GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 6, 1997, 523-532

ON THE NUMBER OF REPRESENTATIONS OF POSITIVE

INTEGERS BY A DIRECT SUM OF BINARY QUADRATIC FORMS WITH DISCRIMINANT −23

G. LOMADZE

Abstract. Explicit exact formulas are obtained for the number of representations of positive integers by some direct sums of quadratic forms F_{1} = x_{1}^{2} + x_{1}x_{2} + 6x_{2}^{2} and Φ_{1} = 2x_{1}^{2} + x_{1}x_{2} + 3x_{2}^{2}.

Let

F_{1} = x_{1}^{2} + x_{1}x_{2} + 6x_{2}^{2} and Φ_{1} = 2x_{1}^{2} + x_{1}x_{2} + 3x_{2}^{2}.

These are primitive reduced binary quadratic forms with discriminant Δ = −23. For each k ≥ 1, let F_{k} and Φ_{k} denote the direct sums of k copies of F_{1} and Φ_{1}, respectively.

In [1] exact formulas are derived for the number of representations of a positive integer by positive quadratic forms in six variables with integral coefficients, among which there are suitable formulas for the quadratic forms F_{3}, F_{2} ⊕ Φ_{1}, F_{1} ⊕ Φ_{2} and Φ_{3}.

In [2] explicit exact formulas are derived for the arithmetical function r(n; Q), the number of representations of a positive integer n by the quadratic forms Q, for Q = F_{2}, F_{1} ⊕ Φ_{1} and Φ_{2}.

In the present paper we obtain formulas for r(n; Q) when Q = F_{4}, F_{3}⊕Φ_{1}, F_{2} ⊕ Φ_{2}, F_{1} ⊕ Φ_{3} and Φ_{4}.

It should be pointed out that the approach described here enables one to get formulas for r(n; Q) for Q = F_{k}, Φ_{k}, F_{i} ⊕ Φ_{j} (i, j ≥ 1, i + j = k), where k > 4. However the calculations will be very tedious.

In this paper the notation, definitions, and some results from [3] will be mostly used.

523

1991 Mathematics Subject Classification. 11E20, 11E25, 11E76, 11F11.

Key words and phrases. Spherical function with respect to the quadratic form, thetaseries, Eisenstein series.

1072-947X/97/1100-0523$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

§ 1. Some Known Results

Let

Q = Q(x_{1}, x_{2}, …, x_{f}) = ∑_{1≤r≤s≤f} b_{rs}x_{r}x_{s}

be a positive quadratic form in f (f is even) variables with integral coefficients b_{rs}. Further let D be the determinant of the quadratic form

2Q = ∑_{r,s=1}^{f} a_{rs}x_{r}x_{s} (a_{rr} = 2b_{rr}, a_{rs} = a_{sr} = b_{rs}, r < s);

A_{rs} the algebraic cofactors of elements a_{rs} in D; Δ the discriminant of the form Q, i.e., Δ = (−1)^{f/2}D; δ = g.c.d.(\frac{A_{rr}}{2} , A_{rs}) (r, s = 1, …, f); N = \frac{D}{δ} the level of the form Q; χ(d) the character of the form Q, i.e., χ(d) = 1 if Δ is a perfect square, but if Δ is not a perfect square and 2 † Δ, then χ(d) = (\frac{d}{|Δ|}) for d > 0 and χ(d) = (−1)^{f/2}χ(−d) for d < 0 (here (\frac{d}{|Δ|}) is the generalized Jacobi symbol). A positive quadratic form in f variables of level N and character χ(d) is called a quadratic form of type (−f/2, N, χ). Finally, let P_{ν} = P_{ν}(x_{1}, x_{2}, …, x_{f}) be a spherical function of order ν (ν is a positive integer) with respect to the quadratic form Q.

In what follows q is an odd prime and z = exp(2πiτ), Im τ > 0.

As is wellknown, to each positive quadratic form Q there corresponds the theta-series

ϑ(τ; Q) = 1 + ∑_{n=1}^{∞} r(n; Q)z^{n}.  (1.1)

We shall formulate the well-known results in the form of the following lemmas.

