More Connections
It struck me clearly the other day that each L2 function can be expressed as the quotient of two L1 functions. For example in the Riemann zeta function (where s = 2) the first term in the product over primes expression is 4/3, which is related to the corresponding first prime 2, through the relationship 1 – 1/ 2 2 . Now this can be expressed as the quotient of the functions, 1/1 2 + 1/2 2 + 1/3 2 + 1/5 2 + … and 1/1 2 + 1/3 2 + 1/5 2 + 1/7 2 + … (where in the latter case all numbers which contain 2 as a shared factor are omitted). So (1/1 2 + 1/2 2 + 1/3 2 + 1/5 2 + …)/(1/1 2 + 1/3 2 + 1/5 2 + 1/7 2 + …) = 1 + ( 1/2 2 ) 1 + ( 1/2 2 ) 2 + ( 1/2 2 ) 3 + … = 4/3 And notice the complementarity here! Whereas the natural numbers represent the base aspect in the L1, they represent the dimensional aspect of number in the L2; and whereas 2 is constant as a dimensional number in the L1 it is constant (as re...