MATHEMATICS

B. Z. MOROZ

ON THE CONTINUABILITY OF THE SCALAR PRODUCT OF HECKE SERIES OF TWO QUADRATIC FIELDS

(Presented by Academician I. M. Vinogradov on 2 I 1964)

1. In the work \((^1)\), E. Hecke introduced characters of magnitudes (Größen-charaktere) in fields of algebraic numbers and studied the properties of \(Z\)-functions constructed with the aid of these characters. The results obtained were applied by Hecke \((^1)\), and later by H. Rademacher and I. P. Kubilius \((^2,^3)\), to multidimensional analytic number theory.

In the present note the scalar product of Hecke \(Z\)-functions is studied for two imaginary quadratic fields. Namely, let \(K_1=R(\sqrt{-d_1})\), \(K_2=R(\sqrt{-d_2})\) be imaginary quadratic fields (\(R\) is the field of rational numbers, \(d_1,d_2>0\), \(d_1\ne d_2\)); \(\eta^{w_1 n}\) and \(\xi^{w_2 m}\) are Hecke characters of the fields \(K_1\) and \(K_2\) (\(w_i\) is the number of units in the field \(K_i\)); \(N_i\) is the norm of divisors in \(K_i\) \((i=1,2)\); consider the function

\[ Z\left(s,\eta^{w_1 n},\xi^{w_2 m}\right) = \sum_{N_1\mathfrak A=N_2\mathfrak B} \frac{\eta^{w_1 n}(\mathfrak A)\xi^{w_2 m}(\mathfrak B)} {(N_1\mathfrak A)^{2s}}; \tag{1} \]

\(\mathfrak A,\mathfrak B\) are integral divisors in the fields \(K_1\) and \(K_2\); \(s\) is a complex variable. It is easy to see that for \(\operatorname{Re}s>1/2\) the series on the right-hand side of (1) converges absolutely and, by virtue of the multiplicativity of the characters \(\eta\) and \(\xi\), decomposes into an absolutely convergent Euler product.

In what follows \(\gamma,\gamma_1,\gamma_2,\ldots\) are absolute constants; the constants occurring in \(O\) depend only on the discriminants of the fields \(K_1\) and \(K_2\).

Theorem 1. For \(|m|+|n|\ne0\), the function (1) is regular in the half-plane \(\operatorname{Re}s>1/2-\gamma\).

Theorem 2. Let \(\widehat{\mathfrak A}\) be a class of divisors of the field \(K_1\), \(\widehat{\mathfrak B}\) a class of divisors of the field \(K_2\), and let \(f_{\widehat{\mathfrak A},\widehat{\mathfrak B}}(X)\) be the number of pairs of prime divisors \((\mathfrak p,\mathfrak q)\) under the condition \(N_1\mathfrak p=N_2\mathfrak q\le X\), \(\mathfrak p\in\widehat{\mathfrak A}\), \(\mathfrak q\in\widehat{\mathfrak B}\). Then the number of such pairs of prime divisors \((\mathfrak p,\mathfrak q)\), with \(\mathfrak p\in\widehat{\mathfrak A}\), \(\mathfrak q\in\widehat{\mathfrak B}\), \(N_1\mathfrak p=N_2\mathfrak q\le X\), \(\varphi_1\le \arg\alpha\le \varphi_2\), \(\tilde\varphi_1\le \arg\beta\le \tilde\varphi_2\), is equal to

\[ \frac{w_1w_2}{4\pi^2} (\varphi_2-\varphi_1)(\tilde\varphi_2-\tilde\varphi_1) f_{\widehat{\mathfrak A},\widehat{\mathfrak B}}(X) + O\left(Xe^{-\gamma_1\sqrt{\log X}}\right) \]

(\(\alpha,\beta\) are Hecke ideal numbers corresponding to the divisors \(\mathfrak p,\mathfrak q\)).

From Theorem 2 one easily obtains the asymptotic uniformity of the distribution of pairs of prime divisors \((\mathfrak p,\mathfrak q)\) with equal norms in similarly expanding contours.

