26. CMB Online first
 Pollack, Paul; Vandehey, Joseph

Some normal numbers generated by arithmetic functions
Let $g \geq 2$. A real number is said to be $g$normal if its base $g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let $\phi$ denote Euler's totient function, let $\sigma$ be the sumofdivisors function, and let $\lambda$ be Carmichael's lambdafunction. We show that if $f$ is any function formed by composing $\phi$, $\sigma$, or $\lambda$, then the number
\[ 0. f(1) f(2) f(3) \dots \]
obtained by concatenating the base $g$ digits of successive $f$values is $g$normal. We also prove the same result if the inputs $1, 2, 3, \dots$ are replaced with the primes $2, 3, 5, \dots$. The proof is an adaptation of a method introduced by Copeland and ErdÅs in 1946 to prove the $10$normality of $0.235711131719\ldots$.
Keywords:normal number, Euler function, sumofdivisors function, Carmichael lambdafunction, Champernowne's number Categories:11K16, 11A63, 11N25, 11N37 

27. CMB 2014 (vol 57 pp. 551)
28. CMB 2014 (vol 57 pp. 495)
 Fujita, Yasutsugu; Miyazaki, Takafumi

JeÅmanowicz' Conjecture with Congruence Relations. II
Let $a,b$ and $c$ be primitive Pythagorean numbers such that
$a^{2}+b^{2}=c^{2}$ with $b$ even.
In this paper, we show that if $b_0 \equiv \epsilon \pmod{a}$
with $\epsilon \in \{\pm1\}$
for certain positive divisors $b_0$ of $b$,
then the Diophantine equation $a^{x}+b^{y}=c^z$ has only the
positive solution $(x,y,z)=(2,2,2)$.
Keywords:exponential Diophantine equations, Pythagorean triples, Pell equations Categories:11D61, 11D09 

29. CMB 2014 (vol 57 pp. 538)
 Ide, Joshua; Jones, Lenny

Infinite Families of $A_4$Sextic Polynomials
In this article we develop a test to determine whether a sextic
polynomial that is irreducible over $\mathbb{Q}$ has Galois group isomorphic
to the alternating group $A_4$. This test does not involve the
computation of resolvents, and we use this test to construct several
infinite families of such polynomials.
Keywords:Galois group, sextic polynomial, inverse Galois theory, irreducible polynomial Categories:12F10, 12F12, 11R32, 11R09 

30. CMB 2014 (vol 57 pp. 485)
 Franc, Cameron; Mason, Geoffrey

Fourier Coefficients of Vectorvalued Modular Forms of Dimension $2$
We prove the following Theorem. Suppose that $F=(f_1, f_2)$ is a $2$dimensional vectorvalued modular form
on $\operatorname{SL}_2(\mathbb{Z})$ whose component functions $f_1, f_2$ have rational Fourier coefficients
with bounded denominators. Then $f_1$ and $f_2$ are classical modular forms on a congruence subgroup of the modular group.
Keywords:vectorvalued modular form, modular group, bounded denominators Categories:11F41, 11G99 

31. CMB 2013 (vol 57 pp. 877)
 Schoen, Tomasz

On Convolutions of Convex Sets and Related Problems
We prove some results concerning covolutions, the
additive energy and sumsets of convex sets and its generalizations. In
particular, we show that if a set $A=\{a_1,\dots,a_n\}_\lt \subseteq
\mathbb R$ has
the property that for every fixed
$1\leqslant d\lt n,$ all differences $a_ia_{id}$, $d\lt i\lt n,$ are distinct, then
$A+A\gg A^{3/2+c}$ for a constant $c\gt 0.$
Keywords:convex sets, additive energy, sumsets Category:11B99 

32. CMB 2013 (vol 57 pp. 845)
 Lei, Antonio

Factorisation of Twovariable $p$adic $L$functions
Let $f$ be a modular form which is nonordinary at $p$. Loeffler has
recently constructed four twovariable $p$adic $L$functions
associated to $f$. In the case where $a_p=0$, he showed that, as in
the onevariable case, Pollack's plus and minus splitting applies to
these new objects. In this article, we show that such a splitting can
be generalised to the case where $a_p\ne0$ using Sprung's logarithmic
matrix.
Keywords:modular forms, padic Lfunctions, supersingular primes Categories:11S40, 11S80 

33. CMB 2013 (vol 57 pp. 381)
 Łydka, Adrian

On Complex Explicit Formulae Connected with the MÃ¶bius Function of an Elliptic Curve
We study analytic properties function $m(z, E)$, which is defined on the upper halfplane as an integral from the shifted $L$function of an elliptic curve. We show that $m(z, E)$ analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for $m(z, E)$ in the strip $\Im{z}\lt 2\pi$.
Keywords:Lfunction, MÃ¶bius function, explicit formulae, elliptic curve Categories:11M36, 11G40 

34. CMB 2013 (vol 56 pp. 827)
 Petridis, Yiannis N.; Raulf, Nicole; Risager, Morten S.

Erratum to ``Quantum Limits of Eisenstein Series and Scattering States''
This paper provides an erratum to Y. N. Petridis,
N. Raulf, and M. S. Risager, ``Quantum Limits
of Eisenstein Series and Scattering States.'' Canad. Math. Bull., published
online 20120203, http://dx.doi.org/10.4153/CMB20112002.
Keywords:quantum limits, Eisenstein series, scattering poles Categories:11F72, 8G25, 35P25 

