You probably mean (I am guessing), that with an appropriate procedure on the right hand side (infinite product) you can recover the elements of the left hand side one-by-one? Is that what you mean? Not only "elements" but also partial sums. Is this the case?
You are right Ark. I will try to clean up the nonsense statements like 'infinite sum series', it really makes a mess of everything.
I need to go back to something much more simple and basic. None of what I am trying to show involves calculus. Everything I am trying to demonstrate is historically from publicly available mathematics that existed 1900 years before calculus was even discovered. None of it requires convergence, divergence, limits, partial sums, etc.
To continue my thought experiment requires a starting point that the entire set of natural numbers (expressed as an infinite series) and the entire set of rational numbers (expressed as an infinite series), each have an equality as an infinite product, where each term of the infinite product is a prime geometric series.
1) We know that the infinite series of all inverse natural numbers can be expressed as an infinite product, where each term of the infinite product is a prime geometric series. (Euler's Theorem 7)
2) I will provide the data and logic to demonstrate that a similar equality exists for the natural numbers. That the infinite series of all natural numbers can also be expressed as an infinite product, where each term of the infinite product is a prime geometric series.
3) I will provide the data and logic to demonstrate that a similar equality exists for the rational numbers. That the infinite series of all rational numbers can also be expressed as an infinite product, where each term of the infinite product is a prime geometric series.
*** Note in the above image: "the sum of all rationals", should be "the infinite series of all rational numbers"
Numbers 2 and 3 above I do not think can be proven algebraically. None the less, I think both are true, and the data and logic simply goes back to the sieve of Eratosthenes. The sieve of Eratosthenes has much more information in it than it just being a way to find prime numbers.
I will put the relevant information together and post here. It may take me a while (Jan 2025, maybe sooner).
Interestingly the data and logic from the sieve of Eratosthenes, Ark has already mentioned in a
blog post here.
The link Ark mentions is:
A periodic table of primes: Research team claims that prime numbers can be predicted
The actual paper is at
The Periodic Table of Primes
You can see from one of the links above it is mentioned as a breakthrough-prime-theory...
It is a breakthrough except, James McCanney made this same breakthrough discovery here:
James McCanney
New Definition of Prime Numbers with Sppn Tables and Proofs by Induction
Except Liu Fengsui, made the same breakthrough discovery here:
Liu Fengsui's Prime Formula
Problem 37. The Liu Fengsui's Prime Formula
Except Gary Croft made the same breakthrough discovery here: 'The Prime Spiral Sieve'
Prime Numbers Demystified by 8-Dimensional Algorithms
And the information that all of the above parties are seeing (breakthrough discovery), is simply already found in the fine details of the information already available over 2000 years ago in the sieve of Eratosthenes.
And it is that same information from the sieve of Eratosthenes, that is the logic and data needed to prove:
2) That the infinite series of all natural numbers can also be expressed as an infinite product, where each term of the infinite product is a prime geometric series.
3) That the infinite series of all rational numbers can also be expressed as an infinite product, where each term of the infinite product is a prime geometric series.
Coincidentally, the same information (data and logic from the sieve of Eratosthenes) appears in Euler's theorem 7.
