A. V. MALYSHEV

ON THE CONNECTION OF THE THEORY OF THE DISTRIBUTION OF ZEROS OF \(L\)-SERIES WITH THE ARITHMETIC OF TERNARY QUADRATIC FORMS

(Presented by Academician I. M. Vinogradov on 16 V 1958)

The present note is a continuation of the note \((^1)\), in which a number of general theorems were formulated on the representation of large numbers \(m\) by positive ternary quadratic forms \(f(x,y,z)\) of odd relatively prime invariants \([\Omega,\Delta]\). In the formulation of these theorems (with the exception of Theorems 1 and 4), in addition to the necessary genus conditions, there entered the condition

\[ \left(\frac{-\Delta m}{q}\right)=1, \tag{1} \]

where \(q\) is a prime number, introduced by the method of proof (and apparently not necessary). It turns out that this condition can be excluded from the formulations of Theorems 2, 3, and 5 of note \((^1)\), if one assumes the validity of the following hypothesis on the zeros of Dirichlet \(L\)-series (which is, evidently, an essential weakening of the extended Riemann hypothesis):

Hypothesis (H). For sufficiently large \(m\), in the region
\[ |s-1|< \frac{(\ln\ln m)^2\ln\ln\ln m}{\sqrt{\ln m}} \]
there are no zeros of Dirichlet \(L\)-functions of the form

\[ L(s)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^s}\quad(\operatorname{Re}s>1), \qquad \chi(n)=\left(\frac{-4\Omega^2\Delta m}{n}\right), \tag{2} \]

where \(\chi(n)\) is a character \((\bmod\,4\Omega^2\Delta m)\).

Theorem. If hypothesis (H) is valid, then the conclusions of Theorems 2, 3, and 5 of note \((^1)\) remain in force if from their formulations one removes the requirement of the existence of a prime number \(q\) satisfying
\[ \left(\frac{-\Delta m}{q}\right)=1 \]
(i.e. if one removes conditions (4), (7), and (12) of note \((^1)\)).

We shall expand the formulation and give an outline of the proof of the assertion corresponding to Theorem 2 of note \((^1)\). Theorems 3 and 5 of \((^1)\) are modified and proved analogously.

Theorem 2a. Let hypothesis (H) be valid. Let \(f(x,y,z)\) be an integral primitive positive ternary quadratic form of odd relatively prime invariants \([\Omega,\Delta]\); let \(m\) be an integer, relatively prime to \(2\Omega\Delta\), for which the congruence \(f(x,y,z)\equiv m\pmod{8\Omega\Delta}\) is solvable. Denote by \(t(f,m)\) the number of primitive representations of the number \(m\) by the form \(f\). Then there exist constants \(m_0\), \(\chi>0\), and \(\chi'>0\), depending only on \(\Omega\Delta\), such that for \(m\ge m_0\)

\[ \chi h(-\Delta m)<t(f,m)<\chi' h(-\Delta m), \tag{3} \]

where \(h(-\Delta m)\) is the number of classes of properly primitive positive binary quadratic forms of determinant \(\Delta m\).

The path of proof of this theorem is close to the proof of the corresponding theorem (¹). Here, too, the arithmetic of quaternions (²) is used. Only more precise estimates of divisor sums are required, carried out by the method of (³).

1°. The estimate (3) from above is trivial (see, for example, (⁴)). Therefore we prove the estimate (3) only from below.

2°. From hypothesis (H), by arguments usual in the theory of \(L\)-functions, we derive that there exists a prime number \(q\), not dividing \(2\Omega\Delta m\), with the condition

\[ \left(\frac{-\Delta m}{q}\right)=1,\qquad q\leq \varkappa_1\exp\left[\mu_1\frac{\sqrt{\ln m}}{\ln\ln m}\right]. \tag{4} \]

The constants \(\varkappa>0\) and \(\mu>0\) here and below depend only on \(\Omega\Delta\).

