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1. CMB Online first

Dobrowolski, Edward
A note on Lawton's theorem
We prove Lawton's conjecture about the upper bound on the measure of the set on the unit circle on which a complex polynomial with a bounded number of coefficients takes small values. Namely, we prove that Lawton's bound holds for polynomials that are not necessarily monic. We also provide an analogous bound for polynomials in several variables. Finally, we investigate the dependence of the bound on the multiplicity of zeros for polynomials in one variable.

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

2. CMB Online first

Le Fourn, Samuel
Nonvanishing of central values of $L$-functions of newforms in $S_2 (\Gamma_0 (dp^2))$ twisted by quadratic characters
We prove that for $d \in \{ 2,3,5,7,13 \}$ and $K$ a quadratic (or rational) field of discriminant $D$ and Dirichlet character $\chi$, if a prime $p$ is large enough compared to $D$, there is a newform $f \in S_2(\Gamma_0(dp^2))$ with sign $(+1)$ with respect to the Atkin-Lehner involution $w_{p^2}$ such that $L(f \otimes \chi,1) \neq 0$. This result is obtained through an estimate of a weighted sum of twists of $L$-functions which generalises a result of Ellenberg. It relies on the approximate functional equation for the $L$-functions $L(f \otimes \chi, \cdot)$ and a Petersson trace formula restricted to Atkin-Lehner eigenspaces. An application of this nonvanishing theorem will be given in terms of existence of rank zero quotients of some twisted jacobians, which generalises a result of Darmon and Merel.

Keywords:nonvanishing of $L$-functions of modular forms, Petersson trace formula, rank zero quotients of jacobians
Categories:14J15, 11F67

3. CMB Online first

Pathak, Siddhi
On a conjecture of Livingston
In an attempt to resolve a folklore conjecture of Erdös regarding the non-vanishing at $s=1$ of the $L$-series attached to a periodic arithmetical function with period $q$ and values in $\{ -1, 1\} $, Livingston conjectured the $\bar{\mathbb{Q}}$ - linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston's conjecture for composite $q \geq 4$, highlighting that a new approach is required to settle Erdös's conjecture. We also prove that the conjecture is true for prime $q \geq 3$, and indicate that more ingredients will be needed to settle Erdös's conjecture for prime $q$.

Keywords:non-vanishing of L-series, linear independence of logarithms of algebraic numbers
Categories:11J86, 11J72

4. CMB 2016 (vol 59 pp. 592)

Liu, H. Q.
The Dirichlet Divisor Problem of Arithmetic Progressions
We design an elementary method to study the problem, getting an asymptotic formula which is better than Hooley's and Heath-Brown's results for certain cases.

Keywords:Dirichlet divisor problem, arithmetic progression
Categories:11L07, 11B83

5. CMB 2016 (vol 59 pp. 528)

Jahan, Qaiser
Characterization of Low-pass Filters on Local Fields of Positive Characteristic
In this article, we give necessary and sufficient conditions on a function to be a low-pass filter on a local field $K$ of positive characteristic associated to the scaling function for multiresolution analysis of $L^2(K)$. We use probability and martingale methods to provide such a characterization.

Keywords:multiresolution analysis, local field, low-pass filter, scaling function, probability, conditional probability and martingales
Categories:42C40, 42C15, 43A70, 11S85

6. CMB 2016 (vol 59 pp. 599)

Liu, Zhixin
Small Prime Solutions to Cubic Diophantine Equations II
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 cubic equation $a_1p_1^3+\cdots +a_9p_9^3=n$ has prime solutions satisfying $p_j \ll |n|^{1/3}+\textrm{max}\{|a_j|\}^{8+\varepsilon};$ (ii) if all $a_j$ are positive and $n \gg \textrm{max}\{|a_j|\}^{25+\varepsilon}$, then $a_1p_1^3+\cdots +a_9p_9^3=n$ is soluble in primes $p_j$. This results improve our previous results (Canad. Math. Bull., 56 (2013), 785-794) with the bounds $\textrm{max}\{|a_j|\}^{14+\varepsilon}$ and $\textrm{max}\{|a_j|\}^{43+\varepsilon}$ in place of $\textrm{max}\{|a_j|\}^{8+\varepsilon}$ and $\textrm{max}\{|a_j|\}^{25+\varepsilon}$ above, respectively.

Keywords:small prime, Waring-Goldbach problem, circle method
Categories:11P32, 11P05, 11P55

7. CMB 2016 (vol 59 pp. 624)

Otsubo, Noriyuki
Homology of the Fermat Tower and Universal Measures for Jacobi Sums
We give a precise description of the homology group of the Fermat curve as a cyclic module over a group ring. As an application, we prove the freeness of the profinite homology of the Fermat tower. This allows us to define measures, an equivalent of Anderson's adelic beta functions, in a similar manner to Ihara's definition of $\ell$-adic universal power series for Jacobi sums. We give a simple proof of the interpolation property using a motivic decomposition of the Fermat curve.

Keywords:Fermat curves, Ihara-Anderson theory, Jacobi sums
Categories:11S80, 11G15, 11R18

8. CMB 2015 (vol 58 pp. 869)

Wright, Thomas
Variants of Korselt's Criterion
Under sufficiently strong assumptions about the first term in an arithmetic progression, we prove that for any integer $a$, there are infinitely many $n\in \mathbb N$ such that for each prime factor $p|n$, we have $p-a|n-a$. This can be seen as a generalization of Carmichael numbers, which are integers $n$ such that $p-1|n-1$ for every $p|n$.

