MATHEMATICS

A. V. MALYSHEV

ON THE REPRESENTATION OF INTEGERS BY POSITIVE QUADRATIC FORMS WITH FOUR AND MORE VARIABLES

(Presented by Academician I. M. Vinogradov, 9 IV 1960)

Let
\[ f=f(x_1,\ldots,x_n)=\sum_{\alpha,\beta=1}^{n} a_{\alpha\beta}x_\alpha x_\beta \]
be an integral positive quadratic form, where \(a_{\alpha\alpha}\) \((\alpha=1,\ldots,n)\) and \(a_{\alpha\beta}+a_{\beta\alpha}=2a_{\alpha\beta}\) \((\alpha,\beta=1,\ldots,n;\ \alpha\ne\beta)\) are integers; \(n\ge 4\); \(\det f\equiv \det(a_{\alpha\beta})=d\). Let \(m\) be a positive integer; in the \(n\)-dimensional Euclidean space \(\{x_1,\ldots,x_n\}\) consider the ellipsoid \(f(x_1,\ldots,x_n)=m\). Let \(\Omega=\Omega_{f,m}\) be a region on the surface of this ellipsoid; by the \(f\)-elliptic angle \(\omega=\omega_f(\Omega)\) of the region \(\Omega\) we shall mean the ordinary \(n\)-dimensional solid angle under which, from the origin, one sees the region \(\Omega'\) obtained from \(\Omega\) by means of such a linear transformation as reduces the form \(f\) to \(f_0=x_1^2+\cdots+x_n^2\).

Suppose, in addition, that integers \(g>0\) and \(b_1,\ldots,b_n\) are given. Denote by \(R_{g;b_1,\ldots,b_n}(\Omega_{f,m})\) the number of all integral points \((x_1,\ldots,x_n)\) lying in the elliptical region \(\Omega_{f,m}\) and congruent to \((b_1,\ldots,b_n)\) modulo \(g\); by \(r_{g;b_1,\ldots,b_n}(\Omega_{f,m})\) denote the number of all such points with the additional condition \(\gcd(x_1,\ldots,x_n)=1\). If \(\Omega_{f,m}\) coincides with the surface of the entire ellipsoid \(f(x_1,\ldots,x_n)=m\), then
\[ R_{g;b_1,\ldots,b_n}(\Omega_{f,m})=R_{g;b_1,\ldots,b_n}(f;m) \]
and
\[ r_{g;b_1,\ldots,b_n}(\Omega_{f,m})=r_{g;b_1,\ldots,b_n}(f;m) \]
are, respectively, the number of all integral and of all integral primitive representations \((x_1,\ldots,x_n)\) of the number \(m\) by the form \(f\) with the additional condition
\[ (x_1,\ldots,x_n)\equiv (b_1,\ldots,b_n)\pmod g. \]
If, moreover, \(g=1\), then
\[ R_{g;b_1,\ldots,b_n}(f;m)=R(f;m) \]
and
\[ r_{g;b_1,\ldots,b_n}(f;m)=r(f;m) \]
are, respectively, the number of all integral and the number of all integral primitive representations of the number \(m\) by the form \(f\).

In this note, developing the results of \((^{1,2})\), asymptotic formulas are obtained and investigated for
\[ R_{g;b_1,\ldots,b_n}(\Omega_{f,m}) \quad\text{and}\quad r_{g;b_1,\ldots,b_n}(\Omega_{f,m}). \]

Let
\[ H_{g;b_1,\ldots,b_n}(f;m) = \sum_{q=1}^{\infty} \left\{ \sum_{h\;(\mathrm{mod}\ q)}' q^{-n} S_{g;b_1,\ldots,b_n}(hf;q) e^{-2\pi i\,\frac{mh}{q}} \right\} \tag{1} \]
be the singular series; here \(S_{g;b_1,\ldots,b_n}(f;q)\) is the generalized Gauss sum defined and studied in \((^3)\); the summation
\[ \sum_{h\;(\mathrm{mod}\ q)}' \]
is taken over a reduced system of residues \((\mathrm{mod}\ q)\). The absolute convergence of the infinite series (1) for \(n\ge 4\) follows from Theorem 4 of the note \((^3)\). As usual, the singular series
\[ H_{g;b_1,\ldots,b_n}(f;m) \]
can be represented in the form of an infinite product over prime numbers. The factors of this infinite product, by means of the formulas of \((^3)\), can be expressed in finite form.

through the joint arithmetic invariants of the form \(f\) and of the residue class \((b_1,\ldots,b_n)\pmod g\). We do not give the formulas, because of their bulkiness.

