.

Therefore no number will measure B, A.

Therefore A, B are prime to one another. Q. E. D."

Can Box arithmetic derive the related Fundamental Identity of Arithmetic somehow "more directly" from the principle of inclusion? If that can be demostrated, it would be very exciting!

[YawnGod]

Heh. You got a mention by Terrence Howard in the Joe Rogan/Terrence Howard/Eric Weinstein interview.
Fun times!

[MichaelKolczynski]

Will we be introducing the corresponding mset structure that represents these rational numbers?

[acortis]

Norman, we are missing MF240 ... when I saw the first slide I thought, "And where is this coming from?!?!" 😂

[davidkeirsey]

YES YES YES. No boolean AND function of a "free variable" e.g. 1/B(x) < 1/2^181 < 1/(2*(the order number of the Monster Group)) ---> is greater than 2/(2^360). Hint: the binary representation of order number of largest Finite Simple Sporadic Group: the Monster Group is exactly 180 boolean bits. The Sporadic Groups have NO PARAMETERS. They are Whole Number "CONSTANTS" and form an informational framework and a Total Order on the Whole Number line. I.e, forms a Framework for Finite Physics. Finite Information Limits via the 15 SuperSingular Primes and the Supersingular Values of the Primes.

[Kraflyn]

according to the Standard Wisdom, no further new operations are available to play with...

[cotasamnemano]

What is your critique on Homotopy Type theory as an attempt for the foundation of mathematics?

[Kraflyn]

isn't this just the Peano arithmetic? Numbers are sets... call them boxes...