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26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. CMB 2012 (vol 56 pp. 814)

Petridis, Yiannis N.; Raulf, Nicole; Risager, Morten S.
Quantum Limits of Eisenstein Series and Scattering States
We identify the quantum limits of scattering states for the modular surface. This is obtained through the study of quantum measures of non-holomorphic Eisenstein series away from the critical line. We provide a range of stability for the quantum unique ergodicity theorem of Luo and Sarnak.

Keywords:quantum limits, Eisenstein series, scattering poles
Categories:11F72, 58G25, 35P25

33. 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

34. CMB 2012 (vol 56 pp. 544)

Gauthier, P. M.
Universally Overconvergent Power Series via the Riemann Zeta-function
The Riemann zeta-function is employed to generate universally overconvergent power series.

Keywords:overconvergence, zeta-function
Categories:30K05, 11M06

35. 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, Shafarevich-Tate group
Categories:11G35, 14G25

36. 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 $|A+A| \leq K|A|$ 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 Roth--Meshulam theorem due to Liu and Spencer.

Keywords:Fourier analysis, Freiman's theorem, capset problem
Category:11B25

37. 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

38. CMB 2011 (vol 56 pp. 500)

Browning, T. D.
The Lang--Weil 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

39. 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

40. 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 $P(Y)=Y^{n}-\alpha XY^{n-1}-\alpha^n,$ 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

41. CMB 2011 (vol 56 pp. 148)

Oukhaba, Hassan; Viguié, Stéphane
On the Gras Conjecture for Imaginary Quadratic Fields
In this paper we extend K. Rubin's methods to prove the Gras conjecture for abelian extensions of a given imaginary quadratic field $k$ and prime numbers $p$ that divide the number of roots of unity in $k$.

Keywords:elliptic units, Stark units, Gras conjecture, Euler systems
Categories:11R27, 11R29, 11G16

42. CMB 2011 (vol 55 pp. 842)

Sairaiji, Fumio; Yamauchi, Takuya
The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey-Jarden Conjecture
Frey and Jarden asked if any abelian variety over a number field $K$ has the infinite Mordell-Weil rank over the maximal abelian extension $K^{\operatorname{ab}}$. In this paper, we give an affirmative answer to their conjecture for the Jacobian variety of any smooth projective curve $C$ over $K$ such that $\sharp C(K^{\operatorname{ab}})=\infty$ and for any abelian variety of $\operatorname{GL}_2$-type with trivial character.

Keywords:Mordell-Weil rank, Jacobian varieties, Frey-Jarden conjecture, abelian points
Categories:11G05, 11D25, 14G25, 14K07

43. CMB 2011 (vol 56 pp. 283)

Coons, Michael
Transcendental Solutions of a Class of Minimal Functional Equations
We prove a result concerning power series $f(z)\in\mathbb{C}[\mkern-3mu[z]\mkern-3mu]$ 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, Mahler-type functional equation
Categories:11B37, 11B83, , 11J91

44. 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

45. 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, Radon-Hurwitz number
Category:11E25

46. 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

47. CMB 2011 (vol 55 pp. 774)

Mollin, R. A.; Srinivasan, A.
Pell Equations: Non-Principal 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 non-square integer $D>1$. We also provide a simple criterion for the solvability of the Pell equation $x^2-Dy^2=-1$ in terms of congruence conditions modulo $D$.

Keywords:Pell's equation, continued fractions, central norms
Categories:11D09, 11A55, 11R11, 11R29

48. 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

49. 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

50. 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, Eisenstein-Kronecker series, $L$-series of $K3$-surfaces, $l$-adic representations, Livné criterion, Rankin-Cohen brackets
Categories:11, 14D, 14J
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