On the Representation of Numbers by the Direct Sums of Some Quaternary Quadratic Forms

2001 ◽  
Vol 8 (1) ◽  
pp. 87-95
Author(s):  
N. Kachakhidze
Keyword(s):  
Quadratic Forms ◽  
Cusp Forms ◽  
Arbitrary Integer ◽  
Direct Sums ◽  
Positive Integers

Abstract The systems of bases are constructed for the spaces of cusp forms S 2m (Γ0(5), χ m ) and S 2m (Γ0(13), χ m ) for an arbitrary integer m ≥ 2. Formulas are obtained for the number of representations of a positive integers by the direct sums of some quaternary quadratic forms.

scholarly journals Cusp Forms in and the Number of Representations of Positive Integers by Some Direct Sum of Binary Quadratic Forms with Discriminant

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Barış Kendirli
Keyword(s):  
Quadratic Forms ◽  
Cusp Forms ◽  
Binary Quadratic Forms ◽  
Direct Sum ◽  
Positive Integers

A basis of is given and the formulas for the number of representations of positive integers by some direct sum of the quadratic forms , , are determined.

scholarly journals The Bases of and the Number of Representation of Integers

2013 ◽  
Vol 2013 ◽  
pp. 1-34
Author(s):  
Barış Kendirli
Keyword(s):  
Quadratic Forms ◽  
Fundamental Theorem ◽  
Class Group ◽  
Equivalence Classes ◽  
Explicit Formulas ◽  
Direct Sums ◽  
Representation Of Integers ◽  
Positive Integers

Following a fundamental theorem of Hecke, some bases of and are determined, and explicit formulas are obtained for the number of representations of positive integers by all possible direct sums (111 different combinations) of seven quadratic forms from the class group of equivalence classes of quadratic forms with discriminant −71 whose representatives are .

On the Representation of Numbers by the Direct Sums of Some Binary Quadratic Forms

1998 ◽  
Vol 5 (1) ◽  
pp. 55-70
Author(s):  
N. Kachakhidze
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Cusp Forms ◽  
Binary Quadratic Forms ◽  
Direct Sums

Abstract The systems of bases are constructed for the spaces of cusp forms Sk (Γ0(3), χ) (k≥6), Sk (Γ0(7), χ) (k≥3) and Sk (Γ0(11), χ) (k≥3). Formulas are obtained for the number of representation of a positive integer by the sum of k binary quadratic forms of the kind , of the kind and of the kind .

On the Number of Representations of Positive Integers by a Direct Sum of Binary Quadratic Forms with Discriminant –23

1997 ◽  
Vol 4 (6) ◽  
pp. 523-532
Author(s):  
G. Lomadze
Keyword(s):  
Quadratic Forms ◽  
Binary Quadratic Forms ◽  
Direct Sums ◽  
Direct Sum ◽  
Positive Integers

Abstract Explicit exact formulas are obtained for the number of representations of positive integers by some direct sums of quadratic forms and .

On the Number of Representations of Integers by Quadratic Forms in Twelve Variables

1998 ◽  
Vol 5 (6) ◽  
pp. 545-564
Author(s):  
G. Lomadze
Keyword(s):  
Quadratic Forms ◽  
Explicit Formulas ◽  
Representations Of Integers ◽  
Positive Integers

Abstract A way of finding exact explicit formulas for the number of representations of positive integers by quadratic forms in 12 variables with integral coefficients is suggested.

On Some Entire Modular Forms of Weights and for the Congruence Group Γ0(4N)

1996 ◽  
Vol 3 (5) ◽  
pp. 485-500
Author(s):  
G. Lomadze
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Congruence Group ◽  
Positive Integers

Abstract Entire modular forms of weights and for the congruence group Γ0(4N) are constructed, which will be useful for revealing the arithmetical sense of additional terms in formulas for the number of representations of positive integers by quadratic forms in 7 and 9 variables.

Ternary Quadratic Forms and Eta Quotients

2015 ◽  
Vol 58 (4) ◽  
pp. 858-868 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Quadratic Forms ◽  
Fourier Coefficients ◽  
Arithmetic Progressions ◽  
Dedekind Eta Function ◽  
Ternary Quadratic Forms ◽  
Sum Formula ◽  
Image Position ◽  
Positive Integers

AbstractLet denote the Dedekind eta function. We use a recent productto- sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly ten eta quotientssuch that the Fourier coefficients c(n) vanish for all positive integers n in each of infinitely many non-overlapping arithmetic progressions. For example, we show that if we have c(n) = 0 for all n in each of the arithmetic progressions

scholarly journals Integral quadratic forms avoiding arithmetic progressions

2020 ◽  
Vol 16 (10) ◽  
pp. 2141-2148
Author(s):  
A. G. Earnest ◽  
Ji Young Kim
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Arithmetic Progressions ◽  
Integral Quadratic Form ◽  
Positive Integers

For every positive integer [Formula: see text], it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in [Formula: see text] arithmetic progressions. For [Formula: see text], all forms with this property are determined.

The number of representations of integers by generalized Bell ternary quadratic forms

2020 ◽  
pp. 1-15
Author(s):  
Kyoungmin Kim ◽  
Yeong-Wook Kwon
Keyword(s):  
Quadratic Form ◽  
Class Number ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Cusp Forms ◽  
Ternary Quadratic Form ◽  
Closed Formula ◽  
Integral Quadratic Form ◽  
Ternary Quadratic Forms ◽  
Representations Of Integers

For a positive definite ternary integral quadratic form [Formula: see text], let [Formula: see text] be the number of representations of an integer [Formula: see text] by [Formula: see text]. A ternary quadratic form [Formula: see text] is said to be a generalized Bell ternary quadratic form if [Formula: see text] is isometric to [Formula: see text] for some nonnegative integers [Formula: see text]. In this paper, we give a closed formula for [Formula: see text] for a generalized Bell ternary quadratic form [Formula: see text] with [Formula: see text] and class number greater than [Formula: see text] by using the Minkowski–Siegel formula and bases for spaces of cusp forms of weight [Formula: see text] and level [Formula: see text] with [Formula: see text] consisting of eta-quotients.

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