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51. CMB 2011 (vol 54 pp. 316)

Mazhouda, Kamel
The Saddle-Point Method and the Li Coefficients
In this paper, we apply the saddle-point method in conjunction with the theory of the Nörlund-Rice integrals to derive precise asymptotic formula for the generalized Li coefficients established by Omar and Mazhouda. Actually, for any function $F$ in the Selberg class $\mathcal{S}$ and under the Generalized Riemann Hypothesis, we have $$ \lambda_{F}(n)=\frac{d_{F}}{2}n\log n+c_{F}n+O(\sqrt{n}\log n), $$ with $$ c_{F}=\frac{d_{F}}{2}(\gamma-1)+\frac{1}{2}\log(\lambda Q_{F}^{2}),\ \lambda=\prod_{j=1}^{r}\lambda_{j}^{2\lambda_{j}}, $$ where $\gamma$ is the Euler's constant and the notation is as below.

Keywords:Selberg class, Saddle-point method, Riemann Hypothesis, Li's criterion
Categories:11M41, 11M06

52. CMB 2011 (vol 54 pp. 330)

Mouhib, A.
Sur la borne inférieure du rang du 2-groupe de classes de certains corps multiquadratiques
Soient $p_1,p_2,p_3$ et $q$ des nombres premiers distincts tels que $p_1\equiv p_2\equiv p_3\equiv -q\equiv 1 \pmod{4}$, $k = \mathbf{Q} (\sqrt{p_1}, \sqrt{p_2}, \sqrt{p_3}, \sqrt{q})$ et $\operatorname{Cl}_2(k)$ le $2$-groupe de classes de $k$. A. Fröhlich a démontré que $\operatorname{Cl}_2(k)$ n'est jamais trivial. Dans cet article, nous donnons une extension de ce résultat, en démontrant que le rang de $\operatorname{Cl}_2(k)$ est toujours supérieur ou égal à $2$. Nous démontrons aussi, que la valeur $2$ est optimale pour une famille infinie de corps $k$.

Keywords:class group, units, multiquadratic number fields
Categories:11R29, 11R11

53. CMB 2010 (vol 53 pp. 654)

Elliott, P. D. T. A.
Variations on a Paper of Erdős and Heilbronn
It is shown that an old direct argument of Erdős and Heilbronn may be elaborated to yield a result of the current inverse type.

Categories:11L07, 11P70

54. CMB 2010 (vol 54 pp. 39)

Chapman, S. T.; García-Sánchez, P. A.; Llena, D.; Marshall, J.
Elements in a Numerical Semigroup with Factorizations of the Same Length
Questions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions.

Keywords:numerical monoid, numerical semigroup, non-unique factorization
Categories:20M14, 20D60, 11B75

55. CMB 2010 (vol 53 pp. 661)

Johnstone, Jennifer A.; Spearman, Blair K.
Congruent Number Elliptic Curves with Rank at Least Three
We give an infinite family of congruent number elliptic curves each with rank at least three.

Keywords:congruent number, elliptic curve, rank
Category:11G05

56. CMB 2010 (vol 53 pp. 385)

Achter, Jeffrey D.
Exceptional Covers of Surfaces
Consider a finite morphism $f: X \rightarrow Y$ of smooth, projective varieties over a finite field $\mathbf{F}$. Suppose $X$ is the vanishing locus in $\mathbf{P}^N$ of $r$ forms of degree at most $d$. We show that there is a constant $C$ depending only on $(N,r,d)$ and $\deg(f)$ such that if $|{\mathbf{F}}|>C$, then $f(\mathbf{F}): X(\mathbf{F}) \rightarrow Y(\mathbf{F})$ is injective if and only if it is surjective.

Category:11G25

57. CMB 2010 (vol 53 pp. 571)

Trifković, Mak
Periods of Modular Forms and Imaginary Quadratic Base Change
Let $f$ be a classical newform of weight $2$ on the upper half-plane $\mathcal H^{(2)}$, $E$ the corresponding strong Weil curve, $K$ a class number one imaginary quadratic field, and $F$ the base change of $f$ to $K$. Under a mild hypothesis on the pair $(f,K)$, we prove that the period ratio $\Omega_E/(\sqrt{|D|}\Omega_F)$ is in $\mathbb Q$. Here $\Omega_F$ is the unique minimal positive period of $F$, and $\Omega_E$ the area of $E(\mathbb C)$. The claim is a specialization to base change forms of a conjecture proposed and numerically verified by Cremona and Whitley.