Lemma 1 ([3], p. 874, 875, 817; see also [4], p. 15). Any positive quadratic form Q of type (−k, q, 1), 2|k, k > 2, corresponds to one and the same Eisenstein series

E(τ; Q) = 1 + ∑_{n=1}^{∞}(ασ_{k−1}(n)z^{n} + βσ_{k−1}(n)z^{qn}),

where

α = \frac{i^{k}}{ρ_{k}} \frac{q^{k/2} − i^{k}}{q^{k} − 1} , β = \frac{1}{ρ_{k}} \frac{q^{k} − i^{k}q^{k/2}}{q^{k} − 1} ,  (1.2)
ρ_{k} = (−1)^{k/2} \frac{(k − 1)!}{(2π)^{k}} ζ(k).  (1.3)

ON THE NUMBER OF REPRESENTATIONS OF POSITIVE INTEGERS 525

Lemma 2 ([3], pp.874, 875, 895). If Q is a primitive quadratic form of type (−k, q, 1), 2|k, then the difference ϑ(τ; Q) − E(τ; Q) is a cusp form of type (−k, q, 1).

Lemma 3 ([3], p. 853, Theorem 33). The homogeneous quadratic polynomials in f variables φ_{rs} = x_{r}x_{s} − \frac{1}{f} \frac{A_{rs}}{D} 2Q (r, s = 1, 2, …, f) are spherical functions of second order with respect to the positive quadratic form Q in f variables.

Lemma 4 ([3], p. 855). If Q is a quadratic form of type (−f/2, N, χ) and P_{ν} is the spherical function of order ν with respect to Q, then the generalized multiple theta-series

ϑ(r; Q, P_{ν}) = ∑_{n=1}^{∞} ( ∑_{Q=n} P_{ν})z^{n}

is a cusp form of type (−(f/2 + ν), N, χ).

Lemma 5 ([3], p.846). If the quadratic forms Q_{1} and Q_{2} have the same level N and characters χ_{1}(d) and χ_{2}(d), respectively, then the quadratic form Q_{1} ⊕ Q_{2} will have the level N and the character χ_{1}(d)χ_{2}(d).

§ 2. Some Auxiliary Results

2.1. For the quadratic form F_{1} we have D = 23, A_{11} = 12, A_{22} = 2, i.e., δ = 1, N = 23, Δ = −23, χ = χ(d) = (\frac{d}{23}) if d > 0. For the quadratic form Φ_{1} we have D = 23, A_{11} = 6, A_{22} = 4, i.e., δ = 1, N = 23, Δ = −23, χ = χ(d) = (\frac{d}{23}) if d > 0. Hence F_{1} and Φ_{1} are quadratic forms of type (−1, 23, χ). Thus, by Lemma 5, F_{2}, Φ_{2}, and F_{1} ⊕ Φ_{1} are quadratic forms of type (−2, 23, 1).

For the quadratic form F_{2} we have D = 23^{2}, A_{11} = 12 · 23, A_{22} = 2 · 23. Hence, if in Lemma 3 we put

f = 4, Q = F_{2}, r = s = 1,

then the polynomial

φ_{11} = x_{1}^{2} − \frac{6}{23} F_{2}

will be a spherical function of second order with respect to F_{2}.

For the quadratic form Φ_{2} we have D = 23^{2}, A_{11} = 6 · 23, A_{22} = 4 · 23. Hence, if in Lemma 3 we put

f = 4, Q = Φ_{2}, r = s = 1, and r = s = 2,

then the polynomials

φ_{11} = x_{1}^{2} − \frac{3}{23} Φ_{2} and φ_{22} = x_{2}^{2} − \frac{2}{23} Φ_{2}

will be spherical functions of second order with respect to Φ_{2}.

For the quadratic form F_{1} ⊕ Φ_{1} we have D = 23^{2}, A_{11} = 12 · 23, A_{22} = 2 · 23. Hence, if in Lemma 3 we put

f = 4, Q = F_{1} ⊕ Φ_{1}, r = s = 1, and r = s = 2,

then the polynomials

φ_{11} = x_{1}^{2} − \frac{6}{23} (F_{1} ⊕ Φ_{1}) and φ_{22} = x_{2}^{2} − \frac{1}{23} (F_{1} ⊕ Φ_{1})

will be spherical functions of second order with respect to F_{1} ⊕ Φ_{1}.