3°. Fix a sufficiently large natural number \(k\) such that there are \(>\varkappa_2 n^2\) pairs of integral quaternions \(R_1\) and \(R_2\) of norm \(r=q^k\) with the condition that \(\overline{R_1}R_2\) is primitive. By Theorem 1 of note (¹), there will be (for sufficiently large \(k\)) \(>\varkappa_3 h(-\Delta m r^2)\) primitive vectors \(L\) of norm \(\Delta m r^2\). Arguing analogously to (⁵), choose among them \(n>\varkappa_4 h(-\Delta m r^2)\) inequivalent primitive vectors \(L_i\) of norm \(\Delta m r^2\), for which the equalities

\[ l+L_i=V_iB_i\qquad (i=1,2,\ldots,n), \tag{5} \]

hold, where \(B_i\) are integral primitive quaternions of norm \(r^s\); \(V_i\) are integral quaternions of norm prime to \(r\); \(l=rl'\), where \(l'\) is an integer prime to \(r\); for the integer \(s\) the inequalities

\[ \varkappa_5 m^\rho \leq r^s < \varkappa_5 r m^\rho, \tag{6} \]

hold, where \(0<\rho\leq 1/2\) is a constant depending only on \(\Omega\Delta\).

4°. Let \(w\) be the number of distinct \(B_i\) in the equalities (5), or in their part \(i=1,\ldots,n'\) with the condition \(n'>\varkappa'_4 h(-\Delta m r^2)\). Then

\[ w>\varkappa_6 m^\rho\exp\left[-\mu_2\frac{\sqrt{\ln m}}{\ln\ln m}\right]. \tag{7} \]

This is the main part of the proof. The arguments are close in idea to (⁵, ⁶), but use more precise estimates of divisor sums by the method of (³).

5°. It turns out that among the equalities (5) there are \(>n/2\) such equalities for which, for fixed \(i\), there are \(>\varkappa_7 s\) indices \(t\) with the condition

\[ B_i=C_i^{(t)}R_1R_2A_i^{(t)},\qquad N\bigl(A_i^{(t)}\bigr)=r^{4t}, \tag{8} \]

for otherwise the number \(w_1\) of all distinct quaternions \(B\) of norm \(r^s\) is estimated from above as

\[ w_1<\varkappa_8 m^\rho\exp[-\mu_3\sqrt{\ln m}\ln\ln m], \tag{9} \]

which, for sufficiently large \(m\), contradicts the estimate (7).

6°. Among these \(>n/2\) equalities (5) choose \(>\varkappa_9 h(-\Delta m r^2)\) such equalities

\[ rl'+L_i=V_iB_i\qquad (i=1,\ldots,n_1>\varkappa_9 h(-\Delta m r^2)); \tag{10} \]

that, for some fixed \(t\), for all \(i\)

\[ B_i=C_i^{(t)}R_1R_2A_i^{(t)}. \]

From the equalities (10) we immediately obtain the same number of equalities of the form

\[ rl'+L_i'=R_2D_iR_1,\qquad L_i'=(R_2A_i^{(t)})L_i(R_2A_i^{(t)})^{-1} \qquad (i=1,\ldots,n_1); \tag{11} \]

$L_i'$ are integral vectors of norm $\Delta m r^2$. Among them there will be

\[ > \frac{x_9 h(-\Delta m r^2)}{x_{10} r^2} > \]

more than $x_{11} h(-\Delta m)$ distinct ones. Equality (11) shows that $L_i'$ is divisible on the right by $R_1$, and on the left by $R_2$, and, since $R_1 R_2$ is primitive, $L_i'$ is divisible by $r$, $L_i' = r L_i''$, where the $L_i''$ are already primitive vectors of norm $\Delta m$. But to these more than $x_{11} h(-\Delta m)$ distinct primitive vectors $L_i''$ of norm $\Delta m$ there correspond, one-to-one, primitive representations of the number $m$ by the form $f$.

Theorem 2a is proved.

I express my deep gratitude to Yu. V. Linnik and A. I. Vinogradov for substantial assistance in carrying out this work.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
15 V 1958

REFERENCES

$^{1}$ A. V. Malyshev, DAN, 118, 1078 (1958).
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$^{3}$ A. I. Vinogradov, Yu. V. Linnik, Usp. matem. nauk, 12, No. 4, 277 (1957).
$^{4}$ B. W. Jones, The Arithmetic Theory of Quadratic Forms, N. Y., 1950.
$^{5}$ Yu. V. Linnik, Izv. AN SSSR, ser. matem., 4, 633 (1940).
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