Let ζ(s) = σn-s (Res >1) denote the Riemann zeta function; then, as is well known, , where Bm denotes the mth Bernoulli number, In this paper we investigate the possibility of similar evaluations of the Epstein zeta function ζq(s) at the rational integers s = k> 2. Let

be a positive definite quadratic form and

where the summation is over all pairs of integers except (0, 0). In attempting to evaluate ζq(k) we are guided by Kronecker's first limit formula [11]

where γ is Euler's constant,

is the Dedekind eta-function, and τ is the complex number in the upper half plane, ℋ, associated with Q by the formula

On the basis of (1.3) we would expect a formula involving functions of τ. This formula is stated in Theorem 1, (2.13).

In the paper [5], Rema used the well-known fact that in a Boolean algebra the binary operation d: B × B → B defined by is a “metric“ operation to show that, if D is any dual ideal of ^, then the sets Up = {(x, y): d(x, y) <p}, where p ∈ D, form a base for a uniformity of }, the resulting topological space <B; T[D]> being called an auto-topologized Boolean algebra. Recently, Kent and Atherton [1, 4] exhibited a family of topologies on an arbitrary lattice ℒ defined in terms of ideals and dual ideals. More specifically, if I and D are respectively an ideal and a dual ideal of ℒ, then the T[I:D] topology on ℒ is the topology defined by taking the sets of the form a*⋂b+, where , as sub-base for the open sets. It is these topologies that are studied in this paper.

We show that the theorem stated in the title is a corollary to a result of K. A. Zaretskii [5] and a theorem of G. Birkhoff [1]. The construction we use further shows that all groups with cardinal less than or equal to the cardinal of the given group are simultaneously realised as maximal subgroups of the same semigroup of binary relations ℬx. For finite or countable groups, when Xmay be taken to be finite or countable, respectively, and for an entirely different method of proof, the paper of J. S. Montague and R. J. Plemmons [3] should be consulted. For two further proofs of the theorem of the title to this note, this time for any X, see also R. J. Plemmons and B. M. Schein [4] and A. H. Clifford [2].

Let p be an odd prime and let f(x) be a complex-valued function such that f(x+p) = f(x) for all integers x. Write e(x) = exp(2πix/p), and define l/x by , where We consider the sum

where is the Legendre symbol. The sum is zero if as is clear on replacing x by bjax. Salié has found a result which can be written in the form

when h2 ≡ 4ab(modp).

In the study of the structure of regular semigroups, it is customary to impose several conditions restricting the behaviour of ideals, idempotents or elements. In a few instances, one may represent them as subdirect products of some much more restricted types of regular semigroups, e.g., completely (0-) simple semigroups, bands, semilattices, etc. In particular, studying the structure of completely regular semigroups, one quickly distinguishes certain special cases of interest when these semigroups are represented as semilattices of completely simple semigroups. In fact, this semilattice of semigroups may be built in a particular way, idempotents may form a subsemigroup, ℋ may be a congruence, and so on.

Let х denote a primitive character modulo k. Using two different representations for Dirichlet L-functions, Apostol [1] recently derived a representation for

involving the sums

where m is a positive integer. Furthermore, if х(r) = (r|p)the residue class character modulo the odd prime p, he derived a representation for Mm(х) involving the sums

Let d(n) denote the number of positive divisors of. A long time ago, Erdös and Mirsky [1] raised the question whether the equation d(n) = d(n+l) holds for infinitely many n. It does not seem easy to settle this problem, and in the present note we give a partial result.

An ideal extension of one semigroup by another is determined by a partial homomorphism into the translational hull of the first semigroup [3, §2, Theorem 5]. In most ins tances, the development of the theory of ideal extensions has been hindered by inadequate knowledge of the translational hull; it is our purpose here to characterize certain basic structures in the translational hull of an arbitrary inverse semigroup.

In this paper, we shall be concerned with the solution of triple integral equations of the type

where M−1 is the inverse Mellin transform, nis a positive integer, and — 1 < Res < 0. The use of these equations will be illustrated by their application to two well-known problems in the mathematical theory of elasticity and further applications will be reported later.

Lowengrub [l] has considered simultaneous dual integral equations of the form

where i = 1,2 …n, I1= {x:0 ≦ x x <1}, I2 {= x:0 ≦ x >1}, the cIJ are constants, the f1(x) are known functions and the functions φ(x) are to be determined.

denotes the modified operator of the Hankel transform with the inversion formula

In a sequence of two papers which appeared in 1968 and 1969 Herbert Abels [1, 2] has developed, from a method originated by Gerstenhaber [6], a means for extending the study of properly discontinuous groups of transformations to that of proper transformation groups in general. We recall that, if G is a Hausdorff locally compact group of transformations of a locally compact space X, then the action of Gis proper when, for any two compact subsets K and L, the subset G(K, L) = {g ɛ G: gL∩K # 0} of G is compact (see [3], p. 55). In what follows all groups and spaces will be Hausdorff and locally compact. If H is a closed subgroup of G, then it is clear that the property just defined is possessed by the action of H as a group of left translations of G.

The similarities between martingale convergence theory and pointwise ergodic theory are now well known [5, 7, 9, 10]. In [5] the similarity between the proofs of the Hopf– Dunford–Schwartz individual ergodic theorem and the martingale convergence theorem is systematically exploited to produce very general ” maximal ergodic ” inequalities for certain sequences of contractions on L1-spaces. A different approach by Rota [10] and Rao [9] leads to a unified convergence theory for martingales and Abel limits. Bishop [1] has produced ” upcrossing” inequalities which yield both theChacon-Ornstein theorem [4] and the martingale convergence theorem.

A well-known product, referred to as the Dirichlet convolution product, is generalized to arithmetic functions defined on an order in a Cayley division algebra. Factorization results for orders, multiplicative functions and analogues of the Moebius inversion formula are discussed.

After Dieudonné [3], many authors have considered locally convex topological vector spaces that admit a fundamental sequence of compact, precompact,… subsets. Their work has been essentially to identify them as strong duals of Fréchet-Montel spaces, under suitable additional assumptions (barrelledness or evaluability). However, it seems that these spaces have never been characterized without additional assumptions. That is the aim of the present paper.