2.2. It is easy to verify that the equation F_{1} = n (a) has two integral solutions for n = 1: x_{1} = ±1, x_{2} = 0;

(b) has no integral solutions for n = 2, 3, 5;

(c) has two integral solutions for n = 4: x_{1} = ±2, x_{2} = 0.

Also it is easy to verify that the equation Φ_{1} = n

(a) has no integral solutions for n = 1, 5;

(b) has two integral solutions for n = 2: x_{1} = ±1, x_{2} = 0;

(c) has two integral solutions for n = 3: x_{1} = 0, x_{2} = ±1.

(d) has two integral solutions for n = 4: x_{1} = ±1, x_{2} = ∓1.

Hence, according to (1.1), we have

ϑ(τ; F_{1}) = 1 + 2z + 2z^{4} + 0z^{5} + ⋯,  (2.1)
ϑ(τ; Φ_{1}) = 1 + 2z^{2} + 2z^{3} + 2z^{4} + 0z^{5} + · · · .  (2.2)

From (2.1) it follows that

ϑ(τ; F_{2}) = ϑ^{2}(τ; F_{1}) = 1 + 4z + 4z^{2} + 4z^{4} + 8z^{5} + ⋯,  (2.3)

whence

ϑ(τ; F_{3}) = ϑ(τ; F_{2})ϑ(τ; F_{1}) = 1 + 6z + 12z^{2} + 8z^{3} + 6z^{4} + 24z^{5} + · · · . (2.4)

From (2.2) it follows that

ϑ(τ; Φ_{2}) = ϑ^{2}(τ; Φ_{1}) = 1 + 4z^{2} + 4z^{3} + 8z^{4} + 8z^{5} + ⋯,  (2.5)

whence

ϑ(τ; Φ_{3}) = ϑ(τ; Φ_{2})ϑ(τ; Φ_{1}) = 1 + 6z^{2} + 6z^{3} + 18z^{4} + 24z^{5} + · · · .(2.6)

From (2.1) and (2.2) it follows that

ϑ(τ; F_{1} ⊕ Φ_{1}) = ϑ(τ; F_{1})ϑ(τ; Φ_{1}) =
= 1 + 2z + 2z^{2} + 6z^{3} + 8z^{4} + 4z^{5} + · · · .  (2.7)

§ 3. Formulas for r(n; F_{4}), r(n; Φ_{4}), r(n; F_{3} ⊕ Φ_{1}), r(n; F_{2} ⊕ Φ_{2}), r(n; F_{1} ⊕ Φ_{3})

Theorem 1. The system of generalized fourfold theta-series

ϑ(τ; F_{2}, φ_{11}) = \frac{1}{23} ∑_{n=1}^{∞} ( ∑_{F_{2}=n} 23x_{1}^{2} − 6n)z^{n},  (3.1)

ϑ(τ; F_{1} ⊕ Φ_{1}, φ_{11}) = \frac{1}{23} ∑_{n=1}^{∞} ( ∑_{F_{1}⊕Φ_{1}=n} 23x_{1}^{2} − 6n)z^{n},  (3.2)

ϑ(τ; F_{1} ⊕ Φ_{1}, φ_{22}) = \frac{1}{23} ∑_{n=1}^{∞} ( ∑_{F_{1}⊕Φ_{1}=n} 23x_{2}^{2} − n)z^{n},  (3.3)

ϑ(τ; Φ_{2}, φ_{11}) = \frac{1}{23} ∑_{n=1}^{∞} ( ∑_{Φ_{2}=n} 23x_{1}^{2} − 3n)z^{n},  (3.4)

ϑ(τ; Φ_{2}, φ_{22}) = \frac{1}{23} ∑_{n=1}^{∞} ( ∑_{Φ_{2}=n} 23x_{2}^{2} − 2n)z^{n}  (3.5)

is a basis of the space S_{4}(23, 1) (the space of cusp forms of type (−4, 23, 1)).

