Congruence Properties Research Articles

On a Congruence Property of the Divisor Function II

Let d(n) be the number of divisors of n and N(k,x) be the number of n ≤ x for which d(n)=0(modk). Then in I it is proved that N(k,x) ∼ [I-ζ(k)/ζ(k-I)]x when k is a prime.

Congruence Properties of Ramanujan's Function τ(n )

Congruence Properties of Ramanujan's Function τ(<i>n</i> )

On the Prime Number Theorems

A few years ago, Watson succeeded in proving an asymptotic congruence property formulated by Ramanujan, namely, the following theorem: If m and Ic are fixed positive integers, there are between n = 1 and n = N only o (N) integers n for which the sum of the (2m 1) -th powers of all divisors of n is not a multiple of k. This implies, in the particular case 2m 1 = 11, that Ramanujan's r(n) is divisible by 691 except for o(N) values of n. Actually, Watson has improved on o(N) by logarithmical factors (for arbitrary m, kc), and has even proved asymptotic formulae for certain summatory functions on which the o-estimates depend 1; asymptotic formulae established by the classical methods of the theory of primes (that is, by contour integrations, based on estimates of 4(s) in a certain domain containing the line a = 1 in its interior). The singularities of the generating Dirichlet series and, correspondingly, the proof of the asymptotic formulae are similar to those occurring in Landau's result,2 according to which the number of the positive integers which are less than x and are representable as a sum of two squares is asymptotically proportional to x/ (log x) '. The following considerations, which deal with the general analytical background of asymptotic laws of this type, seem to be warranted in view of Hardy's recent presentation 3 of the results just meiitioned. The asymptotic formula, given by Hardy in his discussion of Landau's problem, is false, the passage from the order of the singularity (at s = 1) to the order of the summatory function being erroneous. This can be seen, without any Tauberian argument, from Abelian reasons alone. (Among all the possible orders, Landau's exponent, 1, represents the only case in which Hardy's calculation becomes correct.) However, my quarrel is not with this lapsus calatmi (which does not occur in Watson's paper), but with the method of approach. In fact, Hardy refers (as Watson did) to the parallel problem of

Some properties of rectilinear congruences

Some properties of rectilinear congruences

Congruence properties of self-contained balanced sets

Let the set b1b2, b3, …., bk; of k non-negative integers be denoted by Bk. Let ξ;Bk denote the set ξb1 ξb2, ξb3, …. ξbk; ξ being any integer &gt; 1. Without loss of generality, we can suppose that b ≦bi+1.

Note on a Paper by H. Hilton: "On Some Properties of Rectilinear Congruences"

Note on a Paper by H. Hilton: "On Some Properties of Rectilinear Congruences"

On Some Properties of Rectilinear Congruences

On Some Properties of Rectilinear Congruences

Congruence Properties of G-Functions

Denoting the sum of the products of the first n natural numbers taken r at a time by the symbol G(n, r), I have shown thatwith the initial values andIn general it was shown thatDefining G (x, r) by the fundamental relation:for all values of x, we get

Congruence Properties of Partitions

Congruence Properties of Partitions

Some properties of line congruences

In a previous paper the writer has obtained various formulae for surfaces in higher space by means of Cayley's functional method; and recently the same method has been applied to surfaces on a quadric form in [5] with the object of investigating the properties of line congruences in [3]. The purpose of the present note is two-fold: to indicate a few applications of the formulae mentioned above, and to give direct and (it is believed) novel proofs of some results obtained elsewhere by the functional process. The demonstrations given here differ, save in two cases, from those based on Schumacher's four-dimensional representation which, as it stands, applies only to congruences without double rays and therefore lacks generality. It was James who first noticed that a congruence without singularities has in general a finite number of double rays and is accordingly defined by five, instead of four, independent characters. These characters have been considered in previous work; but for present purposes it is convenient to describe them afresh.

Congruence Properties of Ramanujan's Function τ(n )

Proceedings of the London Mathematical SocietyVolume s2-31, Issue 1 p. 1-10 Papers Congruence Properties of Ramanujan's Function τ(n) J. R. Wilton, J. R. WiltonSearch for more papers by this author J. R. Wilton, J. R. WiltonSearch for more papers by this author First published: 1930 https://doi.org/10.1112/plms/s2-31.1.1Citations: 17AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes2-31, Issue11930Pages 1-10 RelatedInformation

Some New Theorems in Geometry of a Surface

The elementary properties of generators of a ruled surface, and the existence of a line of striction when the surface is skew, are well known to readers of this journal. We propose to show that many of these properties do not belong exclusively to ruled surfaces; but that a family of curves on any surface possesses a line of striction and a focal curve or envelope, though these are not always real. When the surface is developable, and the “curves” are the generators of one system, the focal curve is the edge of regression. We shall also see that the properties to be established for a family of curves on a surface are analogous to some of the leading properties of congruences of curves in space, the line of striction corresponding to the surface of striction or orthocentric surface, and the focal curve to the focal surface of the latter.

Some New Theorems in Geometry of a Surface

The elementary properties of generators of a ruled surface, and the existence of a line of striction when the surface is skew, are well known to readers of this journal. We propose to show that many of these properties do not belong exclusively to ruled surfaces; but that a family of curves on any surface possesses a line of striction and a focal curve or envelope, though these are not always real. When the surface is developable, and the “curves” are the generators of one system, the focal curve is the edge of regression. We shall also see that the properties to be established for a family of curves on a surface are analogous to some of the leading properties of congruences of curves in space, the line of striction corresponding to the surface of striction or orthocentric surface, and the focal curve to the focal surface of the latter.

Congruence properties of partitions

Congruence properties of partitions

The combinatorics of weighted vector compositions

A vector composition of a vector $\mathbf{\ell}$ is a matrix $\mathbf{A}$ whose rows sum to $\mathbf{\ell}$. We define a weighted vector composition as a vector composition in which the column values of $\mathbf{A}$ may appear in different colors. We study vector compositions from different viewpoints: (1) We show how they are related to sums of random vectors and (2) how they allow to derive formulas for partial derivatives of composite functions. (3) We study congruence properties of the number of weighted vector compositions, for fixed and arbitrary number of parts, many of which are analogous to those of ordinary binomial coefficients and related quantities. Via the Central Limit Theorem and their multivariate generating functions, (4) we also investigate the asymptotic behavior of several special cases of numbers of weighted vector compositions. Finally, (5) we conjecture an extension of a primality criterion due to Mann and Shanks in the context of weighted vector compositions.