scholarly journals On the representations of numbers by binary cubic forms

1985 ◽  
Vol 27 ◽  
pp. 95-98 ◽  
Author(s):  
C. Hooley
Keyword(s):  
Positive Integer ◽  
Recent Work ◽  
Present Note ◽  
Cubic Form ◽  
Image Position ◽  
Cubic Forms ◽  
Theory Of Numbers

There are not a few situations in the theory of numbers where it is desirable to have as sharp an estimate as possible for the number r(n) of representations of a positive integer n by an irreducible binary cubic formA variety of approaches are available for this problem but, as they stand, they are all defective in that they introduce unwanted factors into the estimate. For instance, an estimate involving the discriminant of f(x, y) is obtained if we adopt the Lagrange procedure [5] of using congruences of the type f(σ, 1)≡0, mod n, to reduce the problem to one where n=1. Alternatively, following Oppenheim (vid. [2]), Greaves [3], and others, we may appeal to the theory of factorization of ideals, which leads to unwanted logarithmic factors owing to the involvement of algebraic units. Having had need, however, in some recent work on quartic forms [4] for an estimate without such extraneous imperfections, we intend in the present note to prove thatuniformly with respect to the coefficients of f(x, y), where ds(n) denotes the number of ways of expressing n as a product of s factors.

On a conjecture of Mordell concerning binary cubic forms

1941 ◽  
Vol 37 (4) ◽  
pp. 325-330 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Strict Inequality ◽  
Cubic Form ◽  
Image Position ◽  
Cubic Forms

Let f(x, y) be a binary cubic form with real coefficients and determinant D ≠ 0. In a recent paper, Mordell has proved that there exist integral values of x, y, not both zero, for whichThese inequalities are best possible, since they cannot be satisfied with the sign of strict inequality when f(x, y) is equivalent tofor the case D < 0, or tofor the case D > 0.

Some Results on L-Indistinguishability for SL(r)

1983 ◽  
Vol 35 (6) ◽  
pp. 1075-1109 ◽  
Author(s):  
Freydoon Shahidi
Keyword(s):  
Positive Integer ◽  
Weyl Group ◽  
Recent Work ◽  
Eisenstein Series ◽  
Number Field ◽  
Parabolic Subgroup ◽  
Cusp Form ◽  
Constant Multiple ◽  
Levi Decomposition ◽  
Image Position

Fix a positive integer r. Let AF be the ring of adeles of a number field F. For a parabolic subgroup P of SLr, we fix a Levi decomposition P = MN, and we letLet be the Weyl group of . It follows from a recent work of James Arthur [1,2] (also cf. [3]) that, among the terms appearing in the trace formula for SLr(AF), coming from the Eisenstein series, are those which are a constant multiple (depending only on M and w) of1where σ is a cusp form on M(AF) satisfying wσ ≅ σ,and in the notation of [2, 3]).

Cubic forms representing arithmetic progressions

1959 ◽  
Vol 55 (3) ◽  
pp. 270-273 ◽  
Author(s):  
G. L. Watson
Keyword(s):  
Arithmetic Progressions ◽  
Cubic Form ◽  
Image Position ◽  
Cubic Forms

The following result has recently been proved by Lewis (3), Davenport (2) and Birch (1).There exists an integer n0 such that every cubic form with rational coefficients and at least n0 integral variables represents zero non-trivially.The arguments of Lewis and Birch are simple, and yield also various generalizations of this result. Davenport's proof is complicated, but it shows that the minimal n0 satisfies

On the summation of certain trigonometric series

1945 ◽  
Vol 41 (2) ◽  
pp. 145-160 ◽  
Author(s):  
L. S. Goddard
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Trigonometric Series ◽  
Present Note ◽  
Rational Functions ◽  
Recurrent Relation ◽  
Polygamma Function ◽  
Special Values ◽  
Image Position

In the present note, which is introductory to the following paper, closed expressions, suitable for computational purposes, are found for the sums of the serieswhere α > 1, t = 1, 2, 3, …, and n is a positive integer. In each case a recurrent relation is found giving the values of and for t > 2 in terms of and the series Θκ(α) (κ = 1, 2, …, t), whereWhen κ is even the last series is expressed in closed form in terms of the Bernoullian polynomial φκ(l/α) and, when κ is odd and α is rational, a closed form is found involving the polygamma function Ψ(κ)(z), where The general expressions for and involve Ψ(z) and Ψ′(z) when α is rational, but for special values of α they reduce to a form independent of the Ψ-function. and are independent of n and are expressible as simple rational functions of α.