Proof. I. As said above, F_{2} is a quadratic form of type (−2, 23, 1) and

φ_{11} = x_{1}^{2} − \frac{6}{23} F_{2}

is a spherical function of second order with respect to F_{2}. Hence, by Lemma 4, the theta-series (3.1) is a cusp form of type (−4, 23, 1).

Taking into account (2.3), it is not difficult to verify that the equation F_{2} = n

(a) has four integral solutions for n = 1: x_{1} = ±1, x_{2} = x_{3} = x_{4} = 0; x_{3} = ±1, x_{1} = x_{2} = x_{4} = 0;

(b) has four integral solutions for n = 3: x_{1} = ±1, x_{3} = 1, x_{2} = x_{4} = 0; x_{1} = ±1, x_{3} = −1, x_{2} = x_{4} = 0 for n = 2;

(c) has no integral solutions for n = 3;

(d) has four integral solutions for n = 4: x_{1} = ±2, x_{2} = x_{3} = x_{4} = 0; x_{3} = ±2, x_{1} = x_{2} = x_{4} = 0;

(e) has eight integral solutions for n = 5: x_{1} = ±2, x_{3} = 1, x_{2} = x_{4} = 0; x_{1} = ±2, x_{3} = −1, x_{2} = x_{4} = 0; x_{1} = ±1, x_{3} = 2, x_{2} = x_{4} = 0; x_{1} = ±1, x_{3} = −2, x_{2} = x_{4} = 0.

Hence

ϑ(τ; F_{2}, φ_{11}) = \frac{1}{23}\{(23 · 2 − 6 · 4)z + (23 − 6 · 2)4z^{2} +

+(23 · 4 · 2 − 6 · 4 · 4)z^{4} + (23 · 4 · 4 + 23 · 4 − 6 · 5 · 8)z^{5} + ⋯ } =

= \frac{22}{23} z + \frac{44}{23} z^{2} + \frac{88}{23} z^{4} + \frac{220}{23} z^{5} + · · · .  (3.6)

II. As said above F_{1} ⊕ Φ_{1} is a quadratic form of type (−2, 23, 1) and

φ_{11} = x_{1}^{2} − \frac{6}{23}(F_{1} ⊕ Φ_{1}) and φ_{22} = x_{2}^{2} − \frac{1}{23}(F_{1} ⊕ Φ_{1})

are spherical functions of second order with respect to F_{1} ⊕ Φ_{1}.

Hence, by Lemma 4, the theta-series (3.2) and (3.3) are cusp forms of type (−4, 23, 1).

Taking into account (2.7), it is not difficult to verify that the equation F_{1} ⊕ Φ_{1} = n

(a) has two integral solutions for n = 1: x_{1} = ±1, x_{2} = x_{3} = x_{4} = 0;

(b) has two integral solutions for n = 2: x_{3} = ±1, x_{1} = x_{2} = x_{4} = 0;

(c) has six integral solutions for n = 3: x_{1} = ±1, x_{3} = 1, x_{2} = x_{4} = 0; x_{1} = ±1, x_{3} = −1, x_{2} = x_{4} = 0; x_{1} = x_{2} = x_{3} = 0, x_{4} = ±1;

(d) has eight integral solutions for n = 4: x_{1} = ±1, x_{4} = 1, x_{2} = x_{3} = 0; x_{1} = ±1, x_{4} = −1, x_{2} = x_{3} = 0; x_{1} = ±2, x_{2} = x_{3} = x_{4} = 0; x_{1} = x_{2} = 0, x_{3} = 1, x_{4} = −1; x_{1} = x_{2} = 0, x_{3} = −1, x_{4} = 1;

(e) has four integral solutions for n = 5: x_{1} = ±1, x_{3} = 1, x_{4} = −1, x_{2} = 0; x_{1} = ±1, x_{2} = 0, x_{3} = −1, x_{4} = 1.

