GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 1, 1998, 91-100

ON THE REPRESENTATION OF NUMBERS BY POSITIVE DIAGONAL QUADRATIC FORMS WITH FIVE

VARIABLES OF LEVEL 16

D. KHOSROSHVILI

Abstract. A general formula is derived for the number of representations r(n; f) of a natural number n by diagonal quadratic forms f with five variables of level 16. For f belonging to one-class series, r(n; f) coincides with the sum of a singular series, while in the case of a many-class series an additional term is required, for which the generalized theta-function introduced by T. V. Vepkhvadze [4] is used.

1. Let f = f(x) = f(x_{1}, x_{2}, …, x_{s}) = \frac{1}{2}X′AX = \frac{1}{2} ∑_{j,k=1} a_{jk}x_{j}x_{k} be an integral positive quadratic form. Here and in what follows X is a column-vector, and X′ is a row-vector with components x_{1}, x_{2}, …, x_{s}. Let further r(n; f) denote the number of representations of a natural number n by the form f.

For our discussion we shall need the following results.

As is well known, for each quadratic form f we have the corresponding series

ϑ(τ, f) = 1 + ∑_{n=1}^{∞} r(n, f)Q^{n},  (1)

θ(τ, f) = 1 + ∑_{n=1}^{∞} ρ(n, f)Q^{n},  (2)

where Q = e^{2πiτ} (I_{m}τ > 0) and ρ(n, f) is a singular series. In the cases considered here the sum of the singular series can be calculated by means of the following two lemmas.

91

1991 Mathematics Subject Classification. 1991 Mathematics Subject Classification:

11F11, 11F03, 11F27.

Key words and phrases. Quadratic forms, modular forms, theta-functions.

1072-947X/98/0100-0091$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

Lemma 1 (see [1]). Let 2 ∤ s, Δ = 2^{s}Δ_{0}, nΔ_{0} = 2^{α+γ}v_{1}v_{2} = r^{2}ω,

2^{α}∥n, 2^{γ}∥Δ_{0}, p^{l}∥Δ_{0}, p^{ω}∥n, v_{1} = ∏_{\begin{matrix}p|n \\ p∤2Δ_{0} \end{matrix}} p^{ω} = r_{1}^{2}ω_{1}, v_{1} = ∏_{\begin{matrix}p|Δ_{0}n \\ p|Δ_{0},p>2 \end{matrix}} p^{ω+l} =

r_{2}^{2}ω_{2}, (ω, ω_{1} and ω_{2} are square-free integers).

Then

ρ(n, f) = \frac{2^{2− \frac{s}{2}} π^{1− \frac{s}{2}} (s − 1)!}{Γ(\frac{s}{2})Δ0^{\frac{1}{2}} B_{\frac{s−1}{2}}}n^{\frac{s}{2} −1}r_{1}^{2−s}χ(2)Π\begin{matrix}p|Δ_{0} \\ p>2 \end{matrix}χ(p) ×

×Π_{p|2Δ_{0}}(1 − p^{1−s})^{−1}L(\frac{s − 1}{2}, (−1)^{\frac{s−1}{2}} ω)Π\begin{matrix}p|r_{2} \\ r_{2}>2 \end{matrix}(1 − (\frac{(−1)^{\frac{s−1}{2}} ω}{p} )p^{\frac{1−s}{2}} )×

× ∑_{d|r_{1}} d^{s−2}Π_{p|d}(1 − (\frac{(−1)^{\frac{s−1}{2}} ω}{p} )p^{\frac{1−s}{2}} ),  (3)

where B_{\frac{s−1}{2}} are Bernoulli’s numbers, (\frac{·}{p}) is Jacobi’s symbol, and the values of χ(2) are given in [2] (p. 66, formulas (28)–(33)).

For the case s = 5 the values of L(·, ·) are given in

Lemma 2 (see, e. g., [3]).

