Multiplicative functions and Dirichlet series via Boxes | Math Foundations 242 | N J Wildberger

[@cegh4dim]

Thank you dear professor Wildberger !

[@reisilva2940]

Such a nice concepts in this arithmetic. Greetings from Brazil. Your approach on arithmetic is fantastic, really nice to watch someone to get actually rigorous.

[@Penrose707]

Thanks Professor, hope you've been well lately- a fan for many years now

[@njwildberger]

Always!

[@Penrose707]

@@njwildberger :)

[@reisilva2940]

It's 4:36 am and i'm so glad i watched this. Thanks professor!

[@santerisatama5409]

Thank you, dear professor. This is very interesting and inspiring. Inspiring also in the sense that I can't help being bothered that so far we don't have a rigorous elementary proof of the Fundamental Identity of Arithmetic, but still a relatively fuzzy conjecture based on infinite arithmetic over infinite series. If I understand correctly, Euclid's lemma is based on the principle that a descending sequence is finite. Ifinite regress is banned by the definition 1 (point has no part).

[@Richard-ft6zp]

very interesting idea. There must be a ton of mind boggling identities derivable this way.

[@detectordegados5292]

Professor, I'm a student of logic and I've been quite interested in your proposals and the ultrafinitistic critique of mainstream "pure mathematics", and this series on mset/box arithmetic has been one of the most interesting projects I've seen for some time. As logic is my main interest, I would like to know if you have (or know) a ultrafinitist substitute for current classical logic (especially if it relates to mset arithmetic) you believe could be used for other forms of formal reasoning. I've seen Gajda's "Consistent Ultrafinitist Logic", "Model Theory of Ultrafinitism" by Mannucci and Cherubin and Terui's "Light Affine Set Theory"; what do you think of these proposals? Also, if you pardon me one more question, do you know other papers or materials using mset/box arithmetic for other areas of math such as group theory and topology? Thank you very much, professor, I've been loving your videos! Please continue with the great work! A warm salute from Brazil!!

[@postbodzapism]

To clarify, Euler's totient is not a strictly multiplicative function, since for example φ(4) = 2 and φ(2) φ(2) =1; but this cooking-up-Euler-type-formula business works as soon as the function is "coprimal multiplicative", ie φ(ab) =φ(a) φ(b) when a and b are coprime.

[@thedoublehelix5661]

In number theory, people call "coprimal multiplicative" functions multiplicative

[@santerisatama5409]

@@thedoublehelix5661 What Euclid means by multiplication of coprime units and what people say nowadays can mean very different things. Language of mathematics has broken down and become incoherently fragmented.

[@postbodzapism]

@@thedoublehelix5661 thanks for the information. What I call multiplicative is called 'totally / completely' multiplicative in number theory, I reckoned.

[@santerisatama5409]

An observation of the totient function and the deep structure of coprime numbers. Box arithmetic is based on tally operations in the context of nesting hierarchies. With the minimum alphabet of two elements, say < and >, we can express coprimes so that numerator elements < and > have string length 1, and denominator elements have string length 2. Thus coprimes a/b have the string length magnitude a+2b. Let's first consider the coprimes a/b < 1/1: < = 1/0, = 0/1 and