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1. CMB Online first

Matringe, Nadir
A specialisation of the Bump-Friedberg $L$-function
We study the restriction of the Bump-Friedberg integrals to affine lines $\{(s+\alpha,2s),s\in\mathbb{C}\}$. It has a simple theory, very close to that of the Asai $L$-function. It is an integral representation of the product $L(s+\alpha,\pi)L(2s,\Lambda^2,\pi)$ which we denote by $L^{lin}(s,\pi,\alpha)$ for this abstract, when $\pi$ is a cuspidal automorphic representation of $GL(k,\mathbb{A})$ for $\mathbb{A}$ the adeles of a number field. When $k$ is even, we show that for a cuspidal automorphic representation $\pi$, the partial $L$-function $L^{lin,S}(s,\pi,\alpha)$ has a pole at $1/2$, if and only if $\pi$ admits a (twisted) global period, this gives a more direct proof of a theorem of Jacquet and Friedberg, asserting that $\pi$ has a twisted global period if and only if $L(\alpha+1/2,\pi)\neq 0$ and $L(1,\Lambda^2,\pi)=\infty$. When $k$ is odd, the partial $L$-function is holmorphic in a neighbourhood of $Re(s)\geq 1/2$ when $Re(\alpha)$ is $\geq 0$.

Keywords:automorphic L functions
Categories:11F70, 11F66

2. CMB Online first

Yamagishi, Masakazu
Resultants of Chebyshev Polynomials: The First, Second, Third, and Fourth Kinds
We give an explicit formula for the resultant of Chebyshev polynomials of the first, second, third, and fourth kinds. We also compute the resultant of modified cyclotomic polynomials.

Keywords:resultant, Chebyshev polynomial, cyclotomic polynomial
Categories:11R09, 11R18, 12E10, 33C45

3. CMB Online first

Sands, Jonathan W.
$L$-functions for Quadratic Characters and Annihilation of Motivic Cohomology Groups
Let $n$ be a positive even integer, and let $F$ be a totally real number field and $L$ be an abelian Galois extension which is totally real or CM. Fix a finite set $S$ of primes of $F$ containing the infinite primes and all those which ramify in $L$, and let $S_L$ denote the primes of $L$ lying above those in $S$. Then $\mathcal{O}_L^S$ denotes the ring of $S_L$-integers of $L$. Suppose that $\psi$ is a quadratic character of the Galois group of $L$ over $F$. Under the assumption of the motivic Lichtenbaum conjecture, we obtain a non-trivial annihilator of the motivic cohomology group $H_\mathcal{M}^2(\mathcal{O}_L^S,\mathbb{Z}(n))$ from the lead term of the Taylor series for the $S$-modified Artin $L$-function $L_{L/F}^S(s,\psi)$ at $s=1-n$.

Keywords:motivic cohomology, regulator, Artin L-functions
Categories:11R42, 11R70, 14F42, 19F27

4. CMB 2014 (vol 58 pp. 115)

Mantilla-Soler, Guillermo
Weak Arithmetic Equivalence
Inspired by the invariant of a number field given by its zeta function, we define the notion of weak arithmetic equivalence and show that under certain ramification hypotheses, this equivalence determines the local root numbers of the number field. This is analogous to a result of Rohrlich on the local root numbers of a rational elliptic curve. Additionally, we prove that for tame non-totally real number fields, the integral trace form is invariant under arithmetic equivalence.

Keywords:arithmeticaly equivalent number fields, root numbers
Categories:11R04, 11R42

5. CMB 2014 (vol 58 pp. 160)

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 sum-of-divisors function, and let $\lambda$ be Carmichael's lambda-function. 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 Erdős in 1946 to prove the $10$-normality of $0.235711131719\ldots$.

Keywords:normal number, Euler function, sum-of-divisors function, Carmichael lambda-function, Champernowne's number
Categories:11K16, 11A63, 11N25, 11N37

6. 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 sum-of-divisors function, and let $\lambda$ be Carmichael's lambda-function. 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 Erdős in 1946 to prove the $10$-normality of $0.235711131719\ldots$.

Keywords:normal number, Euler function, sum-of-divisors function, Carmichael lambda-function, Champernowne's number
Categories:11K16, 11A63, 11N25, 11N37

7. CMB 2014 (vol 57 pp. 551)

Kane, Daniel M.; Kominers, Scott Duke
Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions
For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n:=\mathop{\textrm{lcm}}(u_0,u_1,\dots, u_n)$ of the finite arithmetic progression $\{u_k:=u_0+kr\}_{k=0}^n$. We derive new lower bounds on $L_n$ that improve upon those obtained previously when either $u_0$ or $n$ is large. When $r$ is prime, our best bound is sharp up to a factor of $n+1$ for $u_0$ properly chosen, and is also nearly sharp as $n\to\infty$.

