Explicit bounds for residues of Dedekind zeta functions, values of \(L\)-functions at \(s=1\), and relative class numbers (Q5928009)

Let \(L\) be an algebraic number field of degree \(n >1\) and with discriminant \(d_L\), and let \(\zeta(s)\) denote the Dedekind zeta function of \(L\). In [J. Math. Soc. Japan 50, 57-69 (1998; Zbl 1040.11081)], the author proved that the residue at \(s = 1\) of \(\zeta\) is \(\leq (e \log d_L / 2(n-1))^{n-1}\) if \(L\) is totally real; in this paper, he proves this result for general number fields, thereby improving previously known bounds in many cases. His second theorem is that NEWLINE\[NEWLINE|L(1,\chi)|\leq 2 (e \log (d_L f_\chi)/2n)^n,NEWLINE\]NEWLINE where \(\chi\) is a nontrivial primitive ray class character of conductor \(f_\chi\). NEWLINENEWLINENEWLINEThe following lemma is responsible for the simplicity of the proofs of these theorems: NEWLINENEWLINENEWLINELet \(\zeta\) denote Riemann's zeta function, and put \(\Lambda(s) = s(s-1) \pi^{-s/2} \Gamma(s/2) \zeta(s)\) and \(G(s) = \sqrt{\pi} \Gamma((s+1)/2)/s\Gamma(s/2)\). Then \(\Lambda\) and \(G\) are positive, and \(\log \Lambda\) and \(\log G\) are convex on \((0, \infty)\). NEWLINENEWLINENEWLINEAs an application of these estimates, class number bounds for relative class numbers of CM fields are obtained; in particular, the author strengthens a result of \textit{H. M. Stark} [Invent. Math. 23, 135-152 (1974; Zbl 0278.12005)] on lower bounds of the relative class number of CM fields not having quadratic subfields.