ON THE REPRESENTATION OF NUMBERS BY

Georgian Mathematical Journal Volume 16 (2009), Number 1, 81–88

ON THE REPRESENTATION OF NUMBERS BY SOME QUADRATIC FORMS OF LEVEL 12 NIKOLOZ KACHAKHIDZE Abstract. Formulas are obtained for the number of representations of positive integers by quadratic forms Fr = 3(x21 + · · · + x2r ) + x2r+1 + · · · + x29 (1 ≤ r ≤ 9). 2000 Mathematics Subject Classification: 11E25. Key words and phrases: Quadratic forms, representation of integers.

We will use the notation and notions from [1], [2] and [3]. Lomadze in [5] obtained the formula for the number of representations of natural numbers by the sum of nine squares. In the present paper the following method is used: first a function is constructed as a linear combination of theta functions with characteristics. Then applying the corresponding criterion it is shown that this function is a modular form of certain type. Next it is verified that the sufficient number of Fourier coefficients of this function is equal to zero. This implies that the function is identically equal to zero and as a result the identity is obtained. Equating the coefficients in both parts of this identity the corresponding formula is obtained. The present paper describes one of the applications of the results obtained in [2]–[4]. Namely, these results can be used to obtain the formulas for the number of representations of natural numbers by quadratic forms of certain type. As an example the quadratic form with nine variables is considered. It is well-known that to any positive quadratic form there corresponds the Eisenstein series. The difference of theta series of quadratic form and the corresponding Eisenstein series is a cusp form. The set of cusp forms of certain type is a finite-dimensional linear space. Hence the above-mentioned difference can be represented as a linear combination of basis forms. This linear combination gives us the identity, wherefrom equating the coefficients in both parts, we obtain the corresponding formula.
It is easy to verify that Fr is a quadratic form of type 92 , 12, χ , where
r χ(d) = 4·3 ( ·· ) is the Kronecker symbol . Furthermore, |d| !r   X 2πi S(Fr , 3) = exp − Fr (g) = exp(−2πix2 ) 3 g mod 3 x mod 3 !9−r   X 2πi 2 × exp − x = 3(9+r)/2 · (−i)9−r ; 3 x mod 3 X

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82

N. KACHAKHIDZE

!r 2πi S(Fr , 4) = · 3x2 exp − 4 x mod 4 !9−r  X 2πi 2 x = 213 (1 − i)ir . × exp − 4 x mod 4 

X

In what follows let n ∈ N, n = 22s ·32t n21 u, (6, n1 ) = 1, where u is square-free. Lemma 1. a) Let 2 | r E(τ, Fr ) = 1+

∞ X

  A9 (u) C1,9 (u, χ0 ) + (−1)r/2 · 27s+4

n=1



× B1,9 (u) + (−1)

r/2

7t+8− r2

·3



zn,

where      √  X Y u 48u3 · u 1 u −4 7 · L(4, u) q A9 (u) = 1 − d , 1 − 3485π 4 81 3 q d|n1 q|d  128 −1
if u 6≡ 1 (mod 4); (128s − 1) + C1,9 (u, χ0 ) = 1 2 1 + 8 · u if u ≡ 1 (mod 4), 127  2187 −1 t
if 3 | u; (2187 − 1) + B1,9 (u) = 81 · u3 if 3 - u. 1093 e0 (12)). Then ϑ(τ, Fr ) − E(τ, Fr ) ∈ S9/2 (Γ b) Let 2 - r, ∞   X r−1 A9 (3u) C1,9 (u, χ) + (−1) 2 · 27s+4 E(τ, Fr ) = 1+ n=1

  r−1 9−r × B2,9 (u) + (−1) 2 · 37t+ 2 z n , where

√    X Y 3u −4 1296u3 · 3u 7 L(4, 3u) d 1− q A9 (3u) = , 4 3485π q d|n1

128 (128s − 1) + C1,9 (u, χ) = 127



q|d

−1
if u 6≡ −1 (mod 4); 1 2 1 − 8 · u if u ≡ −1 (mod 4),  −1 if 3 - u; 2187 t
B2,9 (u) = (2187 − 1) + 1 u/3 2 + 27 ( 3 if 3 | u. 1093

   e0 (12), 12 . Then ϑ(τ, Fr ) − E(τ, Fr ) ∈ S9/2 Γ |d|

ON THE REPRESENTATION OF NUMBERS

83

After simple computations the proof directly follows from Theorem 1 of [4] and Corollaries 6 and 7 of [2].  It is known (see [5], pp. 246, 247) that π4 11π 4 √ , , L(4, 2) = 25 · 3 28 · 3 2 X  h  h2 h3  (u > 1, u ≡ 1 (mod 4)), − u 4u2 3u3

L(4, 1) = L(4, u) = −2π 4 u−1/2

0