/sci/ - Science & Math

>>15251693
Note that Na[Na,*] is isomorphic, as semirings and semimodules over the naturals, to Na[x[1],x[2],...] and Na[[Na,*]] is isomorphic to Na[[x[1],x[2],...]
here x[i] is identified with e[p] where p is the i'th prime
We can pull the same "arithmetic expression of the unique factorization of naturals" that we see in pic related using this isomorphism
sum[n=1]^oo e[n] = (1 + e[2] + e[4] + e[8] + ...)(1 + e[3] + e[9] + e[27] + ...)(1 + e[5] + e[25] + ...)...
= 1/(1-e[2]) 1/(1-e[3]) 1/(1-e[5]) ...
HOWEVER
The underlying "mathematical machinery" required to establish this is mind-bogglingly trivial when compared to the theory of complex functions: namely we can define the product of arbitrary elements of Na[[Na,*]] simply by appealing to the fact that every solution to X = a * b in natural numbers a,b where X is a natural number has finitely many solutions
The theory of analytic continuation of complex functions is not required