51. CMB 2012 (vol 56 pp. 602)
52. CMB 2012 (vol 56 pp. 829)
 Pollack, Paul

On Mertens' Theorem for Beurling Primes
Let $1 \lt p_1 \leq p_2 \leq p_3 \leq \dots$ be an infinite sequence
$\mathcal{P}$ of real numbers for which $p_i \to \infty$, and associate to
this sequence the \emph{Beurling zeta function} $\zeta_{\mathcal{P}}(s):=
\prod_{i=1}^{\infty}(1p_i^{s})^{1}$. Suppose that for some constant
$A\gt 0$, we have
$\zeta_{\mathcal{P}}(s) \sim A/(s1)$, as $s\downarrow 1$. We prove that
$\mathcal{P}$ satisfies an analogue of a classical theorem of Mertens:
$\prod_{p_i \leq x}(11/p_i)^{1} \sim A \e^{\gamma} \log{x}$, as
$x\to\infty$.
Here $\e = 2.71828\ldots$ is the base of the natural logarithm and
$\gamma = 0.57721\ldots$ is the usual EulerMascheroni constant. This
strengthens a recent theorem of Olofsson.
Keywords:Beurling prime, Mertens' theorem, generalized prime, arithmetic semigroup, abstract analytic number theory Categories:11N80, 11N05, 11M45 

53. CMB 2012 (vol 56 pp. 723)
54. CMB 2012 (vol 56 pp. 520)
 Elbasraoui, Abdelkrim; Sebbar, Abdellah

Equivariant Forms: Structure and Geometry
In this paper we study the notion of equivariant forms introduced in
the authors' previous works. In particular, we completely classify all the
equivariant forms for a subgroup of
$\operatorname{SL}_2(\mathbb{Z})$
by means of the crossratio, the weight
2 modular forms, the quasimodular forms, as well as differential forms
of a Riemann surface and sections of a canonical line bundle.
Keywords:equivariant forms, modular forms, Schwarz derivative, crossratio, differential forms Category:11F11 

55. CMB 2012 (vol 56 pp. 814)
56. CMB 2012 (vol 56 pp. 544)
57. CMB 2011 (vol 56 pp. 225)
 Agashe, Amod

On the Notion of Visibility of Torsors
Let $J$ be an abelian variety and
$A$ be an abelian subvariety of $J$, both defined over $\mathbf{Q}$.
Let $x$ be an element of $H^1(\mathbf{Q},A)$.
Then there are at least two definitions of $x$ being visible in $J$:
one asks that the torsor corresponding to $x$ be isomorphic over $\mathbf{Q}$
to a subvariety of $J$, and the other asks that $x$ be in the kernel
of the natural map $H^1(\mathbf{Q},A) \to H^1(\mathbf{Q},J)$. In this article, we
clarify the relation between the two definitions.
Keywords:torsors, principal homogeneous spaces, visibility, ShafarevichTate group Categories:11G35, 14G25 

58. CMB 2011 (vol 56 pp. 412)
 Sanders, T.

Structure in Sets with Logarithmic Doubling
Suppose that $G$ is an abelian group, $A \subset G$ is finite with $A+A \leq KA$ and $\eta \in (0,1]$ is a parameter.
Our main result is that there is a set $\mathcal{L}$ such that
\begin{equation*}
A \cap \operatorname{Span}(\mathcal{L}) \geq K^{O_\eta(1)}A \quad\text{and}\quad \mathcal{L} = O(K^\eta\log A).
\end{equation*}
We include an application of this result to a generalisation of the RothMeshulam theorem due to Liu and Spencer.
Keywords:Fourier analysis, Freiman's theorem, capset problem Category:11B25 

