UDC 511.52

MATHEMATICS

E. A. ANFERT’EVA, N. G. CHUDAKOV

ON MINIMA OF THE NORM FUNCTION IN IMAGINARY QUADRATIC FIELDS

(Presented by Academician Yu. V. Linnik on 21 III 1968)

Let \(k=Q(\sqrt{-\Delta})\); \(-\Delta\) be the discriminant of \(k\), \(\Delta>0\); \(h(\Delta)\) the number of ideal classes of \(k\); \(\mathfrak a\) an integral ideal of \(k\); \(K_\nu\) \((\nu=1,\ldots,h)\) the ideal classes in \(k\); \(a_\nu=\min N\mathfrak a\) for \(\mathfrak a\in K_\nu\); \(a(\Delta)=\max a_\nu\) \((\nu=1,\ldots,h)\). As is known, \(a_\nu\) is the first coefficient of the reduced form
\[ Q_\nu(x,y)=a_\nu x^2+b_\nu xy+c_\nu y^2, \]
where \(\Delta=4a_\nu c_\nu-b_\nu^2\) and \(N\mathfrak a=Q_\nu(x,y)\) for \(\mathfrak a\in K_\nu\). Yu. V. Linnik showed \((^1)\), on the basis of ergodic considerations, that
\[ A(x)=\sum_{a_\nu\le x}1=\frac{6}{\pi^2}xL(1,\chi_1)(1+o(1)), \tag{1} \]
where \(\chi_1(n)=(-\Delta/n)\), \(x=\lambda\sqrt{\Delta}\), \(\Delta\to\infty\), and \(\Delta^{-1/4+\varepsilon}\le \lambda\le 3^{-1/2}\).

Therefore there can exist only a finite number of values of \(\Delta\) such that
\[ a(\Delta)\le \lambda\sqrt{\Delta}, \]
if \(\lambda<3^{-1/2}\). However, it has not yet been possible to find an effective bound for such values of \(\Delta\), since relation (1) is proved with the aid of Siegel’s ineffective theorem on the real zero of \(L(s,\chi_1)\). A recently published result of Baker \((^2)\), however, makes it possible to obtain effective results concerning the behavior of the quantity \(a(\Delta)\).

In this note the following is proved.

Theorem. For any fixed value \(h=h(\Delta)\) there exist constants \(\varkappa\) and \(\Delta_0\), depending on \(h\), such that for \(\Delta\ge \Delta_0\) the function
\[ a(\Delta)\ge (\lg\Delta)^{-\varkappa}\sqrt{\Delta}. \]
The constants \(\varkappa\) and \(\Delta_0\) are effectively computable.

Proof. Let \(\left(\frac{D}{\Delta}\right)=1\), \(D>0\), \(\chi(n)\) be an even character \(\bmod D\); \(X(n)=\chi(n)\left(\frac{-\Delta}{n}\right)\). Then, arguing analogously to the way this is done in \((^3)\), we obtain for \(\sigma>1\)
\[ L(2s,\chi^2)=\frac12\sum_{\nu=1}^{h}\sum_{(x,y)}' \frac{\chi(Q_\nu(x,y))}{Q_\nu^s(x,y)} = \]
\[ =\zeta(2s)\prod_{p\mid D}\left(1-\frac1{p^{2s}}\right) \sum_{\nu=1}^{h}\chi(a_\nu)a_\nu^{-s} +\sum_{\nu=1}^{h}a_\nu^{-s}\Phi_\nu\left(\frac{b_\nu}{2a_\nu}\right), \tag{2} \]
where
\[ \Phi_\nu(z)= \sum_{y=1}^{\infty}\sum_{x=-\infty}^{+\infty} \frac{\chi(Q_\nu(x,y))} {\left((x+zy)^2+\Delta(4a_\nu)^{-1}y^2\right)^s}. \]
The function \(\Phi_\nu(z)\) is periodic; its period is equal to \(D\), and therefore it can be represented by a Fourier series
\[ \Phi_\nu(z)= \sum_{r=-\infty}^{+\infty} A_r(s,\nu)\exp\frac{2\pi i}{D}rz, \]

where

\[ A_r(s,\nu)=D^{-1}\int_0^D \Phi_\nu(t)\exp\frac{2\pi i}{D}rt\,dt \quad (r=0,\pm1,\pm2,\ldots). \]

