Tuesday, February 15, 2011

Zeta Infinities

Fortunately, with categorical quantum arithmetic one can follow Euler in completely ignoring all inconvenient axioms of infinity. Consider the Riemann zeta function for the ordinary counting numbers. Recall that the Pauli exclusion principle selects out ordinals $n = p_1 p_2 p_3 \cdots p_r$ which are square free. This introduces the Mobius function $\mu (n)$, so that

$Z(s) = \sum_{n} \frac{\mu (n)}{n^{s}}$

This is the inverse of $\zeta (s)$, which we can see as follows. Consider $Z \zeta (s)$, a sum over all square free $n$. Each term is labeled by a finite string of primes $p_i$. Now using a prime $p$, split the entire sum into a part with terms that include $p$ and a part with terms that do not. Then

$Z \zeta (s) = - p^{- \zeta (s)} S + S$

for $S$ the terms that do not include $p$. We take a minus sign out of the first term to account for the change of sign in the Mobius function when the number of prime factors is altered by $1$. So we have

$Z \zeta (s) = \prod_{p} (1 - p^{- \zeta (s)}) = s$

at least when $s > 1$. Since $\zeta (s)$ is a bosonic partition function, this inversion is a form of supersymmetry. In the fermionic case, we are summing over all subsets of the prime numbers. As in M theory, but not string theory, spaces and numbers are built from primes. Knowing the fermions is enough to generate the bosons.

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