176. CMB 2002 (vol 45 pp. 109)
 Hall, R. R.; Shiu, P.

The Distribution of Totatives
D.~H.~Lehmer initiated the study of the distribution of totatives, which
are numbers coprime with a given integer. This led to various problems
considered by P.~Erd\H os, who made a conjecture on such distributions.
We prove his conjecture by establishing a theorem on the ordering of
residues.
Keywords:Euler's function, totatives Categories:11A05, 11A07, 11A25 

177. CMB 2002 (vol 45 pp. 115)
 Luca, Florian

The Number of NonZero Digits of $n!$
Let $b$ be an integer with $b>1$. In this note, we prove that the
number of nonzero digits in the base $b$ representation of $n!$
grows at least as fast as a constant, depending on $b$, times $\log
n$.
Category:11A63 

178. CMB 2001 (vol 44 pp. 385)
 Ballantine, Cristina M.

A Hypergraph with Commuting Partial Laplacians
Let $F$ be a totally real number field and let $\GL_{n}$ be the
general linear group of rank $n$ over $F$. Let $\mathfrak{p}$
be a prime ideal of $F$ and $F_{\mathfrak{p}}$ the completion of $F$
with respect to the valuation induced by $\mathfrak{p}$. We will
consider a finite quotient of the affine building of the group
$\GL_{n}$ over the field $F_{\mathfrak{p}}$. We will view this object
as a hypergraph and find a set of commuting operators whose sum will
be the usual adjacency operator of the graph underlying the hypergraph.
Keywords:Hecke operators, buildings Categories:11F25, 20F32 

179. CMB 2001 (vol 44 pp. 398)
 Cardon, David A.; Ram Murty, M.

Exponents of Class Groups of Quadratic Function Fields over Finite Fields
We find a lower bound on the number of imaginary quadratic extensions
of the function field $\F_q(T)$ whose class groups have an element of
a fixed order.
More precisely, let $q \geq 5$ be a power of an odd prime and let $g$
be a fixed positive integer $\geq 3$. There are $\gg q^{\ell
(\frac{1}{2}+\frac{1}{g})}$ polynomials $D \in \F_q[T]$ with $\deg(D)
\leq \ell$ such that the class groups of the quadratic extensions
$\F_q(T,\sqrt{D})$ have an element of order~$g$.
Keywords:class number, quadratic function field Categories:11R58, 11R29 

180. CMB 2001 (vol 44 pp. 440)
 Hironaka, Eriko

The Lehmer Polynomial and Pretzel Links
In this paper we find a formula for the Alexander polynomial
$\Delta_{p_1,\dots,p_k} (x)$ of pretzel knots and links with
$(p_1,\dots,p_k, \nega 1)$ twists, where $k$ is odd and
$p_1,\dots,p_k$ are positive integers. The polynomial $\Delta_{2,3,7}
(x)$ is the wellknown Lehmer polynomial, which is conjectured to have
the smallest Mahler measure among all monic integer polynomials. We
confirm that $\Delta_{2,3,7} (x)$ has the smallest Mahler measure among
the polynomials arising as $\Delta_{p_1,\dots,p_k} (x)$.
Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups Categories:57M05, 57M25, 11R04, 11R27 

181. CMB 2001 (vol 44 pp. 282)
 Lee, Min Ho; Myung, Hyo Chul

Hecke Operators on Jacobilike Forms
Jacobilike forms for a discrete subgroup $\G \subset \SL(2,\mbb R)$
are formal power series with coefficients in the space of functions on
the Poincar\'e upper half plane satisfying a certain functional
equation, and they correspond to sequences of certain modular forms.
We introduce Hecke operators acting on the space of Jacobilike forms
and obtain an explicit formula for such an action in terms of modular
forms. We also prove that those Hecke operator actions on Jacobilike
forms are compatible with the usual Hecke operator actions on modular
forms.
Categories:11F25, 11F12 

182. CMB 2001 (vol 44 pp. 313)
 Reverter, Amadeu; Vila, Núria

Images of mod $p$ Galois Representations Associated to Elliptic Curves
We give an explicit recipe for the determination of the images
associated to the Galois action on $p$torsion points of elliptic
curves. We present a table listing the image for all the elliptic
curves defined over $\QQ$ without complex multiplication with
conductor less than 200 and for each prime number~$p$.
Keywords:Galois groups, elliptic curves, Galois representation, isogeny Categories:11R32, 11G05, 12F10, 14K02 

183. CMB 2001 (vol 44 pp. 160)
184. CMB 2001 (vol 44 pp. 242)
 Schueller, Laura Mann

The Zeta Function of a Pair of Quadratic Forms
The zeta function of a nonsingular pair of quadratic forms defined over a
finite field, $k$, of arbitrary characteristic is calculated. A.~Weil made
this computation when $\rmchar k \neq 2$. When the pair has even order, a
relationship between the number of zeros of the pair and the number of
places of degree one in an appropriate hyperelliptic function field is
Category:11G25 

