In deriving the approximate functional equation for certain Dirichlet series, one first establishes an identity for the function in terms of a partial sum of the series (e.g. see [1] and [2]). It is the purpose of this note to give a short proof of this identity for Hecke's Dirichlet series [1]. The proof is valid with only a few minor changes for the identity given by Chandrasekharan and Narasimhan [2, Lemma 2] for a much larger class of Dirichlet series. However, the brevity of the paper would be lost if we introduced the necessary terminology and notation.

From the proof of Theorem 2 of [5] it follows that for every positive integer k there exist infinitely many primes p in the arithmetical progression ax + b (x = 0, 1, 2,…), where a and b are relatively prime positive integers, such that the number 2p−1 − 1 has at least k composite factors of the form (p − 1)x + 1. The following question arises:

For any given natural number k, do there exist infinitely many primes p such that the number 2p−1 − 1 has k prime factors of the form(p − 1)x + 1 and p ≡ b (mod a), where a and b are coprime positive integers?

Let H be a group of characters on an (algebraic) abelian group G. In a natural way, we may regard G as a group of characters on H. In this way, we obtain a duality between the two groups G and H. One may pose several problems about this duality. Firstly, one may ask whether there exists a group topology on G for which H is precisely the set of continuous characters. This question has been answered in the affirmative in [1]. We shall say that such a topology is compatible with the duality between G and H. Next, one may ask whether there exists a locally compact group topology on G which is compatible with a given duality and, if so, whether there is more than one such topology. It is this second question (previously considered by other authors, to whom we shall refer below) which we shall consider here.

In the analysis of mixed boundary value problems by Hankel transforms, one often encounters dual integral equations of the form

where I1 = (0, 1), I2 = (1, ∞); w1(x), w2(x) are weight functions, ψ(x) is the unknown function, and f(y), g(y) are functions continuously differentiate on I1 and I2 respectively. Many successful attempts have been made to solve (1.1) and (1.2). These are all discussed in a recent book by Sneddon [7]. As pointed out in a recent paper by Erdogan and Bahar [4], in mixed boundary value problems of semi-infinite domains involving more than one unknown function such as those arising in elastostatics, viscoelasticity, and electrostatics, the formulation will lead to a system of simultaneous dual integral equations which is a generalization of (1.1) and (1.2). These equations may be expressed as follows:

with i = 1, 2, …, n, where we use the notation

Let {Ui, Uij} be an inductive system of normed linear spaces Ui and continuous linear maps uij; Uj → Ui. (We write j ≺ i if uij: Uj → Ui.) An inductive limit of the system with respect to a class of spaces A in and maps f in is a space Uu in Uu and a system ui → Uu of maps in such that (i) whenever j ≺ i, and that (ii) if A is any space in and fi: Ui → A is any system of maps in for which then there is a unique map f: Uu → A in such that fi = fo ui for each i. If is the class of all vector spaces and is the class of linear maps, we obtain the algebraic inductive limit, which we denote simply by U. The usual choice is to take to be the class of locally convex spaces and the class of continuous linear maps; the inductive limit UL then always exists [1, § 16 C]. If is again the continuous linear mappings but contains only normed spaces, the corresponding inductive limit UN may not always exist. However, if in addition we require that contains just contractions (norm-decreasing linear mappings), then an inductive limit Uc will exist if every uij is a contraction [2]. We shall give a condition under which these limits coincide (as far as possible), and consider the corresponding condition for projective limits.

A Riesz operator is a bounded linear operator on a Banach space which possesses a Riesz spectral theory. These operators have been studied in [5] and [6]. In §2 of this paper we characterise Riesz operators in terms of their resolvent operators. In [6] it was shown that every Riesz operator on a Hilbert space can be decomposed into the sum of compact and quasi-nilpotent parts. §3 contains an example to show that these parts cannot, in general, be chosen to commute. In §4 the eigenset of a Riesz operator is defined. It is a sequence of quadruples each of which consists of an eigenvalue, the corresponding spectral projection, index and nilpotent part. This sequence satisfies certain obvious conditions, and the question arises of the existence of a Riesz operator which has such a sequence as its eigenset. We give an example of an eigenset which has no corresponding Riesz operator.

Many of the techniques and notions used to study various important theorems in locally convex spaces are not effective for general linear topological spaces. In [4], a study is made of notionsin general linear topological spaces which can be used to replace barrelled, bornological, and quasi-barrelled spaces. The present paper contains a parallel study in the context of semiconvex spaces.

In accordance with customary notation, Jv(t) denotes the Bessel function of the first kind and order v, jvk its kth positive zero and jv, 0 = 0.

The object of this note is to prove that

In [2], R. C. James proved that a weakly closed subset X of a real Banach space is weakly compact if and only if each continuous linear form attains its supremum on X. He also extended the result to the locally convex case, and, in [5], J. D. Pryce gave a simplified proof of the general result that is recorded below for reference in the sequel.

A free product sixth-group (FPS-group) is, roughly speaking, a free product of groups with a number of additional defining relators, where, if two of these relators have a subword in common, then the length of this subword is less than one sixth of the lengths of either of the two relators.

Britton [1,2] has proved a general algebraic result for FPS-groups and has used this result in a discussion of the word problem for such groups.

Let k be any algebraically closed field, and denote by k((t)) the field of formal power series in one indeterminate t over k. Let

so that K is the field of Puiseux expansions with coefficients in k (each element of K is a formal power series in tl/r for some positive integer r). It is well-known that K is algebraically closed if and only if k is of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous to K which is algebraically closed when k has non-zero characteristic p. In this paper, I prove that the set L of all formal power series of the form Σaitei (where (ei) is well-ordered, ei = mi|nprt, n ∈ Ζ, mi ∈ Ζ, ai ∈ k, ri ∈ Ν) forms an algebraically closed field.