Note on Integers Representable by Binary Quadratic Forms

1975 ◽  
Vol 18 (1) ◽  
pp. 123-125 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Error Term ◽  
Asymptotic Estimate ◽  
Complex Analysis ◽  
Binary Quadratic Form ◽  
Binary Quadratic Forms ◽  
Transcendental Method ◽  
Image Position ◽  
Positive Integers

Let B be the set of positive integers prime to d which are representable by some primitive, positive, integral binary quadratic form of discriminant d. It is the purpose of this note to show that the following asymptotic estimate for the number of integers in B less than or equal to x can be proved using only elementary arguments:(1)where c1 is the positive constant given in (17) below. (Using the deeper methods of complex analysis James [2] has proved this result with the error term ((log x)-1/2) replacing ((log log x)-1). Heupel [1] using a transcendental method as in James [2] improved this to ((log x)-1).)

A proof of Hecke’s formula for binary quadratic forms

2019 ◽  
Vol 16 (02) ◽  
pp. 233-240
Author(s):  
Frank Patane
Keyword(s):  
Quadratic Form ◽  
Explicit Formula ◽  
Quadratic Forms ◽  
Theta Series ◽  
Binary Quadratic Form ◽  
Binary Quadratic Forms

In Mathematische Werke, Hecke defines the operator [Formula: see text] and describes their utility in conjunction with theta series of quadratic forms. In particular, he shows that the image of theta series associated to classes of binary quadratic forms in CL[Formula: see text] is again a theta series associated to a collection of forms in CL[Formula: see text]. We state and prove an explicit formula for the action of [Formula: see text] on a binary quadratic form of negative discriminant.

scholarly journals Kronecker's fundamental limit formula in the theory of numbers and elliptic functions, and similar theorems

1929 ◽  
Vol 125 (797) ◽  
pp. 262-276 ◽  
Keyword(s):  
Number Theory ◽  
Quadratic Form ◽  
Quadratic Forms ◽  
Elliptic Functions ◽  
General Term ◽  
Binary Quadratic Form ◽  
Binary Quadratic Forms ◽  
Fundamental Limit ◽  
Number Of Classes ◽  
Fundamental Formula

1. Many important applications of analysis to number-theory require the study of a function f (s) of a complex variable s = σ + i τ near a singular point s 0 = σ 0 + i τ 0 . The functions f (s) is frequently defined for σ > σ 0 by an infinite series, really d Dirichlet's series, the general term of which is a function of the variables of summation, e. g ., a quadratic form, raised to the power s . Thus the question of finding the number of classes of binary quadratic forms of given determinant, or the number of classes of ideals in a given field, depends upon the residue, Say R, of an appropriate f (s) at a simple pole s 0 . A deeper question then suggested is that of finding lim s → s 0 ( f (s) — R/ s-s 0 ). In particular, Kronecker's fundamental formula arises when f (s) is a homogeneous binary quadratic form in the variables of summation. Thus, let a a (≠ 0), b, c be any constants real or complex which are such that the roots ω 1 , ω 2 of the quadratic form ϕ (x, y) = ax 2 + bxy + cy 2 = a ( x - ω 1 y ) ( x - ω 2 y ) are neither real nor equal. We need only distinguish the two cases (I) I (ω 1 ) > 0, I (ω 2 ) < 0, (II) I (ω 1 ) > 0, I (ω 2 ) > 0, as the others can be included by writing — y for y .

The inhomogeneous minimum of binary quadratic forms

1992 ◽  
Vol 112 (1) ◽  
pp. 7-19 ◽  
Author(s):  
Daniel Berend ◽  
William Moran
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Binary Quadratic Form ◽  
Binary Quadratic Forms ◽  
General Behaviour ◽  
Inhomogeneous Minimum ◽  
The Subject ◽  
Indefinite Binary Quadratic Form

AbstractAn indefinite binary quadratic form ƒ gives rise to a certain function M on the torus. The properties of M, especially those related to its maximum – the so-called inhomogeneous minimum of ƒ – are the subject of numerous papers. Here we continue this study, putting more emphasis on the general behaviour of M.

scholarly journals On the gaps between values of binary quadratic forms

2010 ◽  
Vol 54 (1) ◽  
pp. 25-32
Author(s):  
Jörg Brüdern ◽  
Rainer Dietmann
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Binary Quadratic Form ◽  
Binary Quadratic Forms ◽  
Fixed Distance ◽  
Quantitative Form

AbstractAmong the values of a binary quadratic form, there are many twins of fixed distance. This is shown in quantitative form. For quadratic forms of discriminant −4 or 8 a corresponding result is obtained for triplets.

The minima of a pair of indefinite, harmonic, binary quadratic forms

1945 ◽  
Vol 41 (2) ◽  
pp. 77-96 ◽  
Author(s):  
Kathleen Ollerenshaw
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Binary Quadratic Form ◽  
Binary Quadratic Forms ◽  
Indefinite Binary Quadratic Form

If f is a real, indefinite, binary quadratic form of discriminant d and if κ(f) is the minimum of | f | taken over all integer values of x, y, not both zero, then it is well known that and that this is a ‘best possible’ result.

The order of binary quadratic forms from representation numbers

2016 ◽  
Vol 12 (03) ◽  
pp. 679-690
Author(s):  
A. G. Earnest ◽  
Robert W. Fitzgerald
Keyword(s):  
Quadratic Form ◽  
Explicit Form ◽  
Quadratic Forms ◽  
Class Group ◽  
Binary Quadratic Form ◽  
Partial Answer ◽  
Binary Quadratic Forms ◽  
Form Class ◽  
The Relationship

We investigate the relationship between the numbers of representations of certain integers by a primitive integral binary quadratic form [Formula: see text] of discriminant [Formula: see text] and the order of the class of [Formula: see text] in the form class group of discriminant [Formula: see text], in the case when this order is even. The explicit form of the solutions obtained is used to give a partial answer to a question regarding which multiples of [Formula: see text] can be parameterized in a particular way.

scholarly journals An Inhomogeneous Minimum For Non-Convex Star-Regions With Hexagonal Symmetry

1955 ◽  
Vol 7 ◽  
pp. 337-346 ◽  
Author(s):  
R. P. Bambah ◽  
K. Rogers
Keyword(s):  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Elementary Geometry ◽  
Hexagonal Symmetry ◽  
Real Numbers ◽  
Inhomogeneous Minimum ◽  
Image Position ◽  
Indefinite Binary Quadratic Form

1. Introduction. Several authors have proved theorems of the following type:Let x0, y0 be any real numbers. Then for certain functions f(x, y), there exist numbers x, y such that1.1 x ≡ x0, y ≡ y0 (mod 1),and1.2 .The first result of this type, but with replaced by min , was given by Barnes (3) for the case when the function is an indefinite binary quadratic form. A generalisation of this was proved by elementary geometry by K. Rogers (6).

scholarly journals Minimal theta functions

1988 ◽  
Vol 30 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Hugh L. Montgomery
Keyword(s):  
Functional Equation ◽  
Quadratic Form ◽  
Zeta Function ◽  
Theta Functions ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Epstein Zeta Function ◽  
Image Position

Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.Among such forms, let . The Epstein zeta function of f is denned to beRankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,We prove a corresponding result for theta functions. For real α > 0, letThis function satisfies the functional equation(This may be proved by using the formula (4) below, and then twice applying the identity (8).)

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.

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