35. CMB 2013 (vol 56 pp. 673)
 Ayadi, K.; Hbaib, M.; Mahjoub, F.

Diophantine Approximation for Certain Algebraic Formal Power Series in Positive Characteristic
In this paper, we study rational approximations for certain algebraic power series over a finite field.
We obtain results for irrational elements of strictly positive degree
satisfying an equation of the type
\begin{equation}
\alpha=\displaystyle\frac{A\alpha^{q}+B}{C\alpha^{q}}
\end{equation}
where $(A, B, C)\in
(\mathbb{F}_{q}[X])^{2}\times\mathbb{F}_{q}^{\star}[X]$.
In particular,
we will give, under some conditions on the polynomials $A$, $B$
and $C$, well approximated elements satisfying this equation.
Keywords:diophantine approximation, formal power series, continued fraction Categories:11J61, 11J70 

36. CMB 2012 (vol 57 pp. 105)
 Luca, Florian; Shparlinski, Igor E.

On the Counting Function of Elliptic Carmichael Numbers
We give an upper bound for the number elliptic Carmichael numbers $n \le x$
that have recently been introduced by J. H. Silverman in the case of an elliptic curve without complex multiplication (non CM). We also discuss
several possible ways for further improvements.
Keywords:elliptic Carmichael numbers, applications of sieve methods Categories:11Y11, 11N36 

37. CMB 2012 (vol 56 pp. 570)
 Hoang, Giabao; Ressler, Wendell

Conjugacy Classes and Binary Quadratic Forms for the Hecke Groups
In this paper we give a lower bound
with respect to block length
for the trace of nonelliptic conjugacy classes
of the Hecke groups.
One consequence of our bound
is that there are finitely many
conjugacy classes of a given trace in any Hecke group.
We show that another consequence of our bound
is that
class numbers are finite for
related hyperbolic \( \mathbb{Z}[\lambda] \)binary quadratic forms.
We give canonical class representatives
and calculate class numbers
for some classes of hyperbolic \( \mathbb{Z}[\lambda] \)binary quadratic forms.
Keywords:Hecke groups, conjugacy class, quadratic forms Categories:11F06, 11E16, 11A55 

38. CMB 2012 (vol 56 pp. 695)
 Banks, William D.; Güloğlu, Ahmet M.; Yeager, Aaron M.

Carmichael meets Chebotarev
For any finite Galois extension $K$ of $\mathbb Q$
and any conjugacy class $C$ in $\operatorname {Gal}(K/\mathbb Q)$,
we show that there exist infinitely many Carmichael numbers
composed solely of primes for which the associated class of Frobenius
automorphisms is $C$. This result implies that for every natural
number $n$ there are infinitely many Carmichael numbers of the form
$a^2+nb^2$ with $a,b\in\mathbb Z $.
Keywords:Carmichael numbers, Chebotarev density theorem Categories:11N25, 11R45 

39. CMB 2012 (vol 56 pp. 785)
 Liu, Zhixin

Small Prime Solutions to Cubic Diophantine Equations
Let $a_1, \cdots, a_9$ be nonzero integers and $n$ any integer. Suppose
that $a_1+\cdots+a_9 \equiv n( \textrm{mod}\,2)$ and $(a_i, a_j)=1$ for $1 \leq i \lt j \leq 9$.
In this paper we prove that (i) if $a_j$ are not all of the same sign, then the above cubic
equation has prime solutions satisfying
$p_j \ll n^{1/3}+\textrm{max}\{a_j\}^{14+\varepsilon};$
and (ii) if all $a_j$ are positive and $n \gg \textrm{max}\{a_j\}^{43+\varepsilon}$, then the cubic
equation $a_1p_1^3+\cdots +a_9p_9^3=n$ is soluble in primes $p_j$.
This result is the extension of the linear and quadratic relative problems.
Keywords:small prime, WaringGoldbach problem, circle method Categories:11P32, 11P05, 11P55 

40. CMB 2012 (vol 56 pp. 759)
 Issa, Zahraa; Lalín, Matilde

A Generalization of a Theorem of Boyd and Lawton
The Mahler measure of a nonzero $n$variable polynomial $P$ is the integral of
$\logP$ on the unit $n$torus. A result of Boyd and Lawton says that
the Mahler measure of a multivariate polynomial is the limit of Mahler
measures of univariate polynomials. We prove the analogous
result for different extensions of Mahler measure such as generalized
Mahler measure (integrating the maximum of $\logP$ for possibly
different $P$'s),
multiple Mahler measure (involving products of $\logP$ for possibly
different $P$'s), and higher Mahler measure (involving $\log^kP$).
Keywords:Mahler measure, polynomial Categories:11R06, 11R09 

41. CMB 2012 (vol 56 pp. 844)
 Shparlinski, Igor E.

On the Average Number of SquareFree Values of Polynomials
We obtain an asymptotic formula for the number
of squarefree integers in $N$ consecutive values
of polynomials on average over integral
polynomials of degree at most $k$ and of
height at most $H$, where $H \ge N^{k1+\varepsilon}$
for some fixed $\varepsilon\gt 0$.
Individual results of this kind for polynomials of degree $k \gt 3$,
due to A. Granville (1998),
are only known under the $ABC$conjecture.
Keywords:polynomials, squarefree numbers Category:11N32 

42. CMB 2012 (vol 56 pp. 602)
43. 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 

44. 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 

45. CMB 2012 (vol 56 pp. 723)
46. CMB 2012 (vol 56 pp. 814)
47. CMB 2012 (vol 56 pp. 544)
48. 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 

49. 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 

50. 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 
