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 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.

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 .

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 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 On the quadratic form x 2 —7 y 2

1949 ◽  
Vol 197 (1049) ◽  
pp. 256-268 ◽  
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Binary Quadratic Form ◽  
Real Numbers ◽  
The Right ◽  
Indefinite Binary Quadratic Form

1. Let ƒ( x,y ) = ax 2 + bxy + cy 2 be an indefinite binary quadratic form, with real coefficients a, b, c , and discriminant d = b 2 —4 ac . A well-known theorem of Minkowski asserts that for any real numbers x 0 , y 0 there are integers x, y such that f(x + x0,y + yQ) (!) ♦Various authors (Heinhold 1938, 1939; Davenport 1946; Varnavides 1948 a ) have given results which are more precise than this. In particular, their results give, for a few special quadratic forms, the exact value of the constant, i.e. the least number by which ¼ √ d may be replaced on the right of (1). Perhaps the simplest form for which the existing results do not determine the exact value of the constant is the form x 2 — 7 y 2 with discriminant 28. Here ¼ √ d = ½ √7 and this was improved by Heinhold (1938, p. 683) to ¾. In this paper, we attack the problem by a method based on that of Davenport (1947), and find the exact value of the constant. We prove that it is always possible to satisfy

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).

An Elementary Proof of a Theorem About the Representation of Primes by Quadratic Forms

1954 ◽  
Vol 6 ◽  
pp. 353-363 ◽  
Author(s):  
W. E. Briggs
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Elementary Proof ◽  
Prime Numbers ◽  
Binary Quadratic Form ◽  
First Case

The theorem that every properly primitive binary quadratic form is capable of representing infinitely many prime numbers was first proved completely by H. Weber (5). The purpose of this paper is to give an elementary proof of the case where the form is ax2 + 2bxy + cy2, with a > 0, (a, 2b, c) = 1, and D = b2 — ac not a square. The cases where the form is ax2 + bxy + cy2 with b odd, and the case where the form is ax2+ 2bxy + cy2 with D a square, can be settled very simply once the first case is taken care of, and this is done in a page and a half in the Weber paper.

scholarly journals Composition of binary quadratic forms over number fields

2021 ◽  
Vol 71 (6) ◽  
pp. 1339-1360
Author(s):  
Kristýna Zemková
Keyword(s):  
Number Field ◽  
Quadratic Forms ◽  
Number Fields ◽  
Binary Quadratic Form ◽  
Base Number ◽  
Binary Quadratic Forms ◽  
Quadratic Number Field ◽  
Totally Positive ◽  
Equivalent Conditions ◽  
Narrow Class

Abstract In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class number one. The article contains an explicit description of the correspondence. In the case of totally negative discriminants, equivalent conditions are given for a binary quadratic form to be totally positive definite.

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