M Theory Lesson 253

One zeta type function for a matrix $A$ is given by

$\zeta_A (s) = \textrm{det} (I - \frac{A}{s})$.

For the Weyl circulant $A = (234 \cdots n1)$ in odd prime dimension this takes the form

$\zeta_A (s) = 1 - s^{-p}$

and so the product of all such determinants would look like

$\prod_p (1 - s^{-p})$,

which is strangely similar to the (inverse of the) usual zeta Euler product, except that $p$ and $s$ are interchanged. Observe that this product takes the form

$\prod_p (1 - s^{-p}) = \sum_n \mu (n) s^{- \kappa (n)}$

where $\mu (n)$ is the Mobius function and $\kappa (n)$ is the sum of the prime factors of $n$, otherwise known as the sequence A001414, or the integer logarithm of $n$.