I am interested if there are any heuristics on the parity of the class number of $\mathbb{Q}(\zeta_p)$ where $\zeta_p$ is a primitive root of unity.
Is it true that it is odd infinitely many often? Is the density of primes for which it is odd known or conjectured?
Let $h_p$ be the class number of $\Bbb Q(\zeta_p)$ and $h_p^+$ the class number of the maximal real subfield, then $h_p=h_p^+h_p^-$ for some integer $h_p^-$. Hasse proved that $h_p$ and $h_p^-$ have the same parity.
It is conjectured that the class number of $\Bbb Q(\zeta_p)$ is odd if $p$ is a Sophie-Germain prime, i.e. if $q=\frac{p-1}{2}$ is also prime. (This has been proven by Estes if one assumes that $2$ is a primitive root modulo $p$) and it is conjectured that there are infinitely many Sophie-Germain primes.