Non-homogeneous binary cubic forms

1951 ◽  
Vol 47 (3) ◽  
pp. 457-460 ◽  
Author(s):  
R. P. Bambah
Keyword(s):  
Lower Bound ◽  
Finite Volume ◽  
Upper Bound ◽  
The Other ◽  
Real Numbers ◽  
Cubic Form ◽  
Homogeneous Form ◽  
Other Hand ◽  
Image Position ◽  
Cubic Forms

1. Let f(x1, x2, …, xn) be a homogeneous form with real coefficients in n variables x1, x2, …, xn. Let a1, a2, …, an be n real numbers. Define mf(a1, …, an) to be the lower bound of | f(x1 + a1, …, xn + an) | for integers x1, …, xn. Let mf be the upper bound of mf(a1, …, an) for all choices of a1, …, an. For many forms f it is known that there exist estimates for mf in terms of the invariants alone of f. On the other hand, it follows from a theorem of Macbeath* that no such estimates exist if the regionhas a finite volume. However, for such forms there may be simple estimates for mf dependent on the coefficients of f; for example, Chalk has conjectured that:If f(x,y) is reduced binary cubic form with negative discriminant, then for any real a, b there exist integers x, y such that

The elliptic products of Jacobi and the theory of linear congruences

1926 ◽  
Vol 23 (4) ◽  
pp. 337-355
Author(s):  
P. A. MacMahon
Keyword(s):  
Elliptic Functions ◽  
Linear Congruences ◽  
Image Position ◽  
Theory Of Numbers

In the application of Elliptic Functions to the Theory of Numbers the two formulae of Jacobiare of great importance.

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

Powder Diffraction Data of CdTexSe1-xSolid Solutions

1990 ◽  
Vol 5 (4) ◽  
pp. 206-209 ◽  
Author(s):  
N.A. Razik ◽  
G. Al-Barakati ◽  
S. Al-Heneti
Keyword(s):  
Powder Diffraction ◽  
Diffraction Data ◽  
Hexagonal Wurtzite Structure ◽  
Powder Diffraction Data ◽  
Lattice Constants ◽  
Cubic Form ◽  
Hexagonal Form ◽  
State Diffusion ◽  
Vegard’S Law ◽  
Image Position

AbstractCdTexSe1-xsolid solutions with (x) ranging between zero and one were prepared by solid state diffusion under vacuum and their precise lattice constants and X-ray powder diffraction data were determined. It was found that alloys with 0≤x≤0.4 possess the hexagonal wurtzite structure while those with 0.5≤x≤1.0 have the cubic zincblende structure. The lattice parameters obeyed Vegard's law according to the following formulaeExtrapolated lattice constants were a = 6.066(6) Å for the cubic form of CdSe and a = 4.563(6) Å and c = 7.502 (10) Å for the hexagonal form of CdTe.

scholarly journals On certain infinite integrals involving Struve functions and parabolic cylinder functions

1946 ◽  
Vol 7 (4) ◽  
pp. 171-173 ◽  
Author(s):  
S. C. Mitra
Keyword(s):  
Present Note ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Struve Functions ◽  
Image Position

The object of the present note is to obtain a number of infinite integrals involving Struve functions and parabolic cylinder functions. 1. G. N. Watson(1) has proved thatFrom (1)follows provided that the integral is convergent and term-by-term integration is permissible. A great many interesting particular cases of (2) are easily deducible: the following will be used in this paper.

The Distribution Of Totatives

1955 ◽  
Vol 7 ◽  
pp. 347-357 ◽  
Author(s):  
D. H. Lehmer
Keyword(s):  
Positive Integer ◽  
Prime Factors ◽  
Image Position

This paper is concerned with the numbers which are relatively prime to a given positive integerwhere the p's are the distinct prime factors of n. Since these numbers recur periodically with period n, it suffices to study the ϕ(n) numbers ≤n and relatively prime to n.

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