For the particular case \(K_1=R(\sqrt{-1})\), \(K_2=R(\sqrt{-3})\), Theorems 1 and 2 were formulated in a note by the author \((^6)\)*.

* The plan of proof of the main lemma given in § 3 of note \((^6)\) is unsuccessful and encounters great difficulties which I have not managed to overcome. However, all the theorems and lemmas formulated in \((^6)\) are correct and are special cases of the results of the present note.

1. Theorems 1 and 2 are obtained from the following Lemma 1 by the method described in § 2 of note \((^6)\).

Lemma 1. Let \(\hat{\mathfrak A}\) and \(\hat{\mathfrak B}\) be classes of divisors of the fields \(K_1\) and \(K_2\); let \(N\) be the number of pairs of integral divisors \((\mathfrak A,\mathfrak B)\) satisfying the conditions \(\mathfrak A\in\hat{\mathfrak A}\), \(\mathfrak B\in\hat{\mathfrak B}\), \(N_1\mathfrak A=N_2\mathfrak B\le X\), \(\varphi\le \arg \alpha\le \varphi+\Delta\), \(\tilde\varphi\le \arg \beta\le \tilde\varphi+\tilde\Delta\), where \(\alpha,\beta\) are Hecke ideal numbers corresponding to the divisors \(\mathfrak A,\mathfrak B\). Then

\[ N=h(X,\hat{\mathfrak A},\hat{\mathfrak B})\,\Delta\tilde\Delta+O(X^{1-\gamma_2}). \tag{2} \]

The function \(h(X;\hat{\mathfrak A},\hat{\mathfrak B})\) can be written explicitly.

We give a sketch of the proof of Lemma 1. Following E. Hecke, to each divisor \(\mathfrak A\) of the field \(K_i\) we associate an “ideal number” \(\alpha\)—an algebraic number from some finite extension of \(K_i\). Let \(\hat{\mathfrak A}\) be a class of divisors of the field \(K_i\); there corresponds to it a quadratic form \(f(x,y)\) with integral coefficients such that for every integral divisor \(\mathfrak A\in\hat{\mathfrak A}\) there exist integers \(x,y\) satisfying \(N_i\mathfrak A=f(x,y)\), and, moreover, if in the variables \(x',y'\) the form \(f(x,y)\) is equal to \(x'^2+y'^2\), then \(|\alpha|^2=x'^2+y'^2\), \(\arg\alpha=\arctg y'/x'\). Thus the problem reduces to the asymptotic counting of the number of integral points \((x,y,z,t)\) under the conditions \(f(x,y)=g(z,t)\le X\), \(\tg\varphi\le y'/x'\le \tg(\varphi+\Delta)\), \(\tg\tilde\varphi\le t'/z'\le \tg(\tilde\varphi+\tilde\Delta)\), where \(f(x,y)\) and \(g(z,t)\) are positive definite quadratic forms with integral coefficients, reducible in the variables \((x',y')\) and \((z',t')\) to the form \(x'^2+y'^2\) and \(z'^2+t'^2\). The equation \(f(x,y)=g(z,t)\) is easily reduced to the equation

\[ b^2-ac=Dt^2, \tag{3} \]

where the numbers \(a,b\), and \(c\) must satisfy certain congruences depending on the forms \(f\) and \(g\), the number \(D\) is positive and also depends only on \(f\) and \(g\), and Lemma 1 follows from the following lemma on the distribution of integral points on the hyperboloid (3).

Lemma 2. Denote by \(H_t\) the hyperboloid \(b^2-ac=Dt^2\). Let \(\mathfrak M\) be the set of points \((a,b,c)\in H_t\) for which \(|a|,|b|,|c|<M\); let \(\Omega\) be a region on \(H_t\) with piecewise smooth boundary, \(\Omega\subset\mathfrak M\); denote by \(N\) the number of integral points \((a,b,c)\in\Omega\) such that \(a\equiv\alpha_1\), \(b\equiv\alpha_2\), \(c\equiv\alpha_3\pmod d\); let \(F'\) be the length of the boundary of the region \(\frac1t\Omega\), and let the curvature of the boundary of \(\frac1t\Omega\) be of order \(O(1)\). Then

\[ N=tm_t(\alpha_1,\alpha_2,\alpha_3,d)\,V_{\frac1t\Omega} +O\left(t^{1-\gamma_3}\left(\frac{M}{t}\right)^{\gamma_4}F'^{\gamma_5}\right), \tag{4} \]

where \(V_{\frac1t\Omega}\) is the volume of the cone with vertex at the origin whose base is the region \(\frac1t\Omega\).