Hence,

ϑ(τ; F_{1} ⊕ Φ_{1}, φ_{11}) = \frac{1}{23}\{(23 − 6)2z + (−6 · 2 · 2)z^{2} +

+(23·4−6·3·6)z^{3}+(23·4·2+23·4−6·4·8)z^{4}+(23−6·5·4)z^{5}+ ⋯ }=

= \frac{34}{23} z − \frac{24}{23} z^{2} − \frac{16}{23} z^{3} + \frac{84}{23} z^{4} − \frac{28}{23} z^{5} + ⋯,  (3.7)
ϑ(τ; F_{1} ⊕ Φ_{1}, φ_{22}) =

= \frac{1}{23}\{ − 2z − 2 · 2z^{2} − 3 · 6z^{3} − 4 · 8z^{4} − 5 · 4z^{5} + ⋯ \} =

= −\frac{2}{23} z − \frac{4}{23} z^{2} − \frac{18}{23} z^{3} − \frac{32}{23} z^{4} − \frac{20}{23} z^{5} + · · · .  (3.8)

III. As said above Φ_{2} is a quadratic form of type (−2, 23, 1) and

φ_{11} = x_{1}^{2} − \frac{3}{23} Φ_{2} and φ_{22} = x_{2}^{2} − \frac{2}{23} Φ_{2}

are spherical functions of second order with respect to Φ_{2}. Hence, by Lemma 4, theta-series (3.4) and (3.5) are cusp forms of type (−4, 23, 1).

Taking into account (2.5) it is easy to verify that the equation Φ_{2} = n (a) has no integral solutions for n = 1;

(b) has four integral solutions for n = 2: x_{1} = ±1, x_{2} = x_{3} = x_{4} = 0; x_{3} = ±1, x_{1} = x_{2} = x_{4} = 0; (c) has four integral solutions for n = 3: x_{2} = ±1, x_{1} = x_{3} = x_{4} = 0;

x_{1} = x_{2} = x_{3} = 0, x_{4} = ±1;

(d) has eight integral solutions for n = 4: x_{1} = ±1, x_{3} = 1, x_{2} = x_{4} = 0; x_{1} = ±1, x_{3} = −1, x_{2} = x_{4} = 0; x_{1} = 1, x_{2} = −1, x_{3} = x_{4} = 0; x_{1} = −1, x_{2} = 1, x_{3} = x_{4} = 0; x_{1} = x_{2} = 0, x_{3} = 1, x_{4} = −1; x_{1} = x_{2} = 0, x_{3} = −1, x_{4} = 1;

(e) has eight integral solutions for n = 5: x_{1} = ±1, x_{2} = x_{3} = 0, x_{4} = 1; x_{1} = ±1, x_{2} = x_{3} = 0, x_{4} = −1; x_{1} = x_{4} = 0, x_{2} = ±1, x_{3} = 1; x_{1} = x_{4} = 0, x_{2} = ±1, x_{3} = −1.

Hence

ϑ(τ; Φ_{2}, φ_{11}) = \frac{1}{23}\{(23 · 1 · 2 − 3 · 2 · 4)z^{2} − 3 · 3 · 4z^{3} +

+(23 · 1 · 6 − 3 · 4 · 8)z^{4} + (23 · 1 · 4 − 3 · 5 · 8)z^{5} + ⋯ } =

= \frac{22}{23} z^{2} − \frac{36}{23} z^{3} + \frac{42}{23} z^{4} − \frac{28}{23} z^{5} + ⋯,  (3.9)

ϑ(τ; Φ_{2}, φ_{22}) = \frac{1}{23}\{ − 2 · 2 · 4z^{2} + (23 · 1 · 2 − 2 · 3 · 4)z^{3} +

+(23 · 1 · 2 − 2 · 4 · 8)z^{4} + (23 · 1 · 4 − 2 · 5 · 8)z^{5} + ⋯ } =

= −\frac{16}{23} z^{2} + \frac{22}{23} z^{3} − \frac{18}{23} z^{4} + \frac{12}{23} z^{5} + · · · .  (3.10)

The system of theta-series (3.1)–(3.5) is linearly independent, since the determinant of fifth order whose elements are coefficients in the expansions of these theta-series is different from zero. Thus the theorem is proved, since dim S_{4}(23, 1) = 5 [3, p. 900].

