scholarly journals Asymptotics for some polynomial patterns in the primes

2019 ◽  
Vol 149 (5) ◽  
pp. 1241-1290
Author(s):  
Pierre-Yves Bienvenu
Keyword(s):  
Convex Body ◽  
Quadratic Form ◽  
Quadratic Forms ◽  
Linear Equations ◽  
Linear Forms ◽  
Representation Function ◽  
Von Mangoldt Function ◽  
Asymptotic Formulae ◽  
Image Position ◽  
Linear Correlations

AbstractWe prove asymptotic formulae for sums of the form $$\sum\limits_{n\in {\open z}^d\cap K} {\prod\limits_{i = 1}^t {F_i} } (\psi _i(n)),$$where K is a convex body, each Fi is either the von Mangoldt function or the representation function of a quadratic form, and Ψ = (ψ1, …, ψt) is a system of linear forms of finite complexity. When all the functions Fi are equal to the von Mangoldt function, we recover a result of Green and Tao, while when they are all representation functions of quadratic forms, we recover a result of Matthiesen. Our formulae imply asymptotics for some polynomial patterns in the primes. For instance, they describe the asymptotic behaviour of the number of k-term arithmetic progressions of primes whose common difference is a sum of two squares.The paper combines ingredients from the work of Green and Tao on linear equations in primes and that of Matthiesen on linear correlations amongst integers represented by a quadratic form. To make the von Mangoldt function compatible with the representation function of a quadratic form, we provide a new pseudorandom majorant for both – an average of the known majorants for each of the functions – and prove that it has the required pseudorandomness properties.

scholarly journals Asymptotic formulae for linear equations

1972 ◽  
Vol 13 (2) ◽  
pp. 147-152 ◽  
Author(s):  
Don B. Hinton
Keyword(s):  
Asymptotic Behaviour ◽  
Fourth Order ◽  
Linear Equations ◽  
Order Equation ◽  
Third Order ◽  
Fourth Order Equation ◽  
The Third ◽  
Asymptotic Formulae ◽  
Image Position

Numerous formulae have been given which exhibit the asymptotic behaviour as t → ∞solutions ofwhere F(t) is essentially positive and Several of these results have been unified by a theorem of F. V. Atkinson [1]. It is the purpose of this paper to establish results, analogous to the theorem of Atkinson, for the third order equationand for the fourth order equation

The inhomogeneous minimum of quadratic forms of signature ± 1

1981 ◽  
Vol 89 (2) ◽  
pp. 225-235 ◽  
Author(s):  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Inhomogeneous Minimum ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Image Position

Let Qr be a real indefinite quadratic form in r variables of determinant D ≠ 0 and of type (r1, r2), 0 < r1 < r, r = r1 + r2, S = r1 − r2 being the signature of Qr. It is known (e.g. Blaney (3)) that, given any real numbers c1, c2,…, cr, there exists a constant C depending only on r and s such that the inequalityhas a solution in integers x1, x2, …, xr.

On a conjecture of Watson

1983 ◽  
Vol 94 (1) ◽  
pp. 9-22 ◽  
Author(s):  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Classical Result ◽  
Preceding Paper ◽  
Real Numbers ◽  
Linear Forms ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Image Position

Let Qn be a real indefinite quadratic form in n variables x1, x2,…, xn, of determinant D ≠ 0 and of type (r, s), 0 < r < n, n = r + s. Let σ denote the signature of Qn so that σ = r − s. It is known (see e.g. Blaney(4)) that, given any real numbers c1 c2, …, cn, there exists a constant C depending upon n and σ only such that the inequalityhas a solution in integers x1, x2, …, xn. Let Cr, s denote the infimum of all such constants. Clearly Cr, s = Cs, r, so we need consider non-negative signatures only. For n = 2, C1, 1 = ¼ follows from a classical result of Minkowski on the product of two linear forms. When n = 3, Davenport (5) proved that C2, 1 = 27/100. For all n and σ = 0, Birch (3) proved that Cr, r = ¼. In 1962, Watson(18) determined the values of Cr, s for all n ≥ 21 and for all signatures σ. He proved thatWatson also conjectured that (1·2) holds for all n ≥ 4. Dumir(6) proved Watson's conjecture for n = 4. For n = 5, it was proved by Hans-Gill and Madhu Raka(7, 8). The author (12) has proved the conjecture for σ = 1 and all n. In the preceding paper (13) we proved that C5, 1 = 1. In this paper we prove Watson's conjecture for σ = 2, 3 and 4.

