76. CMB 2009 (vol 52 pp. 3)
 Banks, W. D.

Carmichael Numbers with a Square Totient
Let $\varphi$ denote the Euler function. In this paper, we show that
for all large $x$ there are more than $x^{0.33}$ Carmichael numbers
$n\le x$ with the property that $\varphi(n)$ is a perfect square. We
also obtain similar results for higher powers.
Categories:11N25, 11A25 

77. CMB 2009 (vol 52 pp. 66)
 Dryden, Emily B.; Strohmaier, Alexander

Huber's Theorem for Hyperbolic Orbisurfaces
We show that for compact orientable hyperbolic orbisurfaces, the
Laplace spectrum determines the length spectrum as well as the
number of singular points of a given order. The converse also holds, giving
a full generalization of Huber's theorem to the setting of
compact orientable hyperbolic orbisurfaces.
Keywords:Huber's theorem, length spectrum, isospectral, orbisurfaces Categories:58J53, 11F72 

78. CMB 2009 (vol 52 pp. 63)
 Dietmann, Rainer

Small Zeros of Quadratic Forms Avoiding a Finite Number of Prescribed Hyperplanes
We prove a new upper bound for the smallest zero $\mathbf{x}$
of a quadratic form over a number field with the additional
restriction that $\mathbf{x}$ does not lie in a finite number of $m$ prescribed
hyperplanes. Our bound is polynomial in the height of the quadratic
form, with an exponent depending only on the number of variables but
not on $m$.
Categories:11D09, 11E12, 11H46, 11H55 

79. CMB 2009 (vol 52 pp. 53)
 Cummins, C. J.

Cusp Forms Like $\Delta$
Let $f$ be a squarefree integer and denote by $\Gamma_0(f)^+$ the
normalizer of $\Gamma_0(f)$ in $\SL(2,\R)$. We find the analogues of
the cusp form $\Delta$ for the groups $\Gamma_0(f)^+$.
Categories:11F03, 11F22, 30F35 

80. CMB 2009 (vol 52 pp. 117)
 Poulakis, Dimitrios

On the Rational Points of the Curve $f(X,Y)^q = h(X)g(X,Y)$
Let $q = 2,3$ and $f(X,Y)$, $g(X,Y)$, $h(X)$ be polynomials with
integer coefficients. In this paper we deal with the curve
$f(X,Y)^q = h(X)g(X,Y)$, and we show that under some favourable
conditions it is possible to determine all of its rational points.
Categories:11G30, 14G05, 14G25 

81. CMB 2008 (vol 51 pp. 627)
 Vidanovi\'{c}, Mirjana V.; Tri\v{c}kovi\'{c}, Slobodan B.; Stankovi\'{c}, Miomir S.

Summation of Series over Bourget Functions
In this paper we derive formulas for summation of series involving
J.~Bourget's generalization of Bessel functions of integer order, as
well as the analogous generalizations by H.~M.~Srivastava. These series are
expressed in terms of the Riemann $\z$ function and Dirichlet
functions $\eta$, $\la$, $\b$, and can be brought into closed form in
certain cases, which means that the infinite series are represented
by finite sums.
Keywords:Riemann zeta function, Bessel functions, Bourget functions, Dirichlet functions Categories:33C10, 11M06, 65B10 

82. CMB 2008 (vol 51 pp. 497)
 Borwein, Peter; Choi, KwokKwong Stephen; Mercer, Idris

Expected Norms of ZeroOne Polynomials
Let $\cA_n = \big\{ a_0 + a_1 z + \cdots + a_{n1}z^{n1} : a_j \in \{0, 1 \
} \big\}$, whose elements are called \emf{zeroone polynomials}
and correspond naturally to the $2^n$ subsets of $[n] := \{ 0, 1,
\ldots, n1 \}$. We also let $\cA_{n,m} = \{ \alf(z) \in \cA_n :
\alf(1) = m \}$, whose elements correspond to the ${n \choose m}$
subsets of~$[n]$ of size~$m$, and let $\cB_n = \cA_{n+1} \setminus
\cA_n$, whose elements are the zeroone polynomials of degree
exactly~$n$.
Many researchers have studied norms of polynomials with restricted
coefficients. Using $\norm{\alf}_p$ to denote the usual $L_p$ norm
of~$\alf$ on the unit circle, one easily sees that $\alf(z) = a_0 +
a_1 z + \cdots + a_N z^N \in \bR[z]$ satisfies $\norm{\alf}_2^2 = c_0$
and $\norm{\alf}_4^4 = c_0^2 + 2(c_1^2 + \cdots + c_N^2)$, where $c_k
:= \sum_{j=0}^{Nk} a_j a_{j+k}$ for $0 \le k \le N$.
If $\alf(z) \in \cA_{n,m}$, say $\alf(z) = z^{\beta_1} + \cdots +
z^{\beta_m}$ where $\beta_1 < \cdots < \beta_m$, then $c_k$ is the
number of times $k$ appears as a difference $\beta_i  \beta_j$. The
condition that $\alf \in \cA_{n,m}$ satisfies $c_k \in \{0,1\}$ for $1
\le k \le n1$ is thus equivalent to the condition that $\{ \beta_1,
\ldots, \beta_m \}$ is a \emf{Sidon set} (meaning all differences of
pairs of elements are distinct).
In this paper, we find the average of~$\\alf\_4^4$ over $\alf \in
\cA_n$, $\alf \in \cB_n$, and $\alf \in \cA_{n,m}$. We further show
that our expression for the average of~$\\alf\_4^4$ over~$\cA_{n,m}$
yields a new proof of the known result: if $m = o(n^{1/4})$ and
$B(n,m)$ denotes the number of Sidon sets of size~$m$ in~$[n]$, then
almost all subsets of~$[n]$ of size~$m$ are Sidon, in the sense that
$\lim_{n \to \infty} B(n,m)/\binom{n}{m} = 1$.
Categories:11B83, 11C08, 30C10 

