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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 $n$.

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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 $1$.

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