Keywords:least common multiple, arithmetic progression
Category:11A05

8. CMB 2014 (vol 57 pp. 495)

Fujita, Yasutsugu; Miyazaki, Takafumi
Jeśmanowicz' 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

9. 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

10. CMB 2014 (vol 57 pp. 485)

Franc, Cameron; Mason, Geoffrey
Fourier Coefficients of Vector-valued Modular Forms of Dimension $2$
We prove the following Theorem. Suppose that $F=(f_1, f_2)$ is a $2$-dimensional vector-valued 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:vector-valued modular form, modular group, bounded denominators
Categories:11F41, 11G99

11. CMB 2013 (vol 57 pp. 845)

Lei, Antonio
Factorisation of Two-variable $p$-adic $L$-functions
Let $f$ be a modular form which is non-ordinary at $p$. Loeffler has recently constructed four two-variable $p$-adic $L$-functions associated to $f$. In the case where $a_p=0$, he showed that, as in the one-variable 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, p-adic L-functions, supersingular primes
Categories:11S40, 11S80

12. 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 $a_i-a_{i-d}$, $d\lt i\lt n,$ are distinct, then $|A+A|\gg |A|^{3/2+c}$ for a constant $c\gt 0.$

Keywords:convex sets, additive energy, sumsets
Category:11B99

13. CMB 2013 (vol 57 pp. 381)

Łydka, Adrian
On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve
We study analytic properties function $m(z, E)$, which is defined on the upper half-plane 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:L-function, Möbius function, explicit formulae, elliptic curve
Categories:11M36, 11G40

14. 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 2012-02-03, http://dx.doi.org/10.4153/CMB-2011-200-2.

Keywords:quantum limits, Eisenstein series, scattering poles
Categories:11F72, 8G25, 35P25

15. 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

16. 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

17. 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 non-elliptic 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

18. 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

19. CMB 2012 (vol 56 pp. 785)

Liu, Zhixin
Small Prime Solutions to Cubic Diophantine Equations
Let $a_1, \cdots, a_9$ be non-zero 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, Waring-Goldbach problem, circle method
Categories:11P32, 11P05, 11P55

20. 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 $\log|P|$ 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 $\log|P|$ for possibly different $P$'s), multiple Mahler measure (involving products of $\log|P|$ for possibly different $P$'s), and higher Mahler measure (involving $\log^k|P|$).

Keywords:Mahler measure, polynomial
Categories:11R06, 11R09

21. CMB 2012 (vol 56 pp. 844)

Shparlinski, Igor E.
On the Average Number of Square-Free Values of Polynomials
We obtain an asymptotic formula for the number of square-free integers in $N$ consecutive values of polynomials on average over integral polynomials of degree at most $k$ and of height at most $H$, where $H \ge N^{k-1+\varepsilon}$ 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, square-free numbers
Category:11N32

22. CMB 2012 (vol 56 pp. 602)

Louboutin, Stéphane R.
Resultants of Chebyshev Polynomials: A Short Proof
We give a simple proof of the value of the resultant of two Chebyshev polynomials (of the first or the second kind), values lately obtained by D. P. Jacobs, M. O. Rayes and V. Trevisan.

Keywords:resultant, Chebyshev polynomials, cyclotomic polynomials
Categories:11R09, 11R04

23. 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} $\zeta_{\mathcal{P}}(s):= \prod_{i=1}^{\infty}(1-p_i^{-s})^{-1}$. Suppose that for some constant $A\gt 0$, we have $\zeta_{\mathcal{P}}(s) \sim A/(s-1)$, as $s\downarrow 1$. We prove that $\mathcal{P}$ satisfies an analogue of a classical theorem of Mertens: $\prod_{p_i \leq x}(1-1/p_i)^{-1} \sim A \e^{\gamma} \log{x}$, as $x\to\infty$. Here $\e = 2.71828\ldots$ is the base of the natural logarithm and $\gamma = 0.57721\ldots$ is the usual Euler--Mascheroni 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

24. 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 cross-ratio, 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, cross-ratio, differential forms
Category:11F11

25. CMB 2012 (vol 56 pp. 723)

Bérczes, Attila; Luca, Florian
On the Sum of Digits of Numerators of Bernoulli Numbers
Let $b\gt 1$ be an integer. We prove that for almost all $n$, the sum of the digits in base $b$ of the numerator of the Bernoulli number $B_{2n}$ exceeds $c\log n$, where $c:=c(b)\gt 0$ is some constant depending on $b$.

Keywords:Bernoulli numbers, sums of digits
Category:11B68
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