59. CMB 2011 (vol 56 pp. 510)
 Dubickas, Artūras

Linear Forms in Monic Integer Polynomials
We prove a necessary and sufficient condition on the list of
nonzero integers $u_1,\dots,u_k$, $k \geq 2$, under which a monic
polynomial $f \in \mathbb{Z}[x]$ is expressible by a linear form
$u_1f_1+\dots+u_kf_k$ in monic polynomials $f_1,\dots,f_k \in
\mathbb{Z}[x]$. This condition is independent of $f$. We also show that if
this condition holds, then the monic polynomials $f_1,\dots,f_k$
can be chosen to be irreducible in $\mathbb{Z}[x]$.
Keywords:irreducible polynomial, height, linear form in polynomials, Eisenstein's criterion Categories:11R09, 11C08, 11B83 

60. CMB 2011 (vol 56 pp. 500)
 Browning, T. D.

The LangWeil Estimate for Cubic Hypersurfaces
An improved estimate is provided for the number of $\mathbb{F}_q$rational points
on a geometrically irreducible, projective, cubic hypersurface that is
not equal to a cone.
Keywords:cubic hypersurface, rational points, finite fields Categories:11G25, 14G15 

61. CMB 2011 (vol 56 pp. 251)
 Borwein, Peter; Choi, Stephen K. K.; Ganguli, Himadri

Sign Changes of the Liouville Function on Quadratics
Let $\lambda (n)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured
that
\begin{equation*}
\label{a.1}
\sum_{n\le x} \lambda (f(n)) =o(x)\tag{$*$}
\end{equation*}
for any polynomial $f(x)$ with integer coefficients which is not of
form $bg(x)^2$.
When $f(x)=x$, $(*)$ is equivalent to the prime number theorem.
Chowla's conjecture has been proved for linear functions,
but for degree
greater than 1, the conjecture seems
to be extremely hard and remains wide open.
One can consider a weaker form
of Chowla's conjecture.
Conjecture 1.
[Cassaigne et al.]
If $f(x) \in \mathbb{Z} [x]$ and is not in the form of $bg^2(x)$
for some $g(x)\in \mathbb{Z}[x]$, then $\lambda (f(n))$
changes sign infinitely often.
Clearly, Chowla's conjecture implies Conjecture 1.
Although weaker,
Conjecture 1 is still wide open for polynomials of degree $\gt 1$.
In this article, we study Conjecture 1 for
quadratic polynomials. One of our main theorems is the following.
Theorem 1
Let $f(x) = ax^2+bx +c $ with $a\gt 0$ and $l$
be a positive integer such that $al$ is
not a perfect square. If the
equation $f(n)=lm^2 $ has one solution
$(n_0,m_0) \in \mathbb{Z}^2$, then it has infinitely
many positive solutions $(n,m) \in \mathbb{N}^2$.
As a direct consequence of Theorem 1, we prove the following.
Theorem 2
Let $f(x)=ax^2+bx+c$ with $a \in \mathbb{N}$ and $b,c \in \mathbb{Z}$. Let
\[
A_0=\Bigl[\frac{b+(D+1)/2}{2a}\Bigr]+1.
\]
Then either the binary sequence $\{ \lambda (f(n)) \}_{n=A_0}^\infty$ is
a constant sequence or it changes sign infinitely often.
Some partial results of Conjecture 1 for quadratic polynomials are also proved using Theorem 1.
Keywords:Liouville function, Chowla's conjecture, prime number theorem, binary sequences, changes sign infinitely often, quadratic polynomials, Pell equation Categories:11N60, 11B83, 11D09 

62. CMB 2011 (vol 56 pp. 258)
 Chandoul, A.; Jellali, M.; Mkaouar, M.

The Smallest Pisot Element in the Field of Formal Power Series Over a Finite Field
Dufresnoy and Pisot characterized the smallest
Pisot number of degree $n \geq 3$ by giving explicitly its minimal
polynomial. In this paper, we translate Dufresnoy and Pisot's
result to the Laurent series case.
The
aim of this paper is to prove that the minimal polynomial
of the smallest Pisot element (SPE) of degree $n$ in the field of
formal power series over a finite field
is given by $P(Y)=Y^{n}\alpha XY^{n1}\alpha^n,$ where $\alpha$
is the least element of the finite field $\mathbb{F}_{q}\backslash\{0\}$
(as a finite total ordered set). We prove that the sequence of
SPEs of degree $n$ is decreasing and converges to $\alpha X.$
Finally, we show how to obtain explicit continued fraction
expansion of the smallest Pisot element over a finite field.
Keywords:Pisot element, continued fraction, Laurent series, finite fields Categories:11A55, 11D45, 11D72, 11J61, 11J66 

