scholarly journals Forms Representable by an Integral Positive-Definite Binary Quadratic Form.

1971 ◽  
Vol 29 ◽  
pp. 73
Author(s):  
Kenneth S. Williams
Keyword(s):  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Positive Definite

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).)

On the Distribution of Integer Solutions of f(x, y) = z2 for a Definite Binary Quadratic Form f

1967 ◽  
Vol 10 (5) ◽  
pp. 755-756
Author(s):  
Nobuo Nobusawa
Keyword(s):  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Integer Solutions

Let f be a positive definite binary quadratic form with rational coefficients. We shall call a point (x, y) in E2 with integers x and y a Pythagorean point of f when f(x, y) = z2 is satisfied with some integer z, and shall prove the following theorem.

scholarly journals Petersson inner products of weight-one modular forms

2019 ◽  
Vol 2019 (749) ◽  
pp. 133-159
Author(s):  
Maryna Viazovska
Keyword(s):  
Quadratic Form ◽  
Algebraic Number ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Theta Series ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Arithmetic Properties ◽  
Factorization Formula ◽  
Weight One

Abstract In this paper we study the regularized Petersson product between a holomorphic theta series associated to a positive definite binary quadratic form and a weakly holomorphic weight-one modular form with integral Fourier coefficients. In [18], we proved that these Petersson products posses remarkable arithmetic properties. Namely, such a Petersson product is equal to the logarithm of a certain algebraic number lying in a ring class field associated to the binary quadratic form. A similar result was obtained independently using a different method by W. Duke and Y. Li [5]. The main result of this paper is an explicit factorization formula for the algebraic number obtained by exponentiating a Petersson product.

On a problem about the Epstein zeta-function

1964 ◽  
Vol 60 (4) ◽  
pp. 855-875 ◽  
Author(s):  
Veikko Ennola
Keyword(s):  
Quadratic Form ◽  
Zeta Function ◽  
Special Form ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Simple Pole ◽  
Epstein Zeta Function ◽  
Image Position

1. Letbe a positive definite binary quadratic form with determinant αβ − δ2 = 1. A special form of this kind isWe consider the Epstein zeta-functionthe series converging for . The function Zh(s) can be analytically continued over the whole s-plane and it is regular except for a simple pole with residue π at s = 1.

Further aspects of the evaluation of σ(m. n≠0, 0) (am2 + bnm + cn2)−s

1984 ◽  
Vol 95 (1) ◽  
pp. 5-13 ◽  
Author(s):  
I. J. Zucker ◽  
M. M. Robertson
Keyword(s):  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Complete List ◽  
Reduced Form ◽  
Sum Of Products ◽  
Image Position ◽  
Reduced Forms ◽  
Double Sum

In some previous publications, Zucker and Robertson [13, 14, 15] for certain cases evaluated exactly the double sumwith a, b, c integers. In (1·1) the summation is over all integer values of m and n, both positive and negative, but excluding the case where m and n are simultaneously zero. The term ‘exact’ used here is in the sense introduced by Glasser [5], and means that S may be expressed as a linear sum of products of pairs of Dirichlet L-series. S is then said to be solvable. Whether S may be solved or not depends on the properties of the related binary quadratic form (am2 + bmn + cn2) = (a, b, c). The cases considered in [15] were when a > 0 and d = b2 − 4ac < 0, i.e. (a, b, c) was positive definite. When this is so, Glasser [5] conjectured that S was solvable if and only if (a, b, c) had one reduced form per genus, i.e. the reduced forms of (a, b, c) were disjoint. Zucker and Robertson [15] were in fact able to solve S for all (a, b, c) for which d < 0 and whose reduced forms were disjoint. A complete list of solutions may be found in [6].

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).

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