Category:11F67

58. CMB 2009 (vol 53 pp. 204)

59. CMB 2009 (vol 53 pp. 58)

Dąbrowski, Andrzej; Jędrzejak, Tomasz
Ranks in Families of Jacobian Varieties of Twisted Fermat Curves
In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series.

Keywords:Fermat curve, Jacobian variety, elliptic curve, canonical height
Categories:11G10, 11G05, 11G50, 14G05, 11G30, 14H45, 14K15

60. CMB 2009 (vol 53 pp. 87)

Ghioca, Dragos
Elliptic Curves over the Perfect Closure of a Function Field
We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

Keywords:elliptic curves, heights
Categories:11G50, 11G05

61. CMB 2009 (vol 53 pp. 95)

Ghioca, Dragos
Towards the Full Mordell-Lang Conjecture for Drinfeld Modules
Let $\phi$ be a Drinfeld module of generic characteristic, and let X be a sufficiently generic affine subvariety of $\mathbb{G_a^g}$. We show that the intersection of X with a finite rank $\phi$-submodule of $\mathbb{G_a^g}$ is finite.

Keywords:Drinfeld module, Mordell-Lang conjecture
Categories:11G09, 11G10

62. CMB 2009 (vol 53 pp. 140)

Mukunda, Keshav
Pisot Numbers from $\{ 0, 1 \}$-Polynomials
A \emph{Pisot number} is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a \emph{Salem number} is a real algebraic integer greater than 1, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is derived from a Newman polynomial –- one with $\{0,1\}$-coefficients –- and shows that they form a strictly increasing sequence with limit $(1+\sqrt{5}) / 2$. It has long been known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Newman polynomials.

Categories:11R06, 11R09, 11C08

63. CMB 2009 (vol 53 pp. 187)

Ünver, Sinan
On the Local Unipotent Fundamental Group Scheme
We prove a local, unipotent, analog of Kedlaya's theorem for the pro-p part of the fundamental group of integral affine schemes in characteristic p.

Category:11G25

64. CMB 2009 (vol 53 pp. 102)

Khan, Rizwanur
Spacings Between Integers Having Typically Many Prime Factors
We show that the sequence of integers which have nearly the typical number of distinct prime factors forms a Poisson process. More precisely, for $\delta$ arbitrarily small and positive, the nearest neighbor spacings between integers n with $|\omega(n) - log log n| < (log log n)^{\delta}$ obey the Poisson distribution law.

Category:11K99

65. CMB 2009 (vol 52 pp. 511)

Bonciocat, Anca Iuliana; Bonciocat, Nicolae Ciprian
The Irreducibility of Polynomials That Have One Large Coefficient and Take a Prime Value
We use some classical estimates for polynomial roots to provide several irreducibility criteria for polynomials with integer coefficients that have one sufficiently large coefficient and take a prime value.

Keywords:Estimates for polynomial roots, irreducible polynomials
Categories:11C08, 11R09

66. CMB 2009 (vol 52 pp. 481)

Alaca, Ay\c{s}e; Alaca, \c{S}aban; Williams, Kenneth S.
Some Infinite Products of Ramanujan Type
In his ``lost'' notebook, Ramanujan stated two results, which are equivalent to the identities \[ \prod_{n=1}^{\infty} \frac{(1-q^n)^5}{(1-q^{5n})} =1-5\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{d} d \Big) q^n \] and \[ q\prod_{n=1}^{\infty} \frac{(1-q^{5n})^5}{(1-q^{n})} =\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{n/d} d \Big) q^n. \] We give several more identities of this type.

Keywords:Power series expansions of certain infinite products
Categories:11E25, 11F11, 11F27, 30B10

67. CMB 2009 (vol 52 pp. 583)

Konstantinou, Elisavet; Kontogeorgis, Aristides
Computing Polynomials of the Ramanujan $t_n$ Class Invariants
We compute the minimal polynomials of the Ramanujan values $t_n$, where $n\equiv 11 \mod 24$, using the Shimura reciprocity law. These polynomials can be used for defining the Hilbert class field of the imaginary quadratic field $\mathbb{Q}(\sqrt{-n})$ and have much smaller coefficients than the Hilbert polynomials.

