# Is $h(n) < 2^n$ for all $n$? ($n$th cyclotomic field class number growth)

Is it ever the case that (for prime $n$) the $n$-th cyclotomic field class number, $h(n)$, is greater than $2^n$? (List of class numbers of $n$-th cyclotomic field for prime $n$). For instance, the $163$rd cyclotomic field class number, and also the largest known class number of any prime cyclotomic field, is $10834138978768308207500526544$ is approximately $2^{93}$, the ratio between these two numbers is $93/163=0.57055$. Seeking a smaller example with the $53$rd cyclotomic field, the class number is $4889$, which is approximately $2^{12}$ in comparison with $2^{53}$, and the ratio between exponents is $12/53=0.22642$. Comparing the two ratios between exponents in these two examples does not indicate that the function $h(n)$ is exponential to $n$.

I have no rigorous proof, but the empirical data do not support the claim in the title. If we render $$h(p)$$ as the class number for a given prime $$p$$ then the derived quantity $$\log_2(h(p))/p$$ shows a strong linear correlation coefficient with $$p$$. So it looks like the controlling behavior will be $$2^{cp^2}$$ for some constant $$c$$. The constant $$c$$ will be small, so the class number starts off slowly with a lot of $$1$$ values, but once it gets going jump out of the way! Probably when they get to a prime around 300 you'll see $$\log_2(h(p))/p$$ surpass $$1$$.