63. CMB 2011 (vol 56 pp. 148)
64. CMB 2011 (vol 56 pp. 161)
 Rêgo, L. C.; Cintra, R. J.

An Extension of the Dirichlet Density for Sets of Gaussian Integers
Several measures for the density of sets of integers have been proposed,
such as the asymptotic density, the Schnirelmann density, and the Dirichlet density. There has been some work in the literature on extending some of these concepts of density to higher dimensional sets of integers. In this work, we propose an extension of the Dirichlet density for sets of Gaussian integers and
investigate some of its properties.
Keywords:Gaussian integers, Dirichlet density Categories:11B05, 11M99, 11N99 

65. CMB 2011 (vol 55 pp. 842)
66. CMB 2011 (vol 56 pp. 283)
 Coons, Michael

Transcendental Solutions of a Class of Minimal Functional Equations
We prove a result concerning power series
$f(z)\in\mathbb{C}[\mkern3mu[z]\mkern3mu]$
satisfying a functional equation of the form
$$
f(z^d)=\sum_{k=1}^n
\frac{A_k(z)}{B_k(z)}f(z)^k,
$$
where $A_k(z),B_k(z)\in
\mathbb{C}[z]$. In particular, we show that if $f(z)$ satisfies a
minimal functional equation of the above form with $n\geqslant 2$,
then $f(z)$ is necessarily transcendental. Towards a more complete
classification, the case $n=1$ is also considered.
Keywords:transcendence, generating functions, Mahlertype functional equation Categories:11B37, 11B83, , 11J91 

67. CMB 2011 (vol 56 pp. 70)
 Hrubeš, P.; Wigderson, A.; Yehudayoff, A.

An Asymptotic Bound on the Composition Number of Integer Sums of Squares Formulas
Let $\sigma_{\mathbb Z}(k)$ be the smallest $n$ such that there exists an
identity
\[
(x_1^2 + x_2^2 + \cdots + x_k^2) \cdot (y_1^2 + y_2^2 + \cdots + y_k^2)
= f_1^2 + f_2^2 + \cdots + f_n^2,
\]
with $f_1,\dots,f_n$ being polynomials with integer coefficients in
the variables $x_1,\dots,x_k$ and $y_1,\dots,y_k$. We prove that
$\sigma_{\mathbb Z}(k) \geq \Omega(k^{6/5})$.
Keywords:composition formulas, sums of squares, RadonHurwitz number Category:11E25 

68. CMB 2011 (vol 55 pp. 850)
 Shparlinski, Igor E.; Stange, Katherine E.

Character Sums with Division Polynomials
We obtain nontrivial estimates of quadratic character sums of division polynomials $\Psi_n(P)$, $n=1,2, \dots$, evaluated at a given point $P$ on an elliptic curve over a finite field of $q$ elements. Our bounds are nontrivial if the order of $P$ is at least $q^{1/2 + \varepsilon}$ for some fixed $\varepsilon > 0$. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences that was recently brought up by K. Lauter and the second author.
Keywords:division polynomial, character sum Categories:11L40, 14H52 

69. CMB 2011 (vol 55 pp. 774)
 Mollin, R. A.; Srinivasan, A.

Pell Equations: NonPrincipal Lagrange Criteria and Central Norms
We provide a criterion for the central norm to be
any value in the simple continued fraction expansion of $\sqrt{D}$
for any nonsquare integer $D>1$. We also provide a simple criterion
for the solvability of the Pell equation $x^2Dy^2=1$ in terms of
congruence conditions modulo $D$.
Keywords:Pell's equation, continued fractions, central norms Categories:11D09, 11A55, 11R11, 11R29 

