A New Shuffle Convolution for Multiple Zeta Values

Recently, interest in shuffle algebra has been renewed due to their connections with multiple zeta values. In this paper, we prove a new shuffle convolution that implies a reduction formula for the multiple zeta value z({5,1}ⁿ).Research partially supported by a grant from the Number Theory Foundation.     

Rigid Analytic Uniformization of Curves and the Study of Isogenies

Complex uniformization of curves is a popular tool in Number Theory. There are, however, some arithmetic and computational advantages in the use of p-adic uniformization. This paper compares the two theories and discusses how they can be used to study isogenies, with explicit examples of p-adic uniformization of hyperelliptic curves.      

On the linear combination of <Emphasis Type="Italic">q</Emphasis>-additive functions at prime places

Necessary and sufficient conditions are given for linear combinations of q-ary additive functions to belong to some function classes when the summation is extended to the set of primes. Supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA T46993.     

The Euler–Lagrange theory for Schur's Algorithm: Algebraic exposed points

In this paper the ideas of Algebraic Number Theory are applied to the Theory of Orthogonal polynomials for algebraic measures. The transferring tool are Wall continued fractions. It is shown that any set of closed arcs on the circle supports a quadratic measure and that any algebraic measure is either a Szegö measure or a measure supported by a proper subset of the unit circle consisting of a finite number of closed arcs. Singular parts of algebraic measures are finite sums of point masses.     

Class field theory for open curves over p-adic fields

We introduce the idèle class group for quasi-projective curves over p-adic fields and show that the kernel of the reciprocity map is divisible. This extends Saito’s class field theory for projective curves (Saito in J Number Theory 21:44–80, 1985).     

Book Reviews

Book reviewed in this article: Three and the Shape of Three, by Arthur G. Razzell and K. G. O. Watts. Classroom Guide to Three and the Shape of Three, by Charles F. Linn. Questions About the Oceans, by Harold W. Dubach and Robert W. Taber. Elements of Number Theory, by Anthony J. Pettofrezzo, Professor of Mathematical Sciences, Florida Technological University, Orlando, Florida, and Donald R. Byrkit. Assistant Professor, Faculty of Mathematics and Statistics, The University of West Florida, Pensacola, Florida.     

Asymptotic formulas for certain arithmetic functions

This is an extended summary of a talk given by the last named author at the Czecho-Slovake Number Theory Conference 2005, held at Malenovice in September 2005. It surveys some recent results concerning asymptotics for a class of arithmetic functions, including, e.g., the second moments of the number-of-divisors function d(n) and of the function r(n) which counts the number of ways to write a positive integer as a sum of two squares. For the proofs, reference is made to original articles by the authors published elsewhere. The last named author gratefully acknowledges support from the Austrian Science Fund (FWF) under project Nr. P18079-N12.     

On the representation of integers as sums of an odd number of squares

In a recent work, S. Cooper (J. Number Theory 103:135–162, [1988]) conjectured a formula for r _2k+1(p ²), the number of ways p ² can be expressed as a sum of 2k+1 squares. Inspired by this conjecture, we obtain an explicit formula for r _2k+1(n ²),n≥1. Dedicated to Srinivasa Ramanujan.     

Padé approximations to the logarithm II: Identities, recurrences, and symbolic computation

Combinatorial identities that were needed in [25] are proved, mostly with C. Schneider’s computer algebra package Sigma. The form of the Padé approximation of the logarithm of arbitrary order is stated as a conjecture. 2000 Mathematics Subject Classification Primary—41A21, 05A19, 33F10 Supported by NRF-grant 2047226. Supported by NRF-grant 2053748. Supported by the Austrian Academy of Sciences, by the John Knopfmacher Research Centre for Applicable Analysis and Number Theory, and by the SFB-grant F1305 and the grant P16613-N12 of the Austrian FWF. Supported by NRF-grant 2053756.     