Theorem 2.

r(n; F_{4}) = \frac{24}{53} σ_{3}^{∗}(n) − \frac{240}{11 · 53}( ∑_{F_{2}=n} 23x_{1}^{2} − 6n) +

+\frac{440}{23 · 53}( ∑_{F_{1}⊕Φ_{1}=n} 23x_{1}^{2} − 6n) − \frac{2640}{23 · 53}( ∑_{F_{1}⊕Φ_{1}=n} 23x_{2}^{2} − n) −

−\frac{8144}{23 · 53}( ∑_{Φ_{2}=n} 23x_{1}^{2} − 3n) − \frac{14096}{23 · 53}( ∑_{Φ_{2}=n} 23x_{2}^{2} − 2n),  (I)

r(n; Φ_{4}) = \frac{24}{53} σ_{3}^{∗}(n) − \frac{28}{11 · 53}( ∑_{F_{2}=n} 23x_{1}^{2} − 6n) +

+\frac{16}{23 · 53}( ∑_{F_{1}⊕Φ_{1}=n} 23x_{1}^{2} − 6n) − \frac{96}{23 · 53}( ∑_{F_{1}⊕Φ_{1}=n} 23x_{2}^{2} − n) −

−\frac{512}{23 · 53}( ∑_{Φ_{2}=n} 23x_{1}^{2} − 3n) − \frac{1164}{23 · 53}( ∑_{Φ_{2}=n} 23x_{2}^{2} − 2n),  (II)

r(n; F_{3} ⊕ Φ_{1}) = \frac{24}{53} σ_{3}^{∗}(n) − \frac{28}{11 · 53}( ∑_{F_{2}=n} 23x_{1}^{2} − 6n) +

+\frac{175}{23 · 53}( ∑_{F_{1}⊕Φ_{1}=n} 23x_{1}^{2} − 6n) − \frac{1050}{23 · 53}( ∑_{F_{1}⊕Φ_{1}=n} 23x_{2}^{2} − n) −

−\frac{2685}{23 · 53}( ∑_{Φ_{2}=n} 23x_{1}^{2} − 3n) − \frac{4609}{23 · 53}( ∑_{Φ_{2}=n} 23x_{2}^{2} − 2n),  (III)

r(n; F_{2} ⊕ Φ_{2}) = \frac{24}{53} σ_{3}^{∗}(n) − \frac{28}{11 · 53}( ∑_{F_{2}=n} 23x_{1}^{2} − 6n) +

+\frac{122}{23 · 53}( ∑_{F_{1}⊕Φ_{1}=n} 23x_{1}^{2} − 6n) − \frac{732}{23 · 53}( ∑_{F_{1}⊕Φ_{1}=n} 23x_{2}^{2} − n) −

−\frac{1360}{23 · 53}( ∑_{Φ_{2}=n} 23x_{1}^{2} − 3n) − \frac{2330}{23 · 53}( ∑_{Φ_{2}=n} 23x_{2}^{2} − 2n),  (IV)

r(n; F_{1} ⊕ Φ_{3}) = \frac{24}{53} σ_{3}^{∗}(n) − \frac{28}{11 · 53}( ∑_{F_{2}=n} 23x_{1}^{2} − 6n) +

+\frac{3}{53}( ∑_{F_{1}⊕Φ_{1}=n} 23x_{1}^{2} − 6n) − \frac{18}{53}( ∑_{F_{1}⊕Φ_{1}=n} 23x_{2}^{2} − n) −

−\frac{1201}{23 · 53}( ∑_{Φ_{2}=n} 23x_{1}^{2} − 3n) − \frac{1959}{23 · 53}( ∑_{Φ_{2}=n} 23x_{2}^{2} − 2n),  (V)

where

σ_{3}^{∗}(n) = σ_{3}(n)  if 23 † n,

= σ_{3}(n) + 23^{2}σ_{3}(\frac{n}{23}) if 23|n.