L(2; 1) = \frac{π^{2}}{8} , L(2; 2) = \frac{2^{\frac{1}{2}} π^{2}}{16} ,
L(2; ω) = −\frac{π}{ω^{\frac{3}{2}}} ∑_{1≤h≤ \frac{ω}{2}} h(\frac{h}{ω} ), if ω ≡ 1 (mod 4), ω > 1;
L(2; ω) = \frac{π^{2}}{2ω^{\frac{3}{2}}} \{2 ∑_{1≤h≤ \frac{ω}{4}} h(\frac{h}{ω} ) + ∑_{\frac{ω}{4} <h≤ \frac{ω}{2}} (ω − 2h)(\frac{h}{ω} )\},
if ω ≡ 3 (mod 4);
L(2; ω) = \frac{π^{2}}{4ω^{\frac{3}{2}}} \{ω ∑_{1≤h≤ \frac{ω}{16}} ( \frac{h}{\frac{1}{2}ω} ) + ∑_{\frac{ω}{16} <h≤ \frac{3ω}{16}} (ω − 16h)( \frac{h}{\frac{1}{2}ω} ) −

− 2ω ∑_{\frac{3ω}{16} <h≤ \frac{ω}{4}} ( \frac{h}{\frac{1}{2}ω} )\}, if ω ≡ 2 (mod 8), ω > 2;

L(2; ω) = \frac{π^{2}}{4ω^{\frac{3}{2}}} \{16 ∑_{1≤h≤ \frac{ω}{16}} ( \frac{h}{\frac{1}{2}ω} )+ω ∑_{\frac{ω}{16} <h≤ \frac{3ω}{16}} ( \frac{h}{\frac{1}{2}ω} )+4ω ∑_{\frac{3ω}{16} <h≤ \frac{ω}{4}} ( \frac{h}{\frac{1}{2}ω} )−

− 16ω ∑_{\frac{3ω}{16} <h≤ \frac{ω}{4}} h( \frac{h}{\frac{1}{2}ω} )\}, if ω ≡ 6 (mod 8).

In [4] Vepkhvadze constructed generalized theta-functions with characteristic and spherical functions

ϑ_{gh}(τ; P_{ν}, f) = ∑_{X≡g (mod N)}(−1)^{\frac{h′A(X−g)}{N^{2}}} P_{ν}(X)e^{\frac{πiτX′AX}{N^{2}}} .  (4)

Here g and h are special vectors with respect to the matrix A of form f, i.e.,

Ag ≡ 0 (mod N), Ah ≡ 0 (mod N),

where N is a level of the form f, i.e., the smallest integer for which NA^{−1} is a symmetric integral matrix with even diagonal elements; P_{ν} = P_{ν}(x) = P_{ν}(x_{1}, …, x_{s}) is a spherical function of ν-th order with respect to f.

The properties of functions (4) are investigated in [4], where these functions are used to derive a formula for the number of representations of a quadratic form with seven variables.

In this paper we use the method of [4] to obtain formulas for the number of representations of natural numbers by all positive diagonal quadratic forms with five variables of level 16.

Lemma 3 (see, e.g., [4], Lemma 4). Let k be an arbitrary integral vector, and l a special vector with respect to the matrix A of the form f. Then the equalities

ϑ_{g+Nk,h}(τ; P_{ν}, f)=(−1)^{\frac{h′Ak}{N}} ϑ_{gh}(τ; P_{ν}, f), ϑ_{g,h+2l}(τ; P_{ν}, f)=ϑ_{gh}(τ; P_{ν}, f)

are valid.

For M = (\begin{matrix}α & β \\ γ & δ \end{matrix}) ∈ Γ_{0}(N) denote

v(M) = (i^{\frac{1}{2} η(γ)(sgn δ−1)})^{s+2ν}(sgn δ)^{ν}(i^{( \frac{|δ|−1}{2} )^{2}})^{s+2ν}(\frac{2Δ(sgn δ)β}{|δ|} )(\frac{−1}{|δ|}),(5)

η(γ) = 1 for γ ≥ 0, η(γ) = −1 for γ < 0. By v_{0}(M) we denote v(M) in the case ν = 0.