Let \(p\) be a prime number. We introduce the quantities \(u_p,\tau_p,w_p,T_p,S_p\).

1) \(p^{u_p}\parallel g\).

2) To define \(\tau_p\) and \(w_p\), we distinguish two cases: \(p>2\) and \(p=2\).

a) \(p>2\). Let

\[ f(x_1,\ldots,x_n)\equiv \sum_{\beta=1}^{n} a_\beta^{(p)}p^{\varepsilon_\beta^{(p)}}x_\beta^2 \pmod {p^t}, \]

where \(\varepsilon_\beta^{(p)}\geq 0\), \(a_1^{(p)},\ldots,a_n^{(p)}\) are relatively prime to \(p\); let \((b_1,\ldots,b_n)\equiv (b_1^{(p)},\ldots,b_n^{(p)})\pmod {p^{u_p}}\), where \(b_\alpha^{(p)}=0\) if \(b_\alpha\equiv 0\pmod {p^{u_p}}\); let \(p^{v_\alpha^{(p)}}\parallel b_\alpha^{(p)}\) \((\alpha=1,\ldots,n;\) if \(b_\alpha^{(p)}=0\), then \(v_\alpha^{(p)}=\infty)\); by definition

\[ \tau_p=\min_\alpha \{u_p+v_\alpha^{(p)}+\varepsilon_\alpha^{(p)}\}, \]

\[ p^{w_p}\parallel \bigl(m-f(b_1^{(p)},\ldots,b_n^{(p)})\bigr). \]

b) \(p=2\). Let

\[ f(x_1,\ldots,x_n)=\sum_{\beta=1}^{n'} a_\beta 2^{\varepsilon_\beta^{(1)}}x_\beta^2+ \sum_{\gamma=1}^{n''}2^{\varepsilon_\gamma^{(2)}}\psi_\gamma(y_\gamma,z_\gamma)\pmod {2^t}, \]

where \(n'+2n''=n\); \(\varepsilon_\beta^{(1)}\geq 0\), \(\varepsilon_\gamma^{(2)}\geq -1\); \(a_1,\ldots,a_{n'}\) are odd; \(\psi_\gamma(y_\gamma,z_\gamma)=2a'_\gamma y_\gamma^2+2a''_\gamma y_\gamma z_\gamma+2a'''_\gamma z_\gamma^2\); \(a'_\gamma,a''_\gamma,a'''_\gamma\) are integers; \(a''_\gamma\) is odd \((\gamma=1,\ldots,n'')\); let \((b_1,\ldots,b_n)\equiv (b_1^{(2)},\ldots,b_n^{(2)})\pmod {2^{u_2}}\), where \(b_\alpha^{(2)}=0\) if \(b_\alpha\equiv 0\pmod {2^{u_2}}\); let \(2^{v_\alpha^{(2)}}\parallel b_\alpha^{(2)}\); we put

\[ \tau'=\min_{\alpha\leq n,\; v_\alpha^{(2)}\neq u_2-1} \{u_2+v_\alpha^{(2)}+\varepsilon_\alpha^{(1)}+1\} \]

(if there are no such \(\alpha\), we assume that \(\tau'=\infty\));

\[ \tau''=\min_{\alpha\leq n',\; v_\alpha^{(2)}=u_2+1} \{u_2+v_\alpha^{(2)}+\varepsilon_\alpha^{(1)}+2\}; \]

\[ \tau'''=\min_{\alpha>n'}\{u_2+v_\alpha^{(2)}+\varepsilon_{\left[\frac{\alpha-n'+1}{2}\right]}^{(2)}+1\}; \qquad \tau_2=\min\{\tau',\tau'',\tau'''\}; \]

we define

\[ 2^{w_2}\parallel \bigl(m-f(b_1^{(2)},\ldots,b_n^{(2)})\bigr). \]

