Insights into Mathematics | Euler's Product for the Zeta function via Boxes I | Math Foundations 240 | N J Wildberger @njwildberger | Uploaded 3 months ago | Updated 5 hours ago
Box arithmetic allows us to reformulate some aspects of number theory and put them into a more combinatorial / data theoretic framework. In this video we consider the Euler product for the Riemann zeta function.

We review some basics of Box Arithmetic and counting operations and look at various pleasant algebraic relations using both powers of multiplication and also powers of the next Box operation, which is the caret operation.

Our reformulation of the Euler identity centres around what we call the Fundamental Identity of Arithmetic, a combinatorial analog of the Fundamental Theorem of Arithmetic, concerning the essentially unique factorization of a natural number into primes. We then introduce the Sum operator on Boxes, and show how Euler's identity can be re-interpreted in the Box Arithmetic world.

Video Contents:
00:00 Introduction
5:47 Boxes
10:34 Critical Operations
14:20 Counting Laws
17:21 Binomial Theorem (Caret Form)
20:19 An Ongoing Calculation
23:56 The Fundamental Identity Of Arithmetic
27:32 The Sum Operator S
29:35 Summation Laws
32:40 A Naive Application


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