Abstract Generated abstract
This paper studies class numbers through finite segments of Euler products associated with Dedekind zeta-functions, congruence zeta-functions, and Hecke or Artin L-series. Using zero-free information, including the possible presence of a Siegel zero, it derives formulas expressing the class number of an algebraic number field in terms of the discriminant, regulator, and a truncated Euler product, with corresponding asymptotic relations for the product of the class number and regulator. An analogous argument is applied to function fields over finite fields, yielding an analytic proof of a known estimate for the number of divisor classes. The paper also uses Artin factorization and Brauer’s theorem to obtain a lower bound for the distance of a real Siegel zero from 1 for Hecke L-series.
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MATHEMATICS
A. I. VINOGRADOV
ON THE CLASS NUMBER
(Presented by Academician I. M. Vinogradov, 12 III 1962)
In connection with A. Weil’s theorem \((^1)\) on the zeros of the congruence zeta-function \(Z(s)\) for a curve over a finite field, Brauer’s theorem \((^2)\), which generalizes Siegel’s theorem to arbitrary number fields, and Fogels’ theorem \((^3)\) on the boundary of the zeros for Hecke \(L\)-series, it has become possible to study segments of Euler products of the corresponding zeta-functions and \(L\)-series in relation to the boundary of their zeros. In what follows, by \(\zeta_K(s)\) we shall mean the Dedekind zeta-function of an algebraic number field \(K\) over the field of rational numbers, \(L_K(s,\chi)\) the Hecke or Artin series of this field, and \(Z(s)\) the congruence zeta-function for a curve over a finite field of characteristic \(p\).
Theorem 1 of the paper \((^4)\) admits a generalization to algebraic fields, and by analogy one may obtain the relation
\[ \prod_{N(\mathfrak p)\le D}\left(1-\frac{1}{N(\mathfrak p)^s}\right)\zeta_K(s)(s-1) = \frac{e^{-c}}{\ln D}\,F(s,D), \tag{1} \]
where \(F(s,D)\) is a certain analytic function depending on the boundary of the zeros of \(\zeta_K(s)\). But if for the rational field, as \(s\to 1\), one obtains the limiting equality
\[ \lim_{s\to 1}\zeta_K(s)(s-1)=1, \]
from which one can obtain some information about the segment of the Euler product at the point \(1\), then in the case of an arbitrary algebraic field
\[ \lim_{s\to 1}\zeta_K(s)(s-1)=\mu H, \]
where
\[ \mu=\frac{2^{r_1+r_2}\pi^{r_2}}{\omega}\,\frac{R}{\sqrt{|d|}}; \]
\(d\) is the discriminant of the field; \(R\) is the regulator; \(n=r_1+2r_2\) is the degree of the field; \(\omega\) is the number of principal roots of unity, and in the limiting relation obtained from (1) as \(s\to 1\), the class-number function \(H\) comes to the fore, since the segment of the Euler product
\[ 1< \prod_{N(\mathfrak p)\le D} \left(1-\frac{1}{N(\mathfrak p)}\right)^{-1} < (\ln D)^n \]
is considerably simpler in its structure and has a good trivial upper and lower estimate under the condition that
\[ \ln D=c(n)\ln |d|\,\ln\ln |d|. \]
A detailed proof of equality (1) leads to the following formulas for the class number.
If \(\zeta_K(s)\) has no Siegel zero, then
\[ H= \frac{\omega}{2^{r_1+r_2}\pi^{r_2}}\, \frac{\sqrt{|d|}}{R}\, \frac{e^{-c}}{\ln D}\, \prod_{N(\mathfrak p)\le D} \left(1-\frac{1}{N(\mathfrak p)}\right)^{-1} \left(1+\frac{\theta}{\ln D}\right), \tag{2} \]
where \(c\) is Euler’s constant.