Direct calculations show that

\[ A_0(s,\nu)=D^{-1}\left(\frac{\Delta}{4}\right)^{1/2-s} I(s)\sum_{\nu=1}^{h} a_\nu^{s-1}\sum_{t=1}^{D}\sigma_\nu(t)\zeta\left(2s-1,\frac{t}{D}\right), \tag{3} \]

where

\[ I(s)=\int_{-\infty}^{+\infty}(1+v^2)^{-s}\,dv, \qquad \sigma_\nu(t)=\sum_{l=0}^{D-1}\chi(Q_\nu(l,t)). \]

Using the identity, as \(s\to1+0\),

\[ \zeta\left(2s-1,\frac{t}{D}\right) =(2(s-1))^{-1}-\gamma+\frac{\lg D}{D} -\sum_{m=1}^{D-1}\rho^{mt}\lg(1-\rho^{-m})+O(1), \]

where \(\gamma\) is Euler’s constant, \(\rho\) is a primitive root of the equation \(x^D=1\), we can pass to the limit \(s\to1\) in the identities (2) and (3):

\[ \begin{aligned} L(1,\chi)L(1,X) &=\pi^2\alpha\sum_{\nu=1}^{h}\frac{\chi(a_\nu)}{a_\nu} +D^2\pi\Delta^{-1/2}\sum_{\nu=1}^{h}\lg a_\nu\sum_{t=1}^{D}\sigma_\nu(t) \\ &\quad -2\pi D^{-2}\Delta^{-1/2}\sum_{\mu=1}^{D-1}\lg(1-\rho^{-\mu}) \sum_{\nu=1}^{h}\sum_{t=1}^{D}\sigma_\nu(t)\rho^{t\mu} +O\left(\Delta^{-1/2}\exp\left(\frac{-\pi\sqrt{\Delta}}{a(\Delta)D}\right)\right), \end{aligned} \tag{4} \]

where \(\alpha\) is an algebraic number.

We shall now show that, by choosing \(\chi\), we can obtain the equalities

\[ \sum_{t=1}^{D}\sigma_\nu(t)=0 \quad(\nu=1,\ldots,h). \tag{5} \]

If \(\Delta\) is odd, we put \(D=8\), and \(\chi(n)=(8/n)\); a direct count justifies (5). If \(\Delta\equiv8\) or \(12\pmod{16}\), then among the first \([\lg h/\lg 2]+2\) odd primes we choose \(q\), which is prime to \(\Delta\); this can be done by the well-known theorem of Gauss, since \(2^{i-1}\le h\), where \(i\) is the number of distinct prime divisors of \(\Delta\). Let \(\chi\) be an odd complex character \(\bmod q\); then \(X\) will be an odd complex character \(\bmod q\Delta\). For such characters there exist formulas (5), allowing us to transform the left-hand side of (4), and, using a remark of Chowla (6), we shall also satisfy (5). After simple transformations (4) becomes the inequality

\[ \sum_{\nu=1}^{(q-1)/2} a_\nu \lg\sin\frac{\pi\nu}{q}+\beta\pi \ll \Delta^{-1/2}\exp\frac{-\pi\sqrt{\Delta}}{a(\Delta)q}, \tag{6} \]

where \(\beta,a_\nu\) are algebraic numbers whose heights are \(\ll\Delta^{c_1}\), with \(c_1\) depending only on \(h\); moreover, for sufficiently large \(\Delta\), the number \(\beta\ne0\) (Lemma XV from (4)), and since \(\pi\) is multiplicatively independent of \(\sin\pi\nu/q\) \((\nu=1,\ldots,(q-1)/2)\), all the hypotheses of Baker’s theorem (2) are satisfied for the left-hand side of (6), while the right-hand side contradicts (2) for sufficiently large \(\Delta\), if one puts \(\chi=q+2+\varepsilon\).

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
15 III 1968

REFERENCES

1. Yu. V. Linnik, Ergodic properties of algebraic fields, L., 1967, pp. 126–127.
2. A. Baker, Mathematika, 14, Part I, No. 27 (1967).
3. H. Stark, Michigan Math. J., 14, No. 1 (1967).
4. H. Heilbronn, Quart. J. Math., Oxf. Ser., 5, No. 18 (1934).
5. T. Hasse, Lectures on number theory, Moscow, 1953, p. 422.
6. S. Chowla, Det Kong. Norske vid. selsk. fornnand., 40, No. 12 (1967).