185. CMB 2001 (vol 44 pp. 3)
 Alexandru, Victor; Popescu, Nicolae; Zaharescu, Alexandru

The Generating Degree of $\C_p$
The generating degree $\gdeg (A)$ of a topological commutative ring
$A$ with $\Char A = 0$ is the cardinality of the smallest subset $M$
of $A$ for which the subring $\Z[M]$ is dense in $A$. For a prime
number $p$, $\C_p$ denotes the topological completion of an algebraic
closure of the field $\Q_p$ of $p$adic numbers. We prove that $\gdeg
(\C_p) = 1$, \ie, there exists $t$ in $\C_p$ such that $\Z[t]$ is
dense in $\C_p$. We also compute $\gdeg \bigl( A(U) \bigr)$ where
$A(U)$ is the ring of rigid analytic functions defined on a ball $U$
in $\C_p$. If $U$ is a closed ball then $\gdeg \bigl( A(U) \bigr) =
2$ while if $U$ is an open ball then $\gdeg \bigl( A(U) \bigr)$ is
infinite. We show more generally that $\gdeg \bigl( A(U) \bigr)$ is
finite for any {\it affinoid} $U$ in $\PP^1 (\C_p)$ and $\gdeg \bigl(
A(U) \bigr)$ is infinite for any {\it wide open} subset $U$ of $\PP^1
(\C_p)$.
Category:11S99 

186. CMB 2001 (vol 44 pp. 19)
187. CMB 2001 (vol 44 pp. 97)
188. CMB 2001 (vol 44 pp. 115)
 Roy, Damien

Approximation algÃ©brique simultanÃ©e de nombres de Liouville
The purpose of this paper is to show the limitations of the
conjectures of algebraic approximation. For this, we construct
points of $\bC^m$ which do not admit good algebraic approximations
of bounded degree and height, when the bounds on the degree and the
height are taken from specific sequences. The coordinates of these
points are Liouville numbers.
Category:11J82 

189. CMB 2001 (vol 44 pp. 87)
 Lieman, Daniel; Shparlinski, Igor

On a New Exponential Sum
Let $p$ be prime and let $\vartheta\in\Z^*_p$ be of
multiplicative order $t$ modulo $p$. We consider exponential
sums of the form
$$
S(a) = \sum_{x =1}^{t} \exp(2\pi i a \vartheta^{x^2}/p)
$$
and prove that for any $\varepsilon > 0$
$$
\max_{\gcd(a,p) = 1} S(a) = O( t^{5/6 + \varepsilon}p^{1/8}) .
$$
Categories:11L07, 11T23, 11B50, 11K31, 11K38 

190. CMB 2001 (vol 44 pp. 22)
191. CMB 2001 (vol 44 pp. 12)
 Anisca, Razvan; Ilie, Monica

A Technique of Studying Sums of Central Cantor Sets
This paper is concerned with the structure of the arithmetic sum of a
finite number of central Cantor sets. The technique used to study this
consists of a duality between central Cantor sets and sets of subsums
of certain infinite series. One consequence is that the sum of a finite
number of central Cantor sets is one of the following: a finite union
of closed intervals, homeomorphic to the Cantor ternary set or an
$M$Cantorval.
Category:11B05 

