Summation of Singular Series Corresponding to the Representation of Integers by Some Quadratic Forms in Fourteen Variables

2000 ◽  
Vol 7 (2) ◽  
pp. 355-372
Author(s):  
G. Lomadze
Keyword(s):  
Quadratic Forms ◽  
Singular Series ◽  
Representation Of Integers ◽  
Positive Integers

Abstract A sum of the singular series corresponding to the number of representations of positive integers by some diagonal quadratic forms with integral coefficients is obtained.

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 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 Representation of integers: a nonclassical point of view

2020 ◽  
Vol 12 ◽  
Author(s):  
Bellaouar Djamel ◽  
Boudaoud Abdelmadjid
Keyword(s):  
Point Of View ◽  
Survey Article ◽  
Representation Of Integers ◽  
Positive Integers

In \cite{A.Boudaoud1}, the author asked the following question: Which $n\in \mathbb{N}$ unlimited can be represented as a sum $% n=s+w_{1}w_{2}$, where $s\in \mathbb{Z}$ is a limited integer and $\omega _{1}$, $\omega _{2}$ are two unlimited positive integers? In this survey article we partially answer this question, i.e., we present some families of unlimited positive integers which can be written as the sum of a limited integer and the product of at least two unlimited positive integers.

scholarly journals New Estimates of the Singular Series Corresponding to Positive Quaternary Quadratic Forms

2006 ◽  
Vol 13 (4) ◽  
pp. 687-691
Author(s):  
Guram Gogishvili
Keyword(s):  
Quadratic Form ◽  
General Result ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Precise Estimate ◽  
Definite Integral ◽  
Singular Series ◽  
Prime Factors ◽  
Quaternary Quadratic Form ◽  
Best Estimates

Abstract Let 𝑚 ∈ ℕ, 𝑓 be a positive definite, integral, primitive, quaternary quadratic form of the determinant 𝑑 and let ρ(𝑓,𝑚) be the corresponding singular series. When studying the best estimates for ρ(𝑓,𝑚) with respect to 𝑑 and 𝑚 we proved in [Gogishvili, Trudy Tbiliss. Univ. 346: 72–77, 2004] that where 𝑏(𝑘) is the product of distinct prime factors of 16𝑘 if 𝑘 ≠ 1 and 𝑏(𝑘) = 3 if 𝑘 = 1. The present paper proves a more precise estimate where 𝑑 = 𝑑0𝑑1, if 𝑝 > 2; 𝑕(2) ⩾ –4. The last estimate for ρ(𝑓,𝑚) as a general result for quaternary quadratic forms of the above-mentioned type is unimprovable in a certain sense.

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.

On the Representation of Integers as Sums of Distinct Terms from a Fixed Sequence

1966 ◽  
Vol 18 ◽  
pp. 643-655 ◽  
Author(s):  
Jon Folkman
Keyword(s):  
Fixed Sequence ◽  
Rate Of Growth ◽  
Representation Of Integers ◽  
Image Position ◽  
Positive Integers

Let A = (a1, a2, a3, …) be a sequence of positive integers. We letdenote the set of integers that are sums of distinct terms of A. If P(A) contains all sufficiently large integers, we say that A is complete. We shall show that certain classes of sequences that are characterized by their rate of growth are complete.

Sign in / Sign up

Export Citation Format

Share Document