70. CMB 2011 (vol 55 pp. 435)
 Zelator, Konstantine

A Note on the Diophantine Equation $x^2 + y^6 = z^e$, $e \geq 4$
We consider the diophantine equation $x^2 + y^6 = z^e$, $e \geq 4$.
We show that, when $e$ is a multiple of $4$ or $6$, this equation
has no solutions in positive integers with $x$ and $y$ relatively prime.
As a corollary, we show
that there exists no primitive Pythagorean triangle one of whose
leglengths is a perfect cube, while the hypotenuse length is an
integer square.
Keywords:diophantine equation Category:11D 

71. CMB 2011 (vol 55 pp. 400)
 Sebbar, Abdellah; Sebbar, Ahmed

Eisenstein Series and Modular Differential Equations
The purpose of this paper is to solve various differential
equations having Eisenstein series as coefficients using various tools and techniques. The solutions
are given in terms of modular forms, modular functions, and
equivariant forms.
Keywords:differential equations, modular forms, Schwarz derivative, equivariant forms Categories:11F11, 34M05 

72. CMB 2011 (vol 55 pp. 26)
 Bertin, Marie José

A Mahler Measure of a $K3$ Surface Expressed as a Dirichlet $L$Series
We present another example of a $3$variable polynomial defining a $K3$hypersurface
and having a logarithmic Mahler measure expressed in terms of a Dirichlet
$L$series.
Keywords:modular Mahler measure, EisensteinKronecker series, $L$series of $K3$surfaces, $l$adic representations, LivnÃ© criterion, RankinCohen brackets Categories:11, 14D, 14J 

73. CMB 2011 (vol 55 pp. 67)
74. CMB 2011 (vol 55 pp. 193)
 Ulas, Maciej

Rational Points in Arithmetic Progressions on $y^2=x^n+k$
Let $C$ be a hyperelliptic curve given by the equation $y^2=f(x)$
for $f\in\mathbb{Z}[x]$ without multiple roots. We say that points
$P_{i}=(x_{i}, y_{i})\in C(\mathbb{Q})$ for $i=1,2,\dots, m$ are in
arithmetic progression if the numbers $x_{i}$ for $i=1,2,\dots, m$
are in arithmetic progression.
In this paper we show that there exists a polynomial $k\in\mathbb{Z}[t]$
with the property that on the elliptic curve $\mathcal{E}':
y^2=x^3+k(t)$ (defined over the field $\mathbb{Q}(t)$) we can find four
points in arithmetic progression that are independent in the group
of all $\mathbb{Q}(t)$rational points on the curve $\mathcal{E}'$. In
particular this result generalizes earlier results of Lee and
V\'{e}lez. We also show that if $n\in\mathbb{N}$ is odd,
then there are infinitely many $k$'s with the property that on
curves $y^2=x^n+k$ there are four rational points in arithmetic
progressions. In the case when $n$ is even we can find infinitely
many $k$'s such that on curves $y^2=x^n+k$ there are six rational
points in arithmetic progression.
Keywords:arithmetic progressions, elliptic curves, rational points on hyperelliptic curves Category:11G05 

75. CMB 2011 (vol 54 pp. 748)
 Shparlinski, Igor E.

On the Distribution of Irreducible Trinomials
We obtain new results about the number of trinomials $t^n + at + b$
with integer coefficients in a box $(a,b) \in [C, C+A] \times [D,
D+B]$ that are irreducible modulo a prime $p$. As a byproduct we
show that for any $p$ there are irreducible polynomials of height at
most $p^{1/2+o(1)}$, improving on the previous estimate of
$p^{2/3+o(1)}$ obtained by the author in 1989.
Keywords:irreducible trinomials, character sums Categories:11L40, 11T06 
