M Theory Lesson 248

In lesson 226 we wondered about prime dimensions for which the allowed MUB matrix entries formed a finite field on a prime number of elements. These are precisely the primes of Sophie Germain, that is $p$ such that $2p + 1$ is also prime. Sophie Germain studied them in connection with Fermat's Last Theorem, in particular for the case $x^5 + y^5 = z^5$.

Now let $\phi (n)$ be the number of integers less than or equal to $n$ which are coprime to $n$. This is the totient function of Euler. Germain primes are the primes that solve an equation of the form $\phi (n) = 2p$. The totient function tells us the cardinality of the multiplicative group of integers modulo $n$. It may be expressed as

$\phi (n) = n \prod_{p | n} (1 - \frac{1}{p})$

and thus it is related to the Riemann zeta function by

$\sum_{n=1}^{\infty} \frac{\phi (n)}{n^s} = \frac{\zeta (s-1)}{\zeta (s)}$.

Note that the Germain primes define a subsequence of the $\phi (n)$, and of the prime numbers, which one might view as a natural regularization of the zeta function.