Lemma 4 (see, e.g., [4], Theorem 2). Let f = f(x) be an integral positive quadratic form with an odd number of variables s, Δ the determinant of the matrix A of the form f, and N the level of the form f. Then function (1) is an integral modular form of type ( − \frac{s}{2}, N, v_{0}(M)).

Lemma 5 (see, e.g., [4], Theorem 2). Let f_{k} = f_{k}(x) (k = 1, …, j) be integral positive quadratic forms with the number of variables s, P_{ν}^{(k)} = P_{ν}^{(k)}(x) (k = 1, 2, …, j) the corresponding spherical functions, A_{k} a matrix of the form f_{k}(x), Δ_{k} the determinant of the matrix A_{k}, and N_{k} the level of the form f_{k}. Let further g^{(k)} and h^{(k)} be vectors with even components, and B_{k} arbitrary complex numbers. Then the function

Φ(τ) = ∑_{k=1}^{j} B_{k}ϑ_{g^{(k)}h^{(k)}}(τ; P_{ν}^{(k)}, f_{k})

is an integral modular form of the type ( − (\frac{s}{2} + ν), N, v(M)), where v(M) are determined by formula (5), if and only if the conditions

N_{k}|N, N_{k}^{2}|f_{k}(g^{(k)}), 4N_{k}|\frac{N}{N_{k}} f_{k}(h^{(k)})  (6)

are fulfilled, and for all α and δ satisfying the condition αδ ≡ 1 (mod N) we have

∑_{k=1}^{j} B_{k}ϑ_{αg^{(k)},−h^{(k)}}(τ; P_{ν}^{(k)}, f_{k})(sgn δ)^{ν}(\frac{(−1)^{\frac{s−1}{2}} Δ_{k}}{|δ|} ) =

= (\frac{(−1)^{\frac{s−1}{2} +ν}Δ}{|δ|}) ∑_{k=1}^{j} B_{k}ϑ_{g^{(k)}h^{(k)}}(τ; P_{ν}^{(k)}, f_{k}).  (7)

Lemma 6 (see, e.g., [5], Theorem 4). If all the conditions of Lemma 5 are fulfilled and ν > 0, then the function Φ(τ) is a cusp form of the type ( − (\frac{s}{2} + ν), N, v(M)).

Lemma 7 (see, e.g., [4], Theorem 1). Let F be an integral modular form of the type (−Γ, N, v(M)), where v(M) are determined by formula (5). Then the function F is identically zero if in its expansion into powers Q = e^{2πiτ} the coefficients of Q^{n} are zero for all

n ≤ \frac{r}{12}N ∏_{p|N} (1 + \frac{1}{p}).

2. Positive diagonal quadratic forms with five variables of level 16 are written as

f_{s_{1},s_{2}} = ∑_{j=1}^{s_{1}} x_{j}^{2} + 2 ∑_{j=s_{1}+1}^{s_{2}} x_{j}^{2} + 4 ∑_{j=s_{2}+1}^{5} x_{j}^{2},

where 1 ≤ s_{1} ≤ s_{2} ≤ 4.

Theorem 1. Let f_{1} = 4x_{1}^{2} + 4x_{2}^{2} + 2x_{3}^{2}, P_{1} = x_{3}, g′ = (4, 4, 8), h′ = (2, 2, 4). Then the identity

ϑ(τ; f_{s_{1},s_{2}}) = θ(τ; f_{s_{1},s_{2}}) + Φ(τ; f_{s_{1},s_{2}}),  (8)

holds, where

Φ(τ; f_{1,2}) = \frac{1}{16}ϑ_{gh}(τ; P_{1}, f_{1}),
Φ(τ; f_{2,3}) = Φ(τ; f_{3,4}) = \frac{1}{4}ϑ_{gh}(τ; P_{1}, f_{1}),
Φ(τ; f_{s_{1},s_{2}}) = 0 in other cases.