3) Finally, we put

\[ T_p=\min\{w_p+1+1+(-1)^p,\tau_p\},\qquad S_p=\max\{T_p,u_p\}. \]

To the quadratic form \(f\) we associate the set \(\mathfrak P_f\) of exceptional (herabsetzende) prime numbers (for their definition see (4)). If \(p\in\mathfrak P_f\), then \(n=4\) and \(p\backslash 2^{\,n+1}d\).

Theorem 1. If for every prime number \(p\backslash 2^{\,n+1}dg\) the system of congruences

\[ \begin{gathered} f(x_1,\ldots,x_n)\equiv m\pmod {p^{S_p}},\\ (x_1,\ldots,x_n)\equiv (b_1,\ldots,b_n)\pmod {p^{u_p}}, \end{gathered} \tag{2} \]

then

\[ H_{g;b_1,\ldots,b_n}(f;m)\geqslant \chi_\varepsilon d^{-\frac12(n-1)} g^{-(n-1)}m^{-\varepsilon} \prod_{p\in\mathfrak P_f}p^{-(\frac12 n-1)}\tau_p, \tag{3} \]

where \(\varepsilon>0\) is arbitrary, \(\chi_\varepsilon>0\) is a constant depending only on \(\varepsilon\). In the contrary case

\[ H_{g;b_1,\ldots,b_n}(f;m)=0. \tag{4} \]

Theorem 2. There is the asymptotic formula

\[ R_{g;b_1,\ldots,b_n}(f;m)= \frac{\pi^{\frac n2}m^{\frac n2-1}}{d^{\frac12}\Gamma(n/2)} H_{g;b_1,\ldots,b_n}(f;m) + O\!\left( d^{\frac n4+\frac32} g^{\frac32 n+2} m^{\frac n4-\frac14+\varepsilon} \right), \tag{5} \]

where the constants occurring in \(O\) depend only on \(n\) and \(\varepsilon>0\). Moreover, if for all primes \(p>2^{n+1}dg\) the system of congruences (2) is solvable\(^*\) and if, for some fixed \(\theta>0\),

\[ d^{\frac34 n+\frac32}g^{\frac32 n+1} \prod_{p\in\mathfrak P_f}p^{(\frac12 n-1)}\tau_p = O\!\left(m^{\frac n4-\frac34-\theta}\right), \tag{6} \]

where the constants occurring in \(O\) depend only on \(n\) and \(\theta\), then as \(m\to\infty\) the remainder term in formula (5) is infinitesimal in comparison with the main term.

Theorem 3. Let \(\Omega_{f,m}\) be a convex domain on the surface of the ellipsoid \(f(x_1,\ldots,x_n)=m\) with \(f\)-elliptic solid angle \(\omega\). Then

\[ R_{g;b_1,\ldots,b_n}(\Omega_{f,m})= \frac{\omega}{\omega_0} \frac{\pi^{\frac n2}m^{\frac n2-1}}{d^{\frac12}\Gamma(n/2)} H_{g;b_1,\ldots,b_n}(f;m) + \]

\[ + O\!\left( d^{\frac n4+\frac52} g^{\frac32 n+7} m^{\frac n2-1-\frac{n-3}{4(3n-2)}+\varepsilon} \right), \tag{7} \]

where

\[ \omega_0=\frac{2\pi^{\frac n2}}{\Gamma(n/2)}; \]

the constants occurring in \(O\) depend only on \(n\) and \(\varepsilon>0\). Moreover, if for all primes \(p>2^{n+1}dg\) the system of congruences (2) is solvable and if, for \(\theta>0\),

\[ \omega^{-1}d^{\frac{3n}{4}+\frac32} g^{\frac52 n+4} \prod_{p\in\mathfrak P_f}p^{(\frac12 n-1)}\tau_p = O\!\left(m^{\frac{n-3}{4(3n-2)}-\theta}\right), \tag{8} \]

where the constants occurring in \(O\) depend only on \(n\) and \(\theta\), then as \(m\to\infty\) the remainder term in formula (7) is infinitesimal in comparison with the main term.