1. Theorems on the distribution of integral points on hyperboloids of the form \(b^2-ac=D\) were studied in the works of Yu. V. Linnik \((^4)\) in the case \(D<0\) and of B. F. Skubenko \((^5)\) in the case \(D>0\). However, in the case of arbitrary \(D\), in formulas of type (3) one cannot obtain a remainder term with power saving. In our case this can be done because the free term of the equation \(b^2-ac=Dt^2\) is divisible by a large square \(t^2\). The plan of the proof of Lemma 2 is as follows.

Along with the hyperboloid \(H_t:b^2-ac=Dt^2\), introduce for consideration the hyperboloid \(H_0:b_0^2-a_0c_0=D\); let the number \(D\) be fixed and \(t\to\infty\). As in the works \((^{4,5})\), to each point \((a,b,c)\in H_t\) we associate the matrix

\[ L=\begin{pmatrix} b&-a\\ c&-b\end{pmatrix} \]

and the form \(\varphi(x,y)=ax^2+2bxy+cy^2\) of discriminant \(Dt^2\). We proceed analogously with the points \((a_0,b_0,c_0)\in H_0\).

We shall call an integral point \((a,b,c)\in H_t\) primitive if \((a,b,t)=1\). (In what follows \((a,b,c)\), \(L=\begin{pmatrix} b&-a\\ c&-b\end{pmatrix}\), and \(\varphi(x,y)=ax^2+2bxy+cy^2\) are considered ...

are regarded as a single object.) It can be proved that for every primitive point
\(L=\begin{pmatrix} b & -a \\ c & -b \end{pmatrix}\in H_t\) there exist an integral matrix \(A\) and a point
\(L_0=\begin{pmatrix} b_0 & -a_0 \\ c_0 & -b_0 \end{pmatrix}\in H_0\) such that
\(AL_0A^{-1}t=L\) and \(\det A=t\). Moreover, by subjecting the matrix \(A\) to certain conditions, one can ensure that, under the mapping
\(A\to AL_0A^{-1}t=L\), different \(A\)’s (for fixed \(L_0\)) correspond to different points \(L\).

Thus, in order to count the number of primitive points \(L\in\Omega\) satisfying the congruences

\[ a\equiv \alpha_1,\qquad b\equiv \alpha_2,\qquad c\equiv \alpha_3\pmod d \tag{*} \]

it suffices to count the number of integral matrices
\(A=\begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}\)
of determinant \(t\) for which \(AL_0A^{-1}t\in\Omega\) and the components \(b_{ij}\) lie in such progressions that the congruences (*) are fulfilled. For the number of such matrices one can obtain an asymptotic formula with a power saving in the remainder term, generalizing Lemma 15 of [4].

In the proof of Lemma 2, conversations with B. F. Skubenko were of great help to me. I take this opportunity to express my gratitude to him.

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

Received
26 XII 1963

References

\(^{1}\) E. Hecke, Math. Zs., 1, 357 (1918); 6, 11 (1920).
\(^{2}\) H. Rademacher, Math. Ann., 111, 209 (1935).
\(^{3}\) I. P. Kubilius, Matem. sborn., 31 (73), 3, 507 (1952).
\(^{4}\) Yu. V. Linnik, Vestn. LGU, No. 2, 3 (1955); No. 5, 3 (1955); No. 8, 15 (1955).
\(^{5}\) B. F. Skubenko, Izv. AN SSSR, ser. matem., 26, No. 5, 721 (1962).
\(^{6}\) B. Z. Moroz, DAN, 150, No. 4, 752 (1963).