83. CMB 2008 (vol 51 pp. 561)
84. CMB 2008 (vol 51 pp. 399)
 Meng, Xianmeng

Linear Equations with Small Prime and Almost Prime Solutions
Let $b_1, b_2$ be any integers such that
$\gcd(b_1, b_2)=1$ and $c_1b_1<b_2\leq c_2b_1$, where
$c_1, c_2$ are any given positive constants. Let $n$ be any
integer satisfying $\{gcd(n, b_i)=1$, $i=1,2$. Let $P_k$ denote
any integer with no more than $k$ prime factors, counted according
to multiplicity. In this paper, for almost all $b_2$, we prove (i)
a sharp lower bound for $n$ such that the equation $b_1p+b_2m=n$
is solvable in prime $p$ and almost prime $m=P_k$, $k\geq 3$
whenever both $b_i$ are positive, and (ii) a sharp upper bound for the
least solutions $p, m$ of the above equation whenever $b_i$ are
not of the same sign, where $p$ is a prime and $m=P_k, k\geq 3$.
Keywords:sieve method, additive problem Categories:11P32, 11N36 

85. CMB 2008 (vol 51 pp. 337)
 Bennett, Michael A.

Differences between Perfect Powers
We apply the hypergeometric method of Thue and Siegel to prove
that if $a$ and $b$ are positive integers, then the inequality $
0 < a^x  b^y  < \frac{1}{4} \, \max \{ a^{x/2}, b^{y/2} \}$
has at most a single solution in positive integers $x$ and $y$.
This essentially sharpens a classic result of LeVeque.
Categories:11D61, 11D45 

86. CMB 2008 (vol 51 pp. 172)
87. CMB 2008 (vol 51 pp. 32)
 Choi, Stephen; Zhou, Ping

On Linear Independence of a Certain Multivariate Infinite Product
Let $q,m,M \ge 2$ be positive integers and
$r_1,r_2,\dots ,r_m$ be positive rationals and
consider the following $M$ multivariate infinite products
\[
F_i = \prod_{j=0}^\infty ( 1+q^{(Mj+i)}r_1+q^{2(Mj+i)}r_2+\dots +
q^{m(Mj+i)}r_m)
\]
for $i=0,1,\dots ,M1$.
In this article, we study the linear independence of these infinite products.
In particular, we obtain a lower bound for the dimension of the vector space
$\IQ F_0+\IQ F_1 +\dots + \IQ F_{M1} + \IQ$ over $\IQ$ and show that
among these $M$ infinite products, $F_0, F_1,\dots ,F_{M1}$, at least
$\sim M/m(m+1)$ of them are irrational for fixed $m$ and $M \rightarrow
\infty$.
Category:11J72 

88. CMB 2008 (vol 51 pp. 3)
89. CMB 2008 (vol 51 pp. 134)
90. CMB 2008 (vol 51 pp. 100)
 Petkov, Vesselin

Dynamical Zeta Function for Several Strictly Convex Obstacles
The behavior of the dynamical zeta function $Z_D(s)$ related to
several strictly convex disjoint obstacles is similar to that of the
inverse $Q(s) = \frac{1}{\zeta(s)}$ of the Riemann zeta function
$\zeta(s)$. Let $\Pi(s)$ be the series obtained from $Z_D(s)$ summing
only over primitive periodic rays. In this paper we examine the
analytic singularities of $Z_D(s)$ and $\Pi(s)$ close to the line $\Re
s = s_2$, where $s_2$ is the abscissa of absolute convergence of the
series obtained by the second iterations of the primitive periodic
rays. We show that at least one of the functions $Z_D(s), \Pi(s)$
has a singularity at $s = s_2$.
Keywords:dynamical zeta function, periodic rays Categories:11M36, 58J50 