Asymmetric inequalities for non-homogeneous ternary quadratic forms

1967 ◽  
Vol 63 (2) ◽  
pp. 291-303 ◽  
Author(s):  
Vishwa Chander Dumir
Keyword(s):  
Quadratic Forms ◽  
Real Numbers ◽  
Linear Forms ◽  
Ternary Quadratic Forms ◽  
Image Position

A well-known theorem of Minkowski on the product of two linear forms states that ifare two linear forms with real coefficients and determinant Δ = |αδ − βγ| ≠ 0, then given any real numbers c1, c2 we can find integers x, y such that

scholarly journals Positive values of inhomogeneous quinary quadratic forms of type (4,1)

1981 ◽  
Vol 31 (2) ◽  
pp. 175-188 ◽  
Author(s):  
R. J. Hans-Gill ◽  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Image Position

AbstractHere it is proved that if Q(x, y, z, t, u) is a real indefinite quinary quadratic form of type (4,1) and determinant D, then given any real numbers x0, y0, z0, t0, u0 there exist integers x, y, z, t, u such thatAll critical forms are also obtained.

scholarly journals Theorems Relating to Quadratic Forms and their Discriminant Matrices

1953 ◽  
Vol 10 (1) ◽  
pp. 13-15
Author(s):  
S. Vajda
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Degrees Of Freedom ◽  
Normal Population ◽  
Royal Statistical Society ◽  
Unit Variance ◽  
The Matrix ◽  
Chi Squared ◽  
Image Position ◽  
Latent Roots

In a paper read before the Research Branch of the Royal Statistical Society (Ref. 1, p. 150) the following case was considered:Let the expression be given; introduce, for c, a linear form in and obtainIf the yi are sample values from a normal population with unit variance, then it is known (Ref. 2) that (1) is distributed as where zi varies as chi-squared with one degree of freedom and the li are the latent roots of the matrix of the quadratic form. If these latent roots are f times unity and n—f times zero, then this reduces to a chi-squared distribution with f degrees of freedom.

Integral Bases for Quadratic Forms

1963 ◽  
Vol 15 ◽  
pp. 412-421 ◽  
Author(s):  
J. H. H. Chalk
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Integer Variables ◽  
Inhomogeneous Minimum ◽  
Integral Bases ◽  
Indefinite Quadratic Form ◽  
Homogeneous Minimum ◽  
Indefinite Quadratic ◽  
Image Position

Letbe an indefinite quadratic form in the integer variables x1, . . . , xn with real coefficients of determinant D = ||ars||(n) ≠ 0. The homogeneous minimum MH(Qn) and the inhomogeneous minimum MI(Qn) of Qn(x) are defined as follows :

scholarly journals On the Evaluation of Determinants, the Formation of their Adjugates, and the Practical Solution of Simultaneous Linear Equations

1933 ◽  
Vol 3 (3) ◽  
pp. 207-219 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Quadratic Form ◽  
Least Squares ◽  
Linear Equations ◽  
Statistical Correlation ◽  
Simultaneous Equations ◽  
Normal Equations ◽  
Practical Solution ◽  
Simultaneous Linear Equations ◽  
Reciprocal Matrix ◽  
Image Position

There are various methods in existence for the practical solution of a set of simultaneous equationsSome of these methods are appropriate to special systems, as for example to the axisymmetric “normal equations” of Least Squares. In many applications, however, as in problems of statistical correlation of many variables, it may be desired not merely to solve a given set of equations but to obtain as much knowledge as possible about the system or matrix of coefficients; perhaps to evaluate its determinant and various minors, such as principal minors, possibly also to determine the elements of the adjugate matrix, or the reciprocal matrix. The examination of the sign of successive principal minors of an axisymmetric determinant, in order to find the signature of the corresponding quadratic form, is a case in point; and there are many such applications.

Inhomogeneous minimum of indefinite quadratic forms in six variables: A conjecture of Watson

1983 ◽  
Vol 94 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Famous Conjecture ◽  
Inhomogeneous Minimum ◽  
Indefinite Quadratic Form ◽  
History Of ◽  
Indefinite Quadratic ◽  
Image Position

The famous conjecture of Watson(11) on the minima of indefinite quadratic forms in n variables has been proved for n ≤ 5, n ≥ 21 and for signatures 0 and ± 1. For the details and history of the conjecture the reader is referred to the author's paper(8). In the succeeding paper (9), we prove Watson's conjecture for signature ± 2 and ± 3 and for all n. Thus only one case for n = 6 (i.e. forms of type (1, 5) or (5, 1)) remains to he proved which we do here; thereby completing the case n = 6. This result is also used in (9) for proving the conjecture for all quadratic forms of signature ± 4. More precisely, here we prove:Theorem 1. Let Q6(x1, …, x6) be a real indefinite quadratic form in six variables of determinant D ( < 0) and of type (5, 1) or (1, 5). Then given any real numbers ci, 1 ≤ i ≤ 6, there exist integers x1,…, x6such that

scholarly journals On indefinite ternary quadratic forms

1974 ◽  
Vol 18 (4) ◽  
pp. 388-401 ◽  
Author(s):  
R. T. Worley
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Ternary Quadratic Forms ◽  
Indefinite Ternary Quadratic Form ◽  
Image Position

In a paper [1] with the same title Barnes has shown that if Q(x, y, z) is an indefinite ternary quadratic form of determinant d ≠ 0 then there exist integers x1, y1, z1, x2,···z3 satisfying for which Furthermore, unless Q is equivalent to a multiple of or two other forms Q2, Q3 then the constant ⅔ in (1.2) can be replaced by 1/2.2. For Q1 equality is needed on at least one side of (1.2) while for Q2, Q3 the constant ⅔ can be reduced to 12/25 but no further.

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