Proof. From (2.3), (2.5) follow the relations

ϑ(τ; F_{4}) = ϑ^{2}(τ; F_{2}) = 1 + 8z + 24z^{2} + 32z^{3} + 24z^{4} + 48z^{5} + ⋯, (3.11)
ϑ(τ; Φ_{4}) = ϑ^{2}(τ; Φ_{2}) = 1 + 8z^{2} + 8z^{3} + 32z^{4} + 48z^{5} + ⋯, (3.12)

respectively. The couples of relations (2.2) and (2.4), (2.3) and (2.4), (2.1) and (2.6) lead to

ϑ(τ; F_{3} ⊕ Φ_{1}) = ϑ(τ; F_{3})ϑ(τ; Φ_{1}) =
= 1 + 6z + 14z^{2} + 22z^{3} + 44z^{4} + 76z^{5} + ⋯,  (3.13)

ϑ(τ; F_{2} ⊕ Φ_{2}) = ϑ(τ; F_{2})ϑ(τ; Φ_{2}) =
= 1 + 4z + 8z^{2} + 20z^{3} + 44z^{4} + 64z^{5} + ⋯,  (3.14)
ϑ(τ; F_{1} ⊕ Φ_{3}) = ϑ(τ; F_{1})ϑ(τ; Φ_{3}) =
= 1 + 2z + 6z^{2} + 18z^{3} + 32z^{4} + 60z^{5} + ⋯,  (3.15)

respectively. By Lemma 5, F_{4}, Φ_{4}, F_{3} ⊕ Φ_{1}, F_{2} ⊕ Φ_{2} and F_{1} ⊕ Φ_{3} are quadratic forms of type (−4, 23, 1), to which by Lemma 1 there corresponds one and the same Eisenstein series. For k = 4, from (1.2) we have

α = \frac{1}{ρ_{4}} \frac{23^{2} − 1}{23^{4} − 1} = \frac{1}{ρ_{4}} \frac{1}{23^{2} + 1}, β = \frac{1}{ρ_{4}} \frac{23^{4} − 23^{2}}{23^{4} − 1} = \frac{1}{ρ_{4}} \frac{23^{2}}{23^{2} + 1}

where ρ_{4} = \frac{1}{240} [3, p. 823] . Thus for all these forms we have

E(τ; F_{4}) = E(τ; Φ_{4}) = E(τ; F_{3} ⊕ Φ_{1}) = E(τ; F_{2} ⊕ Φ_{2}) = E(τ; F_{1} ⊕ Φ_{3}) =

= 1 + \frac{24}{53} ∑_{n=1}^{∞}(σ_{3}(n)z^{n} + 23^{2}σ_{3}(n)z^{23n}) =  (3.16)
= 1+ \frac{24}{53} z+ \frac{24 · 9}{53} z^{2}+ \frac{24 · 28}{53} z^{3}+ \frac{24 · 73}{53} z^{4}+ \frac{24 · 126}{53} z^{5}+· · · . (3.16_{1})

(I). By Lemma 2, the difference ϑ(τ; F_{4})−E(τ; F_{4}) is a cusp form of type (−4, 23, 1). Hence, by Theorem 1, there exist numbers c_{1}, …, c_{5} such that

ϑ(τ; F_{4}) − E(τ; F_{4}) = c_{1}ϑ(τ; F_{2}, φ_{11}) + c_{2}ϑ(τ; F_{1} ⊕ Φ_{1}, φ_{11}) +
+c_{3}ϑ(τ; F_{1} ⊕ Φ_{1}, φ_{22}) + c_{4}ϑ(τ; Φ_{2}, φ_{11}) + c_{5}ϑ(τ; Φ_{2}, φ_{22}).

Equating the coefficients of z, z^{2}, …, z^{5} on both sides of this equality and taking into account (3.11), (3.16_{1}) and (3.6)–(3.10), we can find these numbers and obtain

ϑ(τ, F_{4}) = E(τ; F_{4}) − \frac{240 · 23}{11 · 53} ϑ(τ; F_{2}, φ_{11}) +

+ \frac{440}{53} ϑ(τ; F_{1} ⊕ Φ_{1}, φ_{11}) − \frac{2640}{53} ϑ(τ; F_{1} ⊕ Φ_{1}, φ_{22}) −

− \frac{8144}{53} ϑ(τ; Φ_{2}, φ_{11}) − \frac{14096}{53} ϑ(τ; Φ_{2}, φ_{22}).