192. CMB 2000 (vol 43 pp. 304)
 Darmon, Henri; Mestre, JeanFrançois

Courbes hyperelliptiques Ã multiplications rÃ©elles et une construction de Shih
Soient $r$ et $p$ deux nombres premiers distincts, soit $K = \Q(\cos
\frac{2\pi}{r})$, et soit $\F$ le corps r\'esiduel de $K$ en une place
audessus de $p$. Lorsque l'image de $(2  2\cos \frac{2\pi}{r})$
dans $\F$ n'est pas un carr\'e, nous donnons une construction
g\'eom\'etrique d'une extension r\'eguliere de $K(t)$ de groupe de
Galois $\PSL_2 (\F)$. Cette extension correspond \`a un rev\^etement
de $\PP^1_{/K}$ de \og{} signature $(r,p,p)$ \fg{} au sens de [3,
sec.~6.3], et son existence est pr\'edite par le crit\`ere de
rigidit\'e de Belyi, Fried, Thompson et Matzat. Sa construction
s'obtient en tordant la representation galoisienne associ\'ee aux
points d'ordre $p$ d'une famille de vari\'et\'es ab\'eliennes \`a
multiplications r\'eelles par $K$ d\'ecouverte par Tautz, Top et
Verberkmoes [6]. Ces vari\'et\'es ab\'eliennes sont d\'efinies sur un
corps quadratique, et sont isog\`enes \`a leur conjugu\'e galoisien.
Notre construction g\'en\'eralise une m\'ethode de Shih [4], [5], que
l'on retrouve quand $r = 2$ et $r = 3$.
Let $r$ and $p$ be distinct prime numbers, let $K = \Q(\cos
\frac{2\pi}{r})$, and let $\F$ be the residue field of $K$ at a place
above $p$. When the image of $(2  2\cos \frac{2\pi}{r})$ in $\F$ is
not a square, we describe a geometric construction of a regular
extension of $K(t)$ with Galois group $\PSL_2 (\F)$. This extension
corresponds to a covering of $\PP^1_{/K}$ of ``signature $(r,p,p)$''
in the sense of [3, sec.~6.3], and its existence is predicted by the
rigidity criterion of Belyi, Fried, Thompson and Matzat. Its
construction is obtained by twisting the mod $p$ galois representation
attached to a family of abelian varieties with real multiplications by
$K$ discovered by Tautz, Top and Verberkmoes [6]. These abelian
varieties are defined in general over a quadratic field, and are
isogenous to their galois conjugate. Our construction generalises a
method of Shih [4], [5], which one recovers when $r = 2$ and $r = 3$.
Categories:11G30, 14H25 

193. CMB 2000 (vol 43 pp. 380)
194. CMB 2000 (vol 43 pp. 282)
195. CMB 2000 (vol 43 pp. 236)
 Voloch, José Felipe

On a Question of Buium
We prove that $\{(n^pn)/p\}_p \in \prod_p \mathbf{F}_p$, with $p$
ranging over all primes, is independent of $1$ over the integers,
assuming a conjecture in elementary number theory generalizing
the infinitude of Mersenne primes. This answers a question of
Buium. We also prove a generalization.
Category:11A07 

196. CMB 2000 (vol 43 pp. 218)
197. CMB 2000 (vol 43 pp. 239)
 Yu, Gang

On the Number of Divisors of the Quadratic Form $m^2+n^2$
For an integer $n$, let $d(n)$ denote the ordinary divisor function.
This paper studies the asymptotic behavior of the sum
$$
S(x) := \sum_{m\leq x, n\leq x} d(m^2 + n^2).
$$
It is proved in the paper that, as $x \to \infty$,
$$
S(x) := A_1 x^2 \log x + A_2 x^2 + O_\epsilon (x^{\frac32 +
\epsilon}),
$$
where $A_1$ and $A_2$ are certain constants and $\epsilon$ is any
fixed positive real number.
The result corrects a false formula given in a paper of Gafurov
concerning the same problem, and improves the error $O \bigl(
x^{\frac53} (\log x)^9 \bigr)$ claimed by Gafurov.
Keywords:divisor, large sieve, exponential sums Categories:11G05, 14H52 

198. CMB 2000 (vol 43 pp. 115)
 Schmutz Schaller, Paul

Perfect NonExtremal Riemann Surfaces
An infinite family of perfect, nonextremal Riemann surfaces
is constructed, the first examples of this type of surfaces.
The examples are based on normal subgroups of the modular group
$\PSL(2,{\sf Z})$ of level $6$. They provide nonEuclidean
analogues to the existence of perfect, nonextremal positive
definite quadratic forms. The analogy uses the function {\it syst\/}
which associates to every Riemann surface $M$ the length of a systole,
which is a shortest closed geodesic of $M$.
Categories:11H99, 11F06, 30F45 

199. CMB 1999 (vol 42 pp. 441)
 Berrizbeitia, P.; Elliott, P. D. T. A.

Product Bases for the Rationals
A sequence of positive rationals generates a subgroup of finite
index in the multiplicative positive rationals, and group product
representations by the sequence need only a bounded number of
terms, if and only if certain related sequences have densities
uniformly bounded from below.
Categories:11N99, 11N05 

200. CMB 1999 (vol 42 pp. 427)
 Berndt, Bruce C.; Chan, Heng Huat

Ramanujan and the Modular $j$Invariant
A new infinite product $t_n$ was introduced by S.~Ramanujan on the
last page of his third notebook. In this paper, we prove
Ramanujan's assertions about $t_n$ by establishing new connections
between the modular $j$invariant and Ramanujan's cubic theory of
elliptic functions to alternative bases. We also show that for
certain integers $n$, $t_n$ generates the Hilbert class field of
$\mathbb{Q} (\sqrt{n})$. This shows that $t_n$ is a new class
invariant according to H.~Weber's definition of class invariants.
Keywords:modular functions, the Borweins' cubic thetafunctions, Hilbert class fields Categories:33C05, 33E05, 11R20, 11R29 
