The Representation Numbers of Three Octonary Quadratic Forms

Abstract

In this study we use some known convolution sums to find the representation number for each of the three octonary quadratic forms [Formula: see text], [Formula: see text] and [Formula: see text].

Representations by certain octonary quadratic forms with coefficients 1, 3 or 9

“…We now use Lemma 2.1 to prove the following two theorems. 3,4,6,9,12,18,36) together with E 4,χ (q t ) (t = 1, 2, 4) constitute a basis for E 4 (Γ 0 (36)). (c) E 4 (q t ) (k = 1, 2, 3, 4, 6, 9, 12, 18, 36), E 4,χ (q t ) (t = 1, 2, 4) together with A k (q)…”

“…Formulas for N (1 6 , 3 2 ; n), N (1 4 , 3 4 ; n) and N (1 2 , 3 6 ; n) are given by Alaca and Williams [4]. A formula for N (1 4 , 3 4 ; n) is also given by Köklüce [9].…”

Representations by octonary quadratic forms with coefficients 1, 3 or 9

“…, x 8 ) ∈ Z 8 | n = a 1 x with i + j + k + l = 8 under the conditions i ≡ j ≡ k ≡ l ≡ 0 (mod 2) or i ≡ j ≡ k ≡ l ≡ 1 (mod 2) appeared in literature. See [1], [2], [3], [4], [6], [9] and [11]. For convenience, we write (i, j, k, l) to denote an octonary quadratic form given by (1.1), and we write N(1 i , 2 j , 3 k , 6 l ; n) to denote the number of representations of n by the octonary quadratic form (i, j, k, l).…”

Eisenstein Series, Eta Quotients and Their Applications in Number Theory

“…See [31] for a classical history of this research. For contemporary accounts of the subject see [13,14,19,24,33,43,50,57]. In this chapter we apply the results given in Chapter 9 to In the first section, we find formulas for…”