Equating now the coefficients of z^{n} on both sides of this equation, by (1.1), (3.16), and (3.1)–(3.5), we get the desired formula (I).

(II)-(V). Applying the same arguments as above except that (3.11) is replaced by (3.12), (3.13), (3.14), and (3.15) for the quadratic forms Φ_{4}, Φ_{3}⊕ Φ_{1}, F_{2} ⊕ Φ_{2}, and Φ_{1} ⊕ Φ_{3}, respectively, we obtain

ϑ(τ, Φ_{4}) = E(τ; Φ_{4}) − \frac{644}{11 · 53} ϑ(τ; F_{2}, φ_{11}) +

+ \frac{16}{53} ϑ(τ; F_{1} ⊕ Φ_{1}, φ_{11}) − \frac{96}{53} ϑ(τ; F_{1} ⊕ Φ_{1}, φ_{22}) −

− \frac{512}{53} ϑ(τ; Φ_{2}, φ_{11}) − \frac{1164}{53} ϑ(τ; Φ_{2}, φ_{22}),

ϑ(τ, F_{3} ⊕ Φ_{1}) = E(τ; F_{3} ⊕ Φ_{1}) − \frac{644}{11 · 53} ϑ(τ; F_{2}, φ_{11}) +

+ \frac{175}{53} ϑ(τ; F_{1} ⊕ Φ_{1}, φ_{11}) − \frac{1050}{53} ϑ(τ; F_{1} ⊕ Φ_{1}, φ_{22}) −

− \frac{2685}{53} ϑ(τ; Φ_{2}, φ_{11}) − \frac{4609}{53} ϑ(τ; Φ_{2}, φ_{22}),

ϑ(τ, F_{2} ⊕ Φ_{2}) = E(τ; F_{2} ⊕ Φ_{2}) − \frac{644}{11 · 53} ϑ(τ; F_{2}, φ_{11}) +

+ \frac{122}{53} ϑ(τ; F_{1} ⊕ Φ_{1}, φ_{11}) − \frac{732}{53} ϑ(τ; F_{1} ⊕ Φ_{1}, φ_{22}) −

− \frac{1360}{53} ϑ(τ; Φ_{2}, φ_{11}) − \frac{2330}{53} ϑ(τ; Φ_{2}, φ_{22}),

ϑ(τ, F_{1} ⊕ Φ_{3}) = E(τ; F_{1} ⊕ Φ_{3}) − \frac{644}{11 · 53} ϑ(τ; F_{2}, φ_{11}) +

+ \frac{69}{53} ϑ(τ; F_{1} ⊕ Φ_{1}, φ_{11}) − \frac{414}{53} ϑ(τ; F_{1} ⊕ Φ_{1}, φ_{22}) −

− \frac{1201}{53} ϑ(τ; Φ_{2}, φ_{11}) − \frac{1959}{53} ϑ(τ; Φ_{2}, φ_{22}).

From these identities, as above, we get the formulas (II)–(V).

References

1. A. Mirsalikhov, Theory of modular forms and the problem of finding formulas for the number of representations of numbers by positive quadratic forms in six variables. (Russian) Izv. Akad. Nauk Uzbek. SSR, Ser. Fiz.- Mat. 1(1971), 7–10.

2. H. Petersson, Modulfunktionen und quadratische Formen. Springer-Verlag, Berlin, Heidelberg, New-York, 1982.

3. E. Hecke, Mathematische Werke. Zweite Auflage, Vandenhoeck u. Ruprecht, Göttingen, 1970.

4. G. A. Lomadze, On the representations of numbers by sums of quadratic forms x_{1}^{2} + x_{1}x_{2} + x_{2}^{2}. (Russian) Acta Arith. 54(1987), 9–36.

(Received 5.10.1995)

Author’s address: Faculty of Mathematics and Mechanics I. Javakishvili Tbilisi State University 2, University St., Tbilisi 380043 Georgia