Proof. By Lemma 4 the function ϑ(τ; f_{s_{1},s_{2}}) belongs to the space of integral modular forms of the type ( − \frac{5}{2}, 16, v_{0}(M)), where the system of multiplicators v_{0}(M) is calculated by formula (5). Therefore by Siegel’s theorem the function θ(τ; f_{s_{1},s_{2}}) also belongs to this space.

It is easy to verify that the function Φ(τ; f_{s_{1},s_{2}}) satisfies conditions (6) of Lemma 5.

If αδ ≡ 1 (mod 16), then αδ ≡ 1 (mod 4), i.e., either α ≡ 1 (mod 4) and δ ≡ 1 (mod 4) or α ≡ −1 (mod 4) and δ ≡ −1 (mod 4).

In our case condition (7) of Lemma 5 is written as

ϑ_{αg,−h}(τ; P_{1}, f_{1})(sgn δ)(\frac{−2^{8}}{|δ|} ) = (\frac{2^{10}}{|δ|} )ϑ_{gh}(τ; P_{1}, f_{1})  (9)

and we must check it.

1. Let α ≡ 1 (mod 4) and δ ≡ 1 (mod 4). It is easy to verify that

(sgn δ)(\frac{−2^{8}}{|δ|} ) = (\frac{2^{10}}{|δ|} )

and since αg = Nk_{1} +g with as an integral vector k_{1}, together with Lemma 3 this implies the validity of (9).

2. We now set α ≡ −1 (mod 4) and δ ≡ −1 (mod 4). Since

(sgn δ)(\frac{−2^{8}}{|δ|} ) = −(\frac{2^{10}}{|δ|} )

and αg = Nk_{2} − g, where k_{2} is an integral vector, and, as is easy to verify, ϑ_{−g,h}(τ; P_{1}, f_{1}) = −ϑ_{g,h}(τ; P_{1}, f_{1}), Lemma 3 implies (9). From (9) it follows that the function ϑ_{gh}(τ; P_{1}, f_{1}) satisfies conditions (7) of Lemma 5 as well. Hence, by Lemmas 5 and 6, the function ϑ_{gh}(τ; P_{1}, f_{1}) is a cusp form of the type ( − \frac{5}{2}, 16, v_{0}(M)).

Therefore due to Lemma 7 the function

ψ(τ; f_{s_{1},s_{2}}) = ϑ(τ; f_{s_{1},s_{2}}) − θ(τ; f_{s_{1},s_{2}}) − Φ(τ; f_{s_{1},s_{2}})  (10)

will be identically zero if in its expansion into powers of Q = e^{2πiτ} all coefficients of Q^{n} for n ≤ 5 are zero.

Let n = 2^{α}m (2 ∤ m, α ≥ 0), 2^{10−s_{1}−s_{2}}n = r^{2}ω, m = r_{1}^{2}ω_{1}, ω and ω_{1} be square-free integers. Then by formulas (2) and (3) we have

θ(τ; f_{s_{1},s_{2}}) = 1 + ∑_{n=1}^{∞} ρ(n; f_{s_{1},s_{2}})Q^{n},

where

ρ(n; f_{s_{1},s_{2}}) = \frac{2^{\frac{3α+s_{1}+s_{2}}{2} +2}ω1^{\frac{3}{2}}}{π^{2}} ∑_{d|r_{1}} d^{3} ∏_{p|d} (1 − (\frac{ω}{p} )p^{−2})L(2; ω)χ(2). (11)

The values of L(2, ω) are given by Lemma 2. Introduce the notation χ_{s_{1},s_{2}}(2) for the values of χ(2) corresponding to the quadratic form f_{s_{1},s_{2}}. Using formulas (28)–(33) from [3], we obtain