Categories:11R29, 33E05, 11R20

68. CMB 2009 (vol 52 pp. 186)

Broughan, Kevin A.
Extension of the Riemann $\xi$-Function's Logarithmic Derivative Positivity Region to Near the Critical Strip
If $K$ is a number field with $n_k=[k:\mathbb{Q}]$, and $\xi_k$ the symmetrized Dedekind zeta function of the field, the inequality $$\Re\,{\frac{ \xi_k'(\sigma + {\rm i} t)}{\xi_k(\sigma + {\rm i} t)}} > \frac{ \xi_k'(\sigma)}{\xi_k(\sigma)}$$ for $t\neq 0$ is shown to be true for $\sigma\ge 1+ 8/n_k^\frac{1}{3}$ improving the result of Lagarias where the constant in the inequality was 9. In the case $k=\mathbb{Q}$ the inequality is extended to $\si\ge 1$ for all $t$ sufficiently large or small and to the region $\si\ge 1+1/(\log t -5)$ for all $t\neq 0$. This answers positively a question posed by Lagarias.

Keywords:Riemann zeta function, xi function, zeta zeros
Categories:11M26, 11R42

69. CMB 2009 (vol 52 pp. 195)

Garaev, M. Z.; Garcia, V. C.; Konyagin, S. V.
The Waring Problem with the Ramanujan $\tau$-Function, II
Let $\tau(n)$ be the Ramanujan $\tau$-function. We prove that for any integer $N$ with $|N|\ge 2$ the diophantine equation $$\sum_{i=1}^{148000}\tau(n_i)=N$$ has a solution in positive integers $n_1, n_2,\ldots, n_{148000}$ satisfying the condition $$\max_{1\le i\le 148000}n_i\ll |N|^{2/11}e^{-c\log |N|/\log\log |N|},$$ for some absolute constant $c>0.$

Categories:11B13, 11F30

70. CMB 2009 (vol 52 pp. 237)

Ghioca, Dragos
Points of Small Height on Varieties Defined over a Function Field
We obtain a Bogomolov type of result for the affine space defined over the algebraic closure of a function field of transcendence degree $1$ over a finite field.

Keywords:heights, Bogomolov conjecture
Categories:11G50, 11G25, 11G10

71. CMB 2009 (vol 52 pp. 53)

Cummins, C. J.
Cusp Forms Like $\Delta$
Let $f$ be a square-free integer and denote by $\Gamma_0(f)^+$ the normalizer of $\Gamma_0(f)$ in $\SL(2,\R)$. We find the analogues of the cusp form $\Delta$ for the groups $\Gamma_0(f)^+$.

Categories:11F03, 11F22, 30F35

72. CMB 2009 (vol 52 pp. 117)

Poulakis, Dimitrios
On the Rational Points of the Curve $f(X,Y)^q = h(X)g(X,Y)$
Let $q = 2,3$ and $f(X,Y)$, $g(X,Y)$, $h(X)$ be polynomials with integer coefficients. In this paper we deal with the curve $f(X,Y)^q = h(X)g(X,Y)$, and we show that under some favourable conditions it is possible to determine all of its rational points.

Categories:11G30, 14G05, 14G25

73. CMB 2009 (vol 52 pp. 3)

Banks, W. D.
Carmichael Numbers with a Square Totient
Let $\varphi$ denote the Euler function. In this paper, we show that for all large $x$ there are more than $x^{0.33}$ Carmichael numbers $n\le x$ with the property that $\varphi(n)$ is a perfect square. We also obtain similar results for higher powers.

Categories:11N25, 11A25

74. CMB 2009 (vol 52 pp. 66)

Dryden, Emily B.; Strohmaier, Alexander
Huber's Theorem for Hyperbolic Orbisurfaces
We show that for compact orientable hyperbolic orbisurfaces, the Laplace spectrum determines the length spectrum as well as the number of singular points of a given order. The converse also holds, giving a full generalization of Huber's theorem to the setting of compact orientable hyperbolic orbisurfaces.

Keywords:Huber's theorem, length spectrum, isospectral, orbisurfaces
Categories:58J53, 11F72

75. CMB 2009 (vol 52 pp. 63)

Dietmann, Rainer
Small Zeros of Quadratic Forms Avoiding a Finite Number of Prescribed Hyperplanes
We prove a new upper bound for the smallest zero $\mathbf{x}$ of a quadratic form over a number field with the additional restriction that $\mathbf{x}$ does not lie in a finite number of $m$ prescribed hyperplanes. Our bound is polynomial in the height of the quadratic form, with an exponent depending only on the number of variables but not on $m$.

Categories:11D09, 11E12, 11H46, 11H55
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