Analysis of Alternative Digit Sets for Nonadjacent Representations

It is known that every positive integer n can be represented as a finite sum of the form ∑_ia_i2ⁱ, where a_i ∈ {0, 1,−1} and no two consecutive a_i’s are non-zero (“nonadjacent form”, NAF). Recently, Muir and Stinson [14, 15] investigated other digit sets of the form {0, 1, x}, such that each integer has a nonadjacent representation (such a number x is called admissible). The present paper continues this line of research. The topics covered include transducers that translate the standard binary representation into such a NAF and a careful topological study of the (exceptional) set (which is of fractal nature) of those numbers where no finite look-ahead is sufficient to construct the NAF from left-to-right, counting the number of digits 1 (resp. x) in a (random) representation, and the non-optimality of the representations if x is different from 3 or −1. This paper was written while the first author was a visitor at the John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Johannesburg. He thanks the centre for its hospitality. He was also supported by the grant S8307-MAT of the Austrian Science Fund. This author is supported by the grant NRF 2053748 of the South African National Research Foundation. The research of this author was done while he was with the John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Johannesburg.     

Some Consequences of Schanuel's Conjecture in Exponential Rings

In last years Schanuel's Conjecture has played a fundamental role in Transcendental Number Theory and in decidability issues.

In this article we investigate algebraic relations among the elements of the exponential field (?, e ^x ) modulo Schanuel's Conjecture. We prove that there are no further relations between π and i assuming Schanuel's Conjecture except the known ones, e ^πi  = ?1 and i ² = ?1. Moreover, modulo Schanuel's Conjecture we prove that the E-subring of ? generated by π is isomorphic to the free exponential ring on π.     

Free groups in the smallest ideal of β G

Let G be an infinite group with cardinality κ embeddable into a direct sum of countable groups and let β G be the Stone-Čech compactification of G as a discrete semigroup. We show that the structural group of the smallest ideal of β G contains copies of the free group on generators. Supported by NRF grants FA2007041200005 and IFR2008041600015, respectively, and The John Knopfmacher Centre for Applicable Analysis and Number Theory.     

Book Vignettes

Ripley, J. A., Jr. and Whitten, R. C., The Elements and Structure of the Physical Sciences; Second Edition. Delaney, C. F. G., Electronics for the Physicist. Chinery, Michael, New York: Franklin Watts, Inc., 1969. Science Dictionary of the Animal World. Dolciani, Mary P.; Wooten, William; Beckenbach, Edwin F. and Chinn, William G., Boston: Houghton Mifflin Company, 1970. Modern School Mathematics: Structure and Method: Course 1 (Teacher's Edition). Pettofrezzo, Anthony J. and Bykrit, Donald R., Elements of Number Theory.     

Large spaces between the zeros of the Riemann zeta-function and random matrix theory II

We develop the theory set out in Part 1 [R.R. Hall, Large spaces between the zeros of the Riemann zeta-function and random matrix theory, J. Number Theory 109 (2004) 240–265] and in particular provide a lower bound (and almost sure evaluation) for Λ(7). The square of this number is rational, as were the previous values, but still rather surprizing.     

