Let $K=\mathbb{Q}(\mu_p)$ with class number $h=h^+h^-$, where as usual $h^+$ is the class number of the maximal real subfield of $K$. My question is whether there is an effective lower bound for $h$ (which I imagine would be given through one for $h^-)$. I've seen upper bounds for $h^-$ (e.g. here), as well as an asymptotic formula (e.g. here).
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5$\begingroup$ Lower bounds are $h^+ \ge 1$ and $h^- \ge (2\pi)^{-p/2}p^{(p-25)/4}$ where the latter holds for $p > 200$ (can be found in Lang, Cyclotomic fields). $\endgroup$– RalphCommented Jan 29, 2012 at 23:21
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$\begingroup$ Related to the original question, are there bounds for the prime-to-$p$ part of $h^-$? $\endgroup$– JonahCommented Feb 4, 2012 at 22:24
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$\begingroup$ I would be interested by the same question for imaginary quadratic fields. Is there anything known? $\endgroup$– BernikovCommented Feb 8, 2012 at 12:51
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2$\begingroup$ @Bernikov: There's a lot to be said on that front. A good starting point is Goldfeld's "THE GAUSS CLASS NUMBER PROBLEM FOR IMAGINARY QUADRATIC FIELDS". $\endgroup$– Cam McLemanCommented Feb 8, 2012 at 13:55
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$\begingroup$ The link in seems to be dead, but using the Wayback Machine on can check that it was a link to the paper M.Ram Murty, Yiannis N Petridis: On Kummer's conjecture; DOI: 10.1006/jnth.2001.2667. $\endgroup$– Martin SleziakCommented Apr 1, 2023 at 7:29
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