Let $\mathbb{Q}(\zeta_n)$ be the $n$th cyclotomic field with $n$ being a power of $2$. What is the best known asymptotic upper bound on the class number of $\mathbb{Q}(\zeta_n)$ as n grows? Can we say something better for at least a fraction of the fields. For example, can we say that a large fraction of these fields have class number bounded asymptotically by a polynomial in $n$.
Any heuristics?
Let $h_n$ denote the class number of $\mathbb{Q}(\zeta_n)$. Then we can decompose $h_n$ into the "plus part" and "minus part": $$h_n = h^+_n h^-_n,$$ where $h^+_n$ is the class number of the maximal real subfield $\mathbb{Q}(\zeta_n + \zeta_n^{-1})$, and $h^-_n$ is defined (somewhat tautologically) to be $h_n / h^+_n$.
The "minus part" $h^-_n$ is well understood, and we even have an asymptotic (see Washington, "Introduction to Cyclotomic Fields," pg. 45): $$\log{h^-_n} \sim \frac{1}{4}\phi(n)\log{n},$$ where $\phi$ is the Euler totient function.
Thus, $h^-$ grows exponentially in $n$, and is not bounded by any polynomial in $n$.
The "plus part" is trickier. There is a conjecture that $h^+_n = 1$ if $n$ is a power of 2. The best known upper bounds on such $h^+_n$ with $n=2^k$ are based on cyclotomic regulators and lower bounds of relative regulators. But this still gives bounds which grow exponentially in $n$, so are quite poor. I can give a more precise answer about this if anyone is interested.
In any event, the class number $h_n = h^+_n h^-_n$ will grow exponentially in $n$.