If, however, \(\zeta_K(s)\) has a real Siegel zero \(\gamma\), then
\[ H=\frac{\omega}{2^{r_1+r_2}\pi^{r_2}}\frac{\sqrt{|d|}}{R}(1-\gamma)e^{-\omega_1(D)} \prod_{N(\mathfrak p)\le D}\left(1-\frac{1}{N(\mathfrak p)}\right)^{-1} \left(1+\frac{\theta}{\ln D}\right) \tag{3} \]
under the condition that \(\ln D \ge c(n)\ln |d|\ln\ln |d|\), and the function \(\omega_1(D)\), for \(\ln D=c(n)\ln |d|\ln\ln |d|\), does not exceed a certain absolute constant. As \(D\to\infty\),
\[ (1-\gamma)e^{-\omega_1(D)} \prod_{N(\mathfrak p)\le D}\left(1-\frac{1}{N(\mathfrak p)}\right)^{-1} \to \chi_K, \]
where \(\chi_K\) is a certain constant of the field \(K\). In particular, if \(K\) is a quadratic field, then
\[ \chi_K=L(1,\chi), \]
and we obtain the known formula for a quadratic field. From relations (2) and (3) one obtains the asymptotic relations, as \(|d|\to\infty\):
\[ \ln HR=\ln\sqrt{|d|}+O(\ln\ln |d|), \]
and in the case of a Siegel zero:
\[ \ln HR=\ln\sqrt{|d|}+\ln(1-\gamma)+O(\ln\ln |d|). \]
Let us note that a relation of type (1) for the congruence zeta-function \(Z(s)\) already assumes the limiting form in the sense of the behavior of the function \(E(s,D)\), by A. Weil’s theorem on the distribution of the zeros of this function on \(\operatorname{Re}s=\frac12\).
If the norm of a prime divisor is denoted by \(|\mathfrak p|\), and \(g\) is the genus of the field, then, analogously to (2), from a formula of type (1) one can obtain:
\[ h=p^g\frac{e^{-c-\omega}}{\ln D} \prod_{|\mathfrak p|\le D}\left(1-\frac{1}{|\mathfrak p|}\right)^{-1} \exp\left(\frac{\theta g\ln D}{\sqrt D}\right), \tag{4} \]
where \(h\) is the number of divisor classes, \(D=p^{f+1/2}\), \(f=1,2,3,\ldots\), \(|\theta|\le 2\). If we put \(D=p^{3/2}\), then from (4) we obtain
\[ h=\chi_0 p^g \exp\left(\frac{\theta g}{\sqrt p}\right), \]
where \(\chi_0\) is an absolute positive constant. Let us note that this relation for \(h\) was known earlier (see, for example, \((^5)\), p. 140). Here another, analytic proof of this fact is given.
Segments of Euler products of Artin and Hecke \(L\)-series are studied similarly. For them one can obtain the relation:
\[ \prod_{N(\mathfrak p)\le D}\left(1-\frac{\chi(\mathfrak p)}{N(\mathfrak p)^s}\right) \frac{L_K(s,\chi)}{s-\gamma}=F_K(s,D); \tag{5} \]
\(\chi\) is a real character, \(\gamma\) is a Siegel zero. From this relation and Brauer’s theorem (2), it is already not difficult to obtain an estimate for \((1-\gamma)\):
\[ 1-\gamma>\frac{c(\varepsilon)}{|dq^{n/2}|^\varepsilon}, \tag{6} \]
where \(\varepsilon>0\) is any positive number; \(c(\varepsilon)>0\) is a constant depending only on \(\varepsilon\); \(q\) is the modulus of the real character \(\chi\); \(n\) is the degree of the field \(K\); \(d\) is the discriminant. It is obtained according to the following scheme: we multiply \(\zeta_K(s)\) and \(L_K(s,\chi)\). This corresponds to a quadratic extension of the field \(K\). Denote the resulting field by \(K_1\). But, by Artin’s theorem,
\[ \zeta_{K_1}(s)=\zeta_K(s)L_K(s,\chi). \tag{7} \]
The zero of the zeta-function is simple, and we assumed that it occurs in the expansion of \(L_K(s,\chi)\); therefore, multiplying the right- and left-hand sides of (7) by \((s-1)\) and letting \(s\to 1\), we obtain
\[ \mu H L_K(s,\chi)=\mu_1 H_1. \]
Applying to this relation Brauer’s theorem \({}^{(2)}\), equality (5), and the theorem on the discriminant of the extended field, we obtain the estimate (6).
Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
7 III 1962
REFERENCES
\({}^{1}\) A. Weil, Sur les courbes algébriques et les variétés qui s’en déduisent, Paris, 1948.
\({}^{2}\) R. Brauer, Am. J. Math., 69, 2, 243 (1947).
\({}^{3}\) E. Fogels, Acta Arithm., 7, No. 3 (1962).
\({}^{4}\) S. Lang, Abelian varieties, N. Y., No. 7, 1959.
\({}^{5}\) A. I. Vinogradov, Dokl. Akad. Nauk SSSR, 143, No. 5 (1962).