χ_{2,3}(2)=\begin{cases}1, & for α = 0 or α = 2; \\ \frac{2^{−\frac{3α}{2}−\frac{1}{2}}}{7}(13·2^{\frac{3α}{2}−\frac{1}{2}} +2 −7(\frac{2}{m})), & for 2 ∤ α, m ≡ 1 (mod 4); \\ \frac{2^{−\frac{3α}{2}+\frac{1}{2}}}{7}(13·2^{\frac{3α}{2}−\frac{3}{2}} +15), & for 2 ∤ α, m ≡ 3 (mod 4); \\ \frac{2^{− \frac{3α}{2} +2}}{7} (13 · 2^{\frac{3α}{2} −3} + 15), & for 2 | α, α > 2. \end{cases} (12)

After calculating the values of ρ(n; f_{2,3}) for all n ≤ 5, by (2), (11) and (12) we have

θ(τ; f_{2,3}) = 1 + 2Q + 6Q^{2} + 12Q^{3} + 16Q^{4} + 28Q^{5} + ….

Formula (1) implies

ϑ(τ; f_{2,3}) = 1 + 4Q + 6Q^{2} + 8Q^{3} + 16Q^{4} + 24Q^{5} + ….

By (4) we obtain

\frac{1}{8}ϑ_{gh}(τ; P_{1}, f_{1}) = ∑_{n=1}^{∞} ( ∑_{\begin{matrix}4n=x_{1}^{2}+x_{2}^{2}+2x_{3}^{2} \\ x_{1}≡1 (mod 4) \\ x_{2}≡1 (mod 4) \\ 2∤x_{3} \end{matrix}}(−1)^{\frac{x_{1}−1}{4} + \frac{x_{2}−1}{4} + \frac{x_{3}−1}{2}} x_{3}Q^{n}) =

= 2Q − 4Q^{3} − 4Q^{5} + ….  (13)

Now it is not difficult to verify that all coefficients of Q^{n} in the expansion into powers of Q of the function ψ(τ; f_{2,3}) determined by (10) are zero for all n ≤ 5. Thus identity (8) is proved for the case, where s_{1} = 2 and s_{2} = 3.

For other values of s_{1} and s_{2}, the theorem is proved similarly. We give here a list of suitable values of χ(2) calculated by means of formulas (28)– (33) from [2]:

χ_{1,1}(2) = \begin{cases}0, & for α = 1 or α = 0, \\ & m ≡ 3 (mod 4); \\ 2, & for α = 0, m ≡ 1 (mod 4); \\ \frac{2^{−\frac{3α}{2}+1}}{7}(5·2^{\frac{3α}{2}} +2−7(\frac{2}{m})), & for 2|α, α > 1, m ≡ 1 (mod 4); \\ \frac{2^{− \frac{3α}{2} +2}}{7} (5 · 2^{\frac{3α}{2} −1} + 15), & for 2|α, α > 1, m ≡ 3 (mod 4); \\ \frac{2^{− \frac{3α}{2} + \frac{7}{2}}}{7} (5 · 2^{\frac{3α}{2} − \frac{5}{2}} + 15), & for 2 ∤ α, α > 1; \end{cases}

χ_{1,2}(2) = \begin{cases}1, & for α = 0 or α = 1, \\ & m ≡ 3 (mod 4) or α = 2; \\ \frac{2^{−\frac{3α}{2}+\frac{1}{2}}}{7}(3·2^{\frac{3α}{2}+\frac{1}{2}} +2−7(\frac{2}{m})), & for 2 ∤ α, m ≡ 1 (mod 4); \\ \frac{2^{− \frac{3α}{2} + \frac{3}{2}}}{7} (3 · 2^{\frac{3α}{2} − \frac{1}{2}} + 15), & for 2 ∤ α, α>1, m≡3 (mod 4); \\ \frac{2^{− \frac{3α}{2} +3}}{7} (3 · 2^{\frac{3α}{2} −2} + 15), & for 2 | α, α > 2; \end{cases}

