The Minimum of An Indefinite Binary Quadratic Form

1947 ◽  
Vol s1-22 (4) ◽  
pp. 261-262 ◽  
Author(s):  
A. M. Macbeath
1955 ◽  
Vol 7 ◽  
pp. 337-346 ◽  
Author(s):  
R. P. Bambah ◽  
K. Rogers

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).


1969 ◽  
Vol 9 (3-4) ◽  
pp. 363-386 ◽  
Author(s):  
P. E. Blanksby

Let us suppose that ƒ(x, y) is an indefinite binary quadratic form that does not represent zero. If P is the real point (x0, y0) then we may define where the infimum is taken over all integral x, y. The inhomogeneous minimum of the form ƒ is defined where the supremum taken over all real points P, need only extend over some complete set of points, incongruent mod 1.


Author(s):  
Daniel Berend ◽  
William Moran

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.


Author(s):  
Kathleen Ollerenshaw

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.


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


1988 ◽  
Vol 30 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Hugh L. Montgomery

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).)


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