The Kummer ratio of the relative class number for prime cyclotomic fields (Q6539293)

Let \(q\) be an odd prime, \(h_1(q)\) be the relative class number of the maximal real subfield \({\mathbb Q}(e^{2\pi i/q})^{+}\), \N\[\NG(q):=2q\left(\frac{q}{4\pi^2}\right)^{(q-1)/4},\qquad R(q):=\frac{h_1(q)}{G(q)}. \N\]\NThe main object of research of the paper under review is the quantity \(R(q)\). Kummer conjectured that \(R(q)\) tends to \(1\), which is equivalent to saying that \(h_1(q)\) is asymptotically equal to \(G(q)\) as \(q\) tends to infinity. \textit{M. R. Murty} and \textit{Y. N. Petridis} [J. Number Theory 90, No. 2, 294--303 (2001; Zbl 0994.11036)] showed that there exists a constant \(c>1\) such that \(R(q)\in (1/c,c)\) holds for a set of primes \(q\) of asymptotic density \(1\). \N\textit{A. Granville} [Invent. Math. 100, No. 2, 321--338 (1990; Zbl 0701.11051)] showed that if the Hardy-Littlewood conjecture and the Elliott-Halberstam conjecture hold, then the sequence \(\{R(q): q~{\text{is prime}}\}\) has \((0,\infty)\) as set of its limit points. The authors' main theorem shows that if for some \(q\ge q_0\), the family of Dirichlet \(L\)-functions \(L(s,\chi)\) with an odd character \(\chi\) modulo \(q\) has no Siegel zero (for example, if \(q\equiv 1\pmod 4\)), then \N\[\N\max\{R(q),R(q)^{-1}\}<e^{0.41}(\log q) \ell(q), \N\]\Nwhere \(\ell(q)\) can be taken to be any monotonic function tending to infinity with \(q\) (and \(q_0\) depends on \(\ell\)). They also show that under the Riemann Hypothesis for every Dirichlet \(L\)-series \(L(s,\chi)\) with an odd character \(\chi\) modulo \(q\) one can remove the \(\ell(q)\) factor for all \(q\ge q_1\), where \(q_1\) is absolute and effectively computable. The proof is achieved by relating \(R(q)\) with sums over primes in progressions and using the Brun-Titchmarsch theorem. The paper ends with some computational results.