91. CMB 2008 (vol 51 pp. 57)
 Dobrowolski, Edward

A Note on Integer Symmetric Matrices and Mahler's Measure
We find a lower bound on the absolute value of the discriminant of
the minimal polynomial of an integral symmetric matrix and apply
this result to find a lower bound on Mahler's measure of related
polynomials and to disprove a conjecture of D. Estes and R. Guralnick.
Keywords:integer matrices, Lehmer's problem, Mahler's measure Categories:11C20, 11R06 

92. CMB 2007 (vol 50 pp. 486)
93. CMB 2007 (vol 50 pp. 594)
 Laubie, François

Ramification des groupes abÃ©liens d'automorphismes des corps $\mathbb F_q(\!(X)\!)$
Soit $q$ une puissance d'un nombre premier
$p$. Dans cette note on \'etablit la g\'en\'eralisation suivante
d'un th\'eor\`eme de Wintenberger : tout sousgroupe ab\'elien
ferm\'e du groupe des $\mathbb F_q$auto\morphismes continus du corps
des s\'eries formelles $\mathbb F_q(\!(X)\!)$ muni de sa filtration
de ramification est un groupe filtr\'e isomorphe au groupe de Galois
d'une extension ab\'elienne d'un corps local {\`a} corps
r\'esiduel $\mathbb F_q$, filtr\'e par les groupes de ramification
de l'extension en num\'erotation inf\'erieure.
Category:11S15 

94. CMB 2007 (vol 50 pp. 409)
95. CMB 2007 (vol 50 pp. 334)
96. CMB 2007 (vol 50 pp. 399)
 Komornik, Vilmos; Loreti, Paola

Expansions in Complex Bases
Beginning with a seminal paper of R\'enyi, expansions in noninteger real bases have been widely
studied in the last
forty years. They turned out to be relevant in
various domains of mathematics, such as the theory of finite
automata, number
theory, fractals or dynamical systems.
Several results were extended by Dar\'oczy and K\'atai
for expansions
in complex bases. We introduce an adaptation of the socalled greedy
algorithm to the complex case, and we
generalize one of their main theorems.
Keywords:noninteger bases, greedy expansions, betaexpansions Categories:11A67, 11A63, 11B85 

97. CMB 2007 (vol 50 pp. 313)
 Tzermias, Pavlos

On CauchyLiouvilleMirimanoff Polynomials
Let $p$ be a prime greater than or equal to 17 and
congruent to
2 modulo 3. We use results of Beukers and Helou on
CauchyLiouvilleMirimanoff
polynomials to show that
the intersection of the Fermat curve of degree $p$ with the
line $X+Y=Z$ in the projective plane
contains no algebraic points of degree
$d$ with $3 \leq d \leq 11$.
We prove a result on
the roots of these polynomials and show that, experimentally,
they seem to satisfy
the conditions of a mild extension of
an irreducibility theorem of P\'{o}lya and Szeg\"{o}.
These conditions are \emph{conjecturally}
also necessary for irreducibility.
Categories:11G30, 11R09, 12D05, 12E10 

98. CMB 2007 (vol 50 pp. 191)
 Drungilas, Paulius; Dubickas, Artūras

Every Real Algebraic Integer Is a Difference of Two Mahler Measures
We prove that every real
algebraic integer $\alpha$ is expressible by a
difference of two Mahler measures of integer polynomials.
Moreover, these polynomials can be chosen in such a way that they
both have the same degree as that of $\alpha$, say
$d$, one of these two polynomials is irreducible and
another has an irreducible factor of degree $d$, so
that $\alpha=M(P)bM(Q)$ with irreducible polynomials
$P, Q\in \mathbb Z[X]$ of degree $d$ and a
positive integer $b$. Finally, if $d \leqslant 3$, then one can take $b=1$.
Keywords:Mahler measures, Pisot numbers, Pell equation, $abc$conjecture Categories:11R04, 11R06, 11R09, 11R33, 11D09 

99. CMB 2007 (vol 50 pp. 234)
 Kuo, Wentang

A Remark on a Modular Analogue of the SatoTate Conjecture
The original SatoTate Conjecture concerns the angle distribution
of the eigenvalues arising from nonCM elliptic curves. In this paper,
we formulate a modular analogue of the SatoTate Conjecture and prove
that the angles arising from nonCM holomorphic Hecke
eigenforms with nontrivial central characters are not distributed
with respect to the SateTate measure
for nonCM elliptic curves. Furthermore, under a reasonable conjecture,
we prove that the expected distribution is uniform.
Keywords:$L$functions, Elliptic curves, SatoTate Categories:11F03, 11F25 

100. CMB 2007 (vol 50 pp. 215)
 Kloosterman, Remke

Elliptic $K3$ Surfaces with Geometric MordellWeil Rank $15$
We prove that the elliptic surface
$y^2=x^3+2(t^8+14t^4+1)x+4t^2(t^8+6t^4+1)$ has geometric MordellWeil
rank $15$. This completes a list of Kuwata, who gave explicit examples
of elliptic $K3$surfaces with geometric MordellWeil ranks
$0,1,\dots, 14, 16, 17, 18$.
Categories:14J27, 14J28, 11G05 
