Corrigenda and addition to ``Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than '

An error in the program for verifying the Ankeny-Artin-Chowla (AAC) conjecture is reported. As a result, in the case of primes which are , the AAC conjecture has been verified using a different multiple of the regulator of the quadratic field than was meant. However, since any multiple of this regulator is suitable for this purpose, provided that it is smaller than , the main result that the AAC conjecture is true for all the primes which are , remains valid.

As an addition, we have verified the AAC conjecture for all the primes between and , with the corrected program.

     

关于两种连分式加速收敛方法等价性的一般猜想的证明

本文证明了［２］提出的两种连分式加速收敛方法等价性的一般猜想是正确的。     

Hilbert's Tenth Problem and Mazur's Conjecture for large subrings of

We give the first examples of infinite sets of primes such that Hilbert's Tenth Problem over has a negative answer. In fact, we can take to be a density 1 set of primes. We show also that for some such there is a punctured elliptic curve over such that the topological closure of in has infinitely many connected components.

     

Some Diophantine equations of the form

Let be a prime. It is shown that each of the two Diophantine equations or has integral solutions.

     

矩阵连分式序列的收敛性

本文研究了矩阵连分式的性质，获得了关于矩阵连分式序列收敛性的一些结果．     

convergence of the reconstruction formula for the potential function

It is known that the potential function of the Sturm-Liouville problem can be reconstructed from the nodal data by a pointwise limit. We show that this convergence is in fact .

     

Some characterizations of the automorphisms of

We present some nonlinear characterizations of the automorphisms of the operator algebra and the function algebra by means of their spectrum preserving properties.

     

On an asymptotic series and corresponding continued fraction for a gamma function ratio

Recurrence relations for the coefficients in the asymptotic expansion of a gamma function ratio are derived and a property of these coefficients is proved. The Stieltjes fraction for the series is given and a characteristic of the partial numerators is explained. A connection between the continued fraction and the error of a particular least squares approximation problem is discussed.     

Specialisation and Reduction of Continued Fractions of Formal Power Series

We discuss and illustrate the behaviour of the continued fraction of a formal power series under specialisation of parameters or their reduction modulo p and sketch some applications of the reduction theorem here proved.To Jean-Louis Nicolas in celebration of his sixtieth birthday2000 Mathematics Subject Classification: Primary—11J70, 11A55, 11J68     

Computational estimation of the constant

The paper describes a computational estimation of the constant characterizing the bounds of . It is known that as

with , while the truth of the Riemann hypothesis would also imply that . In the range , two sets of estimates of are computed, one for increasingly small minima and another for increasingly large maxima of . As increases, the estimates in the first set rapidly fall below and gradually reach values slightly below , while the estimates in the second set rapidly exceed and gradually reach values slightly above . The obtained numerical results are discussed and compared to the implications of recent theoretical work of Granville and Soundararajan.

     

Subgroups of ideal class groups of real quadratic algebraic function fields

Necessary and sufficient condition on real quadratic algebraic function fields K is given for their ideal class groups H(K) to contain cyclic subgroups of order n. And eight series of such real quadratic function fields K are obtained whose ideal class groups contain cyclic subgroups of order n. In particular, the ideal class numbers of these function fields are divisible by n.     

Inequalities for the Gamma function and estimates for the volume of sections of

Let and let be a -dimensional subspace of . We prove that , for and whenever . We also consider and other related cases. We obtain sharp inequalities involving Gamma function in order to get these results.

     

A representation of random upper semicontinuous functions

In this paper a representation of random upper semicontinuous functions in terms of -valued random elements is stated. This fact allows us to consider for the first time a complete and separable metric, the Skorohod one, on a wide class of upper semicontinuous functions. Finally, different relevant concepts of measurability for random upper semicontinuous functions are studied and the relationships between them are analyzed.

     

The closing lemma for nonsingular endomorphisms equivariant under free actions of finite groups

In this paper a closing lemma for nonsingular endomorphisms equivariant under free actions of finite-groups is proved. Hence a recurrent trajectory, as well as all of its symmetric conjugates, of a nonsingular endomorphism equivariant under a free action of a finite group can be closed up simultaneously by an arbitrarily small equivariant perturbation.

     

On Sufficient Conditions for Convergence of Two-Dimensional Continued Fractions

By the method of majorant fractions and equivalent transformations, the analogies of leszyski–Pringsheim criteria for two-dimensional continued fractions are obtained.     

Modular Equations for Ramanujan's Cubic Continued Fraction

In this paper we present two new identities providing relations between Ramanujan's cubic continued fraction G(q) and the two continued fractions G(q⁵) and G(q⁷).     

The nilpotence height of for odd primes

K. G. Monks has recently shown that the element has nilpotence height in the mod Steenrod algebra. Here the method and result are generalized to show that for an odd prime the element has nilpotence height in the mod Steenrod algebra.

     

Easy proofs of Riemann's functional equation for and of Lipschitz summation

We present a new, simple proof, based upon Poisson summation, of the Lipschitz summation formula. A conceptually easy corollary is the functional relation for the Hurwitz zeta function. As a direct consequence we obtain a short, motivated proof of Riemann's functional equation for .

     

Local Rankin-Selberg convolutions for : explicit conductor formula

Let be a non-Archimedean local field and , positive integers. For , let and let be an irreducible supercuspidal representation of . Jacquet, Piatetskii-Shapiro and Shalika have defined a local constant to the and an additive character of . This object is of central importance in the study of the local Langlands conjecture. It takes the form

where is an integer. The irreducible supercuspidal representations of have been described explicitly by Bushnell and Kutzko, via induction from open, compact mod centre, subgroups of . This paper gives an explicit formula for in terms of the inducing data for the . It uses, on the one hand, the alternative approach to the local constant due to Shahidi, and, on the other, the general theory of types along with powerful existence theorems for types in , developed by Bushnell and Kutzko.