Small quotients in Euclidean algorithms

Numbers whose continued fraction expansion contains only small digits have been extensively studied. In the real case, the Hausdorff dimension ?? _M of the reals with digits in their continued fraction expansion bounded by M was considered, and estimates of ?? _M for M???? were provided by Hensley (J. Number Theory 40:336?C358, 1992). In the rational case, first studies by Cusick (Mathematika 24:166?C172, 1997), Hensley (In: Proc. Int. Conference on Number Theory, Quebec, pp. 371?C385, 1987) and Vallée (J. Number Theory 72:183?C235, 1998) considered the case of a fixed bound M when the denominator N tends to ??. Later, Hensley (Pac. J. Math. 151(2):237?C255, 1991) dealt with the case of a bound M which may depend on the denominator N, and obtained a precise estimate on the cardinality of rational numbers of denominator less than N whose digits (in the continued fraction expansion) are less than M(N), provided the bound M(N) is large enough with respect to N. This paper improves this last result of Hensley towards four directions. First, it considers various continued fraction expansions; second, it deals with various probability settings (and not only the uniform probability); third, it studies the case of all possible sequences M(N), with the only restriction that M(N) is at least equal to a given constant M ₀; fourth, it refines the estimates due to Hensley, in the cases that are studied by Hensley. This paper also generalises previous estimates due to Hensley (J. Number Theory 40:336?C358, 1992) about the Hausdorff dimension ?? _M to the case of other continued fraction expansions. The method used in the paper combines techniques from analytic combinatorics and dynamical systems and it is an instance of the Dynamical Analysis paradigm introduced by Vallée (J. Théor. Nr. Bordx. 12:531?C570, 2000), and refined by Baladi and Vallée (J. Number Theory 110:331?C386, 2005).     

The Height of Increasing Trees

Increasing trees have been introduced by Bergeron, Flajolet, and Salvy [1]. This kind of notion covers several well-know classes of random trees like binary search trees, recursive trees, and plane oriented (or heap ordered) trees. We consider the height of increasing trees and prove for several classes of trees (including the above mentioned ones) that the height satisfies EH _n ~ γlogn (for some constant γ > 0) and Var H _n =  O(1) as n → ∞. The methods used are based on generating functions. This research was supported by the Austrian Science Foundation FWF, project S9604, that is part of the Austrian National Research Network "Analytic Combinatorics and Probabilistic Number Theory".     

Fonctions de partitions à parité périodique

Let be the set of positive integers and a subset of . For , let denote the number of partitions of n with parts in . In the paper J. Number Theory 73 (1998) 292, Nicolas et al. proved that, given any and , there is a unique set , such that is even for n>N. Soon after, Ben Saïd and Nicolas (Acta Arith. 106 (2003) 183) considered , and proved that for all k≥0, the sequence is periodic on n. In this paper, we generalise the above works for any formal power series f in with f(0)=1, by constructing a set such that the generating function of is congruent to f modulo 2, and by showing that if f=P/Q, where P and Q are in with P(0)=Q(0)=1, then for all k≥0 the sequence is periodic on n.     

On the index of composition of integers from various sets

Given an integer n ≥ 2, let λ(n) := (log n)/(log γ(n)), where γ(n) = Π _p|n p, stand for the index of composition of n, with λ(1) = 1. We study the distribution function of (λ(n) – 1) log n as n runs through particular sets of integers, such as the shifted primes, the values of a given irreducible cubic polynomial and the shifted powerful numbers. Research supported in part by a grant from NSERC. Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA. Professor M.V. Subbarao passed away on February 15, 2006. Received: 3 March 2006 Revised: 28 October 2006     

Padé approximations to the logarithm III: Alternative methods and additional results

We use C. Schneider’s summation software Sigma and a method due to Andrews-Newton-Zeilberger to reprove results from [5] and [24] as well as to prove new related material. 2000 Mathematics Subject Classification Primary—41A21, 05A19, 33F10 K. Driver: Supported by NRF-Grant 2047226. H. Prodinger: Supported by NRF-Grant 2053748. C. Schneider: Supported by the Austrian Academy of Sciences, by the John Knopfmacher Research Centre for Applicable Analysis and Number Theory, and by the SFB-grant F1305 and the grant P16613-N12 of the Austrian FWF. J. A. C. Weideman: Supported by NRF-Grant FA2005032300018.     

A reciprocity theorem for certain q-series found in Ramanujan’s lost notebook

Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan’s lost notebook. The first proof uses Ramanujan’s ₁ψ₁ summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D15 B. C. Berndt: Research partially supported by grant MDA904-00-1-0015 from the National Security Agency. A. J. Yee: Research partially supported by a grant from The Number Theory Foundation.