It is also possible to obtain the asymptotic formula (7) for nonconvex domains \(\Omega_{f,m}\), assuming, however, that as \(m\) varies their shape does not change (then the constants occurring in \(O\) will depend on the shape of the domain \(\Omega_{f,m}\)).

The asymptotic formulas (5) and (7) can be obtained by refining the arguments in \((^1,^2)\); here Theorem 4 of the note \((^3)\) is used essentially. The concluding parts of Theorems 2 and 3 are derived from the estimate (3).

From Theorem 3 one may derive:

\(^*\) In the contrary case, obviously, \(R_{g;b_1,\ldots,b_n}(f;m)=0\).

Theorem 4. Let \(\Omega_{f,m}\) be a convex domain on the surface of the ellipsoid \(f(x_1,\ldots,x_n)=m\) with \(f\)-elliptic solid angle \(\omega\). Then

\[ r_{g;b_1,\ldots,b_n}(\Omega_{f,m}) = \frac{\omega}{\omega_0} \frac{\pi^{n/2}m^{\,n/2-1}}{d^{1/2}\Gamma(n/2)} G_{g;b_1,\ldots,b_n}(f;m) + O\!\left( d^{\,n/4+5/2}g^{\,3n/2+7} m^{\,n/2-1-\frac{n-3}{4(3n-2)}+\varepsilon} \right), \tag{9} \]

where

\[ G_{g;b_1,\ldots,b_n}(f;m) = \sum_{\substack{\delta^2\mid m\\ \gcd(\delta,g)=1}} \frac{\mu(\delta)}{\delta^{\,n-2}}\, H_{g;b_1\delta^{-1}(\bmod g),\ldots,b_n\delta^{-1}(\bmod g)} \!\left(f;\frac{m}{\delta^2}\right) \]

is a primitive singular series; the constants entering the \(O\)-term depend only on \(n\) and \(\varepsilon>0\). Moreover, if for all primes \(p>2^{n+1}dg\) the system of congruences (2) is primitively solvable \((\bmod\,p)\)* and if, for \(\theta>0\),

\[ \omega^{-1}d^{\,7n/4+3/2}g^{\,7n/2+5} = O\!\left( m^{\frac{n-3}{4(3n-2)}-\theta} \right), \tag{10} \]

where the constants entering the \(O\)-term depend only on \(n\) and \(\theta\), then as \(m\to\infty\) the remainder term in formula (9) is infinitely small in comparison with the main term.

Theorem 5. Let \(\gcd(m,g,2d)=1\). Then, under the conditions of Theorem 4,

\[ r_{g;b_1,\ldots,b_n}(\Omega_{f,m}) \sim \frac{\omega}{\omega_0}\, \frac{1}{\nu_g(f,m)}\,r(f;m), \tag{11} \]

where \(\nu_g(f,m)\) is the number of primitive \((\bmod\,g)\) solutions of the congruence

\[ f(x_1,\ldots,x_n)\equiv m \pmod g . \tag{12} \]

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

Received
6 IV 1960

REFERENCES

1. A. V. Malyshev, DAN, 114, 25 (1957).
2. A. V. Malyshev, Izv. AN SSSR, ser. matem., 23, 337 (1959).
3. A. V. Malyshev, DAN, 133, No. 5 (1960).
4. V. A. Tartakovskii, Izv. AN SSSR, Otd. fiz.-matem., 111, 165 (1929).

* By \(a^{-1}(\bmod g)\) we denote the number \(a_1(\bmod g)\) for which \(aa_1\equiv1\pmod g\).

** Otherwise, obviously, \(r_{g;b_1,\ldots,b_n}(\Omega_{f,m})=0\).