χ_{1,3}(2) = \begin{cases}1, & for α = 0 or α = 1; \\ \frac{2^{− \frac{3α}{2}}}{7}(5 · 2^{\frac{3α}{2}} +2−7(\frac{2}{m})), & for 2 | α, α>1, m≡3 (mod 4); \\ \frac{2^{− \frac{3α}{2} +1}}{7}(5·2^{\frac{3α}{2} −1}+15), & for 2 | α, α>1, m≡3 (mod 4); \\ \frac{2^{− \frac{3α}{2} + \frac{5}{2}}}{7} (5 · 2^{\frac{3α}{2} − \frac{5}{2}} + 15), & for 2 ∤ α, α > 1; \end{cases}

χ_{1,4}(2) = χ_{2,3}(2) (see (12));

χ_{2,2}(2) = \begin{cases}1, & for 2 | α, α ≥ 0, m ≡ 3 (mod 4); \\ \frac{2^{−\frac{3α}{2}}}{7}(5·2^{\frac{3α}{2}} +2−7(\frac{2}{m})), & for 2 | α, α≥0, m≡1 (mod 4); \\ \frac{2^{− \frac{3α}{2} +1}}{7} (5 · 2^{\frac{3α}{2} −1} + 15), & for 2 | α, α≥0, m≡3 (mod 4); \\ \frac{2^{− \frac{3α}{2} + \frac{5}{2}}}{7} (5 · 2^{\frac{3α}{2} − \frac{5}{2}} +15), & for 2 ∤ α, α > 1; \end{cases}

98 D. KHOSROSHVILI

χ_{2,4}(2) = \begin{cases}1, & for α = 0 or α = 1; \\ \frac{2^{−\frac{3α}{2}−1}}{7}(3·2^{\frac{3α}{2}+2}+2−7(\frac{2}{m})), & for 2 | α, α > 1, \\ & m ≡ 1 (mod 4); \\ \frac{2^{− \frac{3α}{2}}}{7} (3 · 2^{\frac{3α}{2} +1} + 15), & for 2 | α, α > 1, \\ & m ≡ 3 (mod 4); \\ \frac{2^{− \frac{3α}{2} + \frac{3}{2}}}{7} (5 · 2^{\frac{3α}{2} − \frac{1}{2}} + 15), & for 2 ∤ α, α > 1; \end{cases}

χ_{3,3}(2) = \begin{cases}\frac{3}{2}, & for α = 1 or α = 0, m ≡ 1 (mod 4); \\ \frac{1}{2}, & for α = 0, m ≡ 3 (mod 4); \\ \frac{2^{− \frac{3α}{2} −1}}{7} , & for 2 | α, α > 1, m ≡ 1 (mod 4); \\ \frac{2^{− \frac{3α}{2}}}{7} , & for 2 | α, α > 1, m ≡ 3 (mod 4); \\ \frac{2^{− \frac{3α}{2} + \frac{3}{2}}}{7} , & for 2 ∤ α, α > 1; \end{cases}

χ_{3,4}(2) = \begin{cases}1, & for α = 0 or α = 1, \\ & m ≡ 3 (mod 4) or α = 2; \\ \frac{2^{−\frac{3α}{2}−\frac{3}{2}}}{7}(27·2^{\frac{3α}{2}−\frac{1}{2}} +2−7(\frac{2}{m})), & for 2 ∤ α, , m ≡ 1 (mod 4); \\ \frac{2^{−\frac{3α}{2}−\frac{3}{2}}}{7}(27·2^{\frac{3α}{2}−\frac{1}{2}} +30), & for 2 ∤ α, α > 1, \\ & m ≡ 3 (mod 4); \\ \frac{2^{− \frac{3α}{2} +1}}{7} (9 · 2^{\frac{3α}{2} −3} + 5), & for 2 | α, α > 2; \end{cases}

χ_{4,4}(2) = \begin{cases}1, & for α = 0; \\ \frac{3 · 2^{− \frac{3α}{2} + \frac{1}{2}}}{7} (2^{\frac{3α}{2} − \frac{1}{2}} + 5), & for 2 ∤ α; \\ \frac{2^{− \frac{3α}{2} −2}}{7}(3·2^{\frac{3α}{2}+2}+2−7(\frac{2}{m})), & for 2 | α, m ≡ 1 (mod 4); \\ \frac{3 · 2^{− \frac{3α}{2} −1}}{7} (2^{\frac{3α}{2} +1} + 5), & for 2 | α, m ≡ 3 (mod 4). \end{cases}