Keywords:Carmichael number, pseudoprime, Korselt's Criterion, primes in arithmetic progressions
Category:11A51

9. CMB 2015 (vol 58 pp. 704)

Benamar, H.; Chandoul, A.; Mkaouar, M.
On the Continued Fraction Expansion of Fixed Period in Finite Fields
The Chowla conjecture states that, if $t$ is any given positive integer, there are infinitely many prime positive integers $N$ such that $\operatorname{Per} (\sqrt{N})=t$, where $\operatorname{Per} (\sqrt{N})$ is the period length of the continued fraction expansion for $\sqrt{N}$. C. Friesen proved that, for any $k\in \mathbb{N}$, there are infinitely many square-free integers $N$, where the continued fraction expansion of $\sqrt{N}$ has a fixed period. In this paper, we describe all polynomials $Q\in \mathbb{F}_q[X] $ for which the continued fraction expansion of $\sqrt {Q}$ has a fixed period, also we give a lower bound of the number of monic, non-squares polynomials $Q$ such that $\deg Q= 2d$ and $ Per \sqrt {Q}=t$.

Keywords:continued fractions, polynomials, formal power series
Categories:11A55, 13J05

10. CMB 2015 (vol 58 pp. 774)

Hanson, Brandon
Character Sums over Bohr Sets
We prove character sum estimates for additive Bohr subsets modulo a prime. These estimates are analogous to classical character sum bounds of Pólya-Vinogradov and Burgess. These estimates are applied to obtain results on recurrence mod $p$ by special elements.

Keywords:character sums, Bohr sets, finite fields
Categories:11L40, 11T24, 11T23

11. CMB 2015 (vol 58 pp. 858)

Williams, Kenneth S.
Ternary Quadratic Forms and Eta Quotients
Let $\eta(z)$ $(z \in \mathbb{C},\;\operatorname{Im}(z)\gt 0)$ denote the Dedekind eta function. We use a recent product-to-sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly 10 eta quotients \[ f(z):=\eta^{a(m_1)}(m_1 z)\cdots \eta^{{a(m_r)}}(m_r z)=\sum_{n=1}^{\infty}c(n)e^{2\pi i nz},\quad z \in \mathbb{C},\;\operatorname{Im}(z)\gt 0, \] such that the Fourier coefficients $c(n)$ vanish for all positive integers $n$ in each of infinitely many non-overlapping arithmetic progressions. For example, it is shown that for $f(z)=\eta^4(z)\eta^{9}(4z)\eta^{-2}(8z)$ we have $c(n)=0$ for all $n$ in each of the arithmetic progressions $\{16k+14\}_{k \geq 0}$, $\{64k+56\}_{k \geq 0}$, $\{256k+224\}_{k \geq 0}$, $\{1024k+896\}_{k \geq 0}$, $\ldots$.

Keywords:Dedekind eta function, eta quotient, ternary quadratic forms, vanishing of Fourier coefficients, product-to-sum formula
Categories:11F20, 11E20, 11E25

12. CMB 2015 (vol 59 pp. 119)

Hu, Pei-Chu; Li, Bao Qin
A Simple Proof and Strengthening of a Uniqueness Theorem for L-functions
We give a simple proof and strengthening of a uniqueness theorem for functions in the extended Selberg class.

Keywords:meromorphic function, Dirichlet series, L-function, zero, order, uniqueness
Categories:30B50, 11M41

13. CMB 2015 (vol 58 pp. 730)

Efrat, Ido; Matzri, Eliyahu
Vanishing of Massey Products and Brauer Groups
Let $p$ be a prime number and $F$ a field containing a root of unity of order $p$. We relate recent results on vanishing of triple Massey products in the mod-$p$ Galois cohomology of $F$, due to Hopkins, Wickelgren, Mináċ, and Tân, to classical results in the theory of central simple algebras. For global fields, we prove a stronger form of the vanishing property.

Keywords:Galois cohomology, Brauer groups, triple Massey products, global fields
Categories:16K50, 11R34, 12G05, 12E30

14. CMB 2015 (vol 58 pp. 548)

Lü, Guangshi; Sankaranarayanan, Ayyadurai
Higher Moments of Fourier Coefficients of Cusp Forms
Let $S_{k}(\Gamma)$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group $SL(2, \mathbb{Z})$. Let $\lambda_f(n)$, $\lambda_g(n)$, $\lambda_h(n)$ be the $n$th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms $f(z) \in S_{k_1}(\Gamma), g(z) \in S_{k_2}(\Gamma), h(z) \in S_{k_3}(\Gamma)$ respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as $\lambda_f(n)^4\lambda_g(n)^2$, $\lambda_g(n)^6$, $\lambda_g(n)^2\lambda_h(n)^4$, and $\lambda_g(n^3)^2$ twisted by the arithmetic function $\lambda_f(n)$.

Keywords:Fourier coefficients of automorphic forms, Dirichlet series, triple product $L$-function, Perron's formula
Categories:11F30, 11F66

15. CMB 2015 (vol 58 pp. 580)

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

16. CMB 2015 (vol 58 pp. 620)

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

17. CMB 2015 (vol 58 pp. 423)

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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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
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