Let d(n; l, k) be the number of positive divisors of n which lie in the arithmetic progression l mod k. Using the complex integration technique the formula

is proved. This formula holds uniformly in l, k and x satisfying 1 ≦ l ≦ k, (lx)1/2 ≦ k ≦ x1-∊ ; the exponent α ≦ 1/3.

For each pair (m, n) of integers such that m+1 ≦ n ≦ 2m+1, we give, by original and straightforward methods, some examples of rings A' such that dim A' = m and dim A'[X] = n.

A linear map A : C(T) → C(S) is called separating if f • g ≡ 0 implies Af • Ag = 0. We describe the general form of such maps and prove that any separating isomorphism is continuous.

Generalizing a result of Soulé we prove that for an elliptic curve E defined over an imaginary quadratic field K with complex multiplication having good ordinary reduction at the prime number p > 3 which is regular for E and the extension F of K contained in K(Ep) the dimensions of the étale K-groups are equal to the numbers predicted by Bloch and Beilinson, i.e.,

We use the general factorisation theorems of Grothendieck, Nikishin and Maurey to characterise coefficient multipliers between Bergman spaces and into the Nevanlinna class.

The 6-dimensional sphere S 6 has an almost complex structure induced by properties of Cayley algebra. With respect to this structure S 6 is a nearly Kaehlerian manifold. We investigate 2-dimensional totally real submanifolds in S 6. We prove that a 2-dimensional totally real submanifold in S 6 is flat.

Let Hp,ϕ be the subspace of Hardy space Hp consisting of those f ∊ Hp(Bn) satisfying where ϕ is a positive decreasing differentiable function on [0, 1) with ϕ(1—) = 0. Concerning image area growth, criteria for f to be of Hp,ϕ are considered extending known results for Hp .

Bruns' Theorem states that the classical integrals of the gravitational three-body problem generate all algebraic integrals. We show that the first step in his proof, together with Ziglin's non-integrability criterion for complex systems, can be used to prove the non-existence of energy independent algebraic integrals in certain real analytic systems. We also show that this aspect of Bruns' argument is purely algebraic: We offer a proof based on elementary differential algebraic methods.

We show that the Hochschild-Serre spectral sequence for a product G x H of abstract or profinite groups splits for G-modules, and that this splitting is not functorial.

We study the set W(𝓛) of Weierstrass points of all positive tensor powers of an invertible sheaf 𝓛 on an irreducible rational curve X with g ≧ 2 ordinary cusps. Using an idea from B. Olsen's study of the analogous question on smooth curves, and an explicit formula for the "theta function" of a cuspidal rational curve, we show that W(𝓛) is never dense on X (in contrast to the case of smooth curves of genus g ≧ 2).

We show that, if [s,t][u, v] = x2 in a free group, x need not be a commutator. We arrive at our example by use of a result of D. Piollet which characterizes solutions of such equations using an algebraic interpretation of the mapping class group of the corresponding surface.

Each polynomial P(x) has a "Lorentz representation", of the form This representation becomes unique if we insist that n equals the degree of P. Motivated partly by questions involving polynomials with integer coefficients, we investigate the relationship between

This paper studies for a number field K and a finite group Γ the cokernel of the residue homomorphism .

A homotopy theoretic version is given of the following result of Conner and Raymond: If the circle acts on a space so that the orbit map induces an injection in homology, then the space fibres over the circle with finite structure group. This homotopical analogue is related to recent results pertaining to the effect of the fundamental group's structure on the Euler characteristic. It is also used in the construction of a compact, simple 7-manifold with trivial Gottlieb group which, together with an infinite dimensional example of Ganea, answers a question of Gottlieb.

The following conjecture of Chowla and Zassenhaus ( 1968) is proved. If f(x) is an integral polynomial of degree ≧ 2 and p is a sufficiently large prime for which f (considered modulo p) is a permutation polynomial of the finite prime field Fp , then for no integer c with 1 ≦ c < p is f(x) + cx a permutation polynomial of Fp .

Let N(n) be the set of all integers that can be written in the form where ∊(n) → 0 as n → ∞, answering a question of P. Erdös and R. L. Graham.

In this brief note, we will show how in principle to find all units in the integral group ring ZG, whenever G is a finite group such that and Z(G) each have exponent 2, 3, 4 or 6. Special cases include the dihedral group of order 8, whose units were previously computed by Polcino Milies [5], and the group discussed by Ritter and Sehgal [6]. Other examples of noncommutative integral group rings whose units have been computed include , but in general very little progress has been made in this direction. For basic information on units in group rings, the reader is referred to Sehgal [7].

One of the questions concerning the Hexagonal Packing Lemma ([1], [3], [4]) is the rate of convergence of Sn . It was suggested in [3] and [4] that Sn = 0(1/n). In the following we prove this conjecture under the additional condition of some "nice" behaviour of the "circle function".

We show that is an integer. Special cases include the theorems of Wilson and Fermat.