Mirror Set, rev-2

After discovering the multifactorials, or Barnes G-functions, a revision is necessary since G-functions are more general. Mirror Set maps pairs of naturals to rationals;





Where m,n are natural, G(m,n)=m!n is multifactorial of m to depth n. Revising and broadening Mirror Set previously defined as M(m)=M(m,2). This form is more general and combines G-functions and Zeta function. It has a ying-yang look and variables do not commute, G(m,n)=/G(n,m). Here are some small values;

M(1,1-4) = (1, 1/2, 1/3, 1/4)
M(2,1-4) = (2, 1/2, 2/9, 1/8)
M(3,1-4) = (6, 3/2, 8/9, 3/4)
M(4,1-4) = (24, 18, 4^4/3, 6^4)

Previously I took M(3,2) as the center. Now M(2,2) =2/4=1/2 takes the center, standing for time/space. {2,2/9}={light,gravity}, why 2/9 for gravity? (3/2,1/2}={strong,weak} is reasonable.

Applying continuity by setting m-->z and summing gives a harmonically weighed G-function. For G=1 it is the Zeta function;






Assuming continuity holds, G may be generalised to a function of 2 complex variables;






Where z!w stands for the multifactorial to depth w, G(z,w). The form of M seems to imply the extension of G but it remains to be seen if it can be made continuous. I think G works like a local correlation function.

Thus naturals take the center stage of ATON, instead of sets. Maybe not surprising. Don't they (mathematicians) consider number theory as the purest?