A simple remark on Kummer's conjecture (Q6595598)

Let \(q\) be an odd prime number. Kummer's conjecture discussed by the authors of the current paper concerns the relative class number \(h_q^{-}\) of the \(q\)-th cyclotomic field:\N\[\Nh_q^{-} \sim 2q \left(\frac{q}{4\pi^2} \right)^{\frac{q-1}{4}},\N\]\Nwhich is equivalent, in terms of Dirichlet \(L\)-functions, to\N\[\N\sum_{\chi (-1) = -1} \log (L(1,\chi )) = o(1),\N\]\Nwhere the sum is taken over the odd Dirichlet characters modulo \(q\). The authors prove that there exist constants \(C\) and \(q_0\) such that, for primes \(q \ge q_0\) satisfying \(q \equiv 1 \pmod{4}\), the inequality\N\[\N\prod_{\chi (-1) =-1} L(1,\chi ) \le (\log q ) (\log \log q)^C\N\]\Nholds. They also obtain a similar estimate for the case \(q \equiv 3 \pmod{4}\).