Theorem 2. Let n = 2^{α}m (α ≥ 0, 2 ∤ m), m = r_{1}^{2}ω_{1}, 1 ≤ s_{1} ≤ s_{2} ≤ 4, 2^{10−s_{1}−s_{2}}n = r^{2}ω (ω and ω_{1} are square-free integers). Then

r(n; f_{s_{1},s_{2}}) = \frac{2^{\frac{3α+s_{1}+s_{2}}{2} +1}ω1^{\frac{3}{2}}}{π^{2}} ∑_{d|r_{1}} d^{3} ∏_{p|d} (1 − (\frac{ω}{p} )p^{−2})L(2; ω)χ(2) +
+ ν_{s_{1},s_{2}}(n),  (14)

where

2ν_{1,2}(n) = ν_{2,3}(n) = ν_{3,4}(n) = 2 ∑_{\begin{matrix}4n=x_{1}^{2}+x_{2}^{2}+2x_{3}^{2} \\ 2∤x_{1}, 2∤x_{2}, 2∤x_{3} \\ x_{1}>0, x_{2}>0, x_{3}>0 \end{matrix}} (\frac{2}{x_{1}x_{2}} )(\frac{−1}{x_{3}})x_{3},

ν_{s_{1},s_{2}}(n) = 0 in other cases.

Proof. By equating the coefficients of equal powers of Q in both parts of identity (8) we obtain

r(n; f_{s_{1},s_{2}}) = ρ(n; f_{s_{1},s_{2}}) + ν_{s_{2},s_{2}}(n),  (15)

where ν_{s_{1},s_{2}}(n) denotes the coefficients of Q^{n} in the expansion of the function Φ(τ; f_{s_{1},s_{2}}) into powers of Q.

When s_{1} = 2 and s_{2} = 3, by (13) we have

ν_{2,3}(n) = ∑_{\begin{matrix}4n=x_{1}^{2}+x_{2}^{2}+2x_{3}^{2} \\ x_{1}≡1 (mod 4) \\ x_{2}≡1 (mod 4) \\ 2∤x_{3} \end{matrix}}(−1)^{\frac{x_{1}−1}{4} + \frac{x_{2}−1}{4} + \frac{x_{3}−1}{2}} x_{3}

i.e.,

ν_{2,3}(n) = 2 ∑_{\begin{matrix}4n=x_{1}^{2}+x_{2}^{2}+x_{3}^{2} \\ 2∤x_{1},2∤x_{2},2∤x_{3} \\ x_{1}>0,x_{2}>0,x_{3}>0 \end{matrix}} (\frac{2}{x_{1}x_{2}} )(\frac{−1}{x_{3}})x_{3}.  (16)

From formulas (11), (15) and (16) it follows that the theorem is valid when s_{1} = 2 and s_{2} = 3. The validity of equality (14) for other values of s_{1} and s_{2} is proved in a similar manner.

References

1. R. I. Beridze, On the summation of a singular Hardy–Littlewood series. (Russian) Bull. Acad. Sci. Georgian SSR 38(1965), No. 3, 529–534.

2. A. I. Malishev, On the representation of integers numbers by positive quadratic forms. (Russian) Trudy Mat. Inst. Steklov. 65(1962), 1–319.

3. T. V. Vepkhvadze, On the representation of numbers by positive quadratic forms with five variables. (Russian) Trudy Tbiliss. Mat. Inst. Razmadze 84(1987), 21–27.

4. T. V. Vepkhvadze, Generalized theta-functions with characteristics and the representation of numbers by quadratic forms. (Russian) Acta Arithmetica 53(1990), 433–451.

(Received 26.06.1995; revised 16.08.1996)

Author’s address: I. Javakhishvili Tbilisi State University Faculty of Physics 1, I. Chavchavadze Ave. Tbilisi 380028 Georgia