On the size of the first factor of the class number of a cyclotomic field (Q913855)

In 1850 Kummer claimed to have found an explicit ``law for the asymptotic growth'' of the first factor \(h_ 1(p)\) of the p-th cyclotomic field. This would take the form \[ h_ 1(p)\approx 2p(\frac{p}{4\pi^ 2})^{(p-1)/4}=G(p), \] say. The author starts with the observation that the above asymptotic relation would be equivalent to \[ \sum^{\infty}_{n=1}\epsilon_ n\frac{\Lambda (n)}{n \log n}=o(1/p), \] where \(\epsilon_ n=\pm 1\) for \(n\equiv \pm 1(mod p)\), and \(\epsilon_ n=0\) otherwise. The principal result of the paper is then that this is not compatible with two well conjectures in prime number theory - namely (i) there are \(\gg x/\log^ 2x\) primes \(p\leq x\) for which \(2p+1\) is also prime, and (ii) for any fixed \(\epsilon >0\), \(a\in {\mathbb{Z}}\setminus \{0\}\) and \(A>0\) we have \[ \sum_{q<x^{1- \epsilon}}| \psi (x;q,a)-\pi (x)/\phi (q)| \quad \ll \quad x(\log x)^{-A}. \] He also shows, subject to a generalization of the first conjecture, that \(h_ 1(p)/G(p)\) is dense in (0,\(\infty)\). To do this he proves a conjecture of \textit{P. Erdős} and \textit{J. L. Nicolas} [Number Theory and Applications, NATO ASI Ser., Ser. C 265, 381-391 (1989; Zbl 0687.10031)], concerning numbers \(r_ i\) for which \(r_ ip+1\) may be simultaneously prime. Finally it is suggested that the true range of variation of \(h_ 1(p)\) may be \[ (\log \log p)^{-1/2+o(1)}\leq h_ 1(p)/G(p)\quad \leq \quad (\log \log p)^{1/2+o(1)}. \]