Expand all Collapse all | Results 51 - 75 of 210 |
51. CMB 2011 (vol 54 pp. 739)
The Infimum in the Metric Mahler Measure Dubickas and Smyth defined the metric Mahler measure on the
multiplicative group of non-zero algebraic numbers.
The definition involves taking an infimum over representations
of an algebraic number $\alpha$ by other
algebraic numbers. We verify their conjecture that the
infimum in its definition is always achieved, and we establish its
analog for the ultrametric Mahler measure.
Keywords:Weil height, Mahler measure, metric Mahler measure, Lehmer's problem Categories:11R04, 11R09 |
52. CMB 2011 (vol 54 pp. 288)
The Resultant of Chebyshev Polynomials Let $T_{n}$ denote the $n$-th
Chebyshev polynomial of the first kind,
and let $U_{n}$ denote the $n$-th
Chebyshev polynomial of the second kind.
We give an explicit formula for the resultant
$\operatorname{res}( T_{m}, T_{n} )$.
Similarly, we give a formula for
$\operatorname{res}( U_{m}, U_{n} )$.
Keywords:resultant, Chebyshev polynomial Categories:11Y11, 68W20 |
53. CMB 2011 (vol 54 pp. 316)
The Saddle-Point Method and the Li Coefficients
In this paper, we apply the saddle-point method in conjunction with
the theory of the NÃ¶rlund-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 |
54. CMB 2011 (vol 54 pp. 330)
Sur la borne infÃ©rieure 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. FrÃ¶hlich a
dÃ©montrÃ© que $\operatorname{Cl}_2(k)$ n'est jamais trivial. Dans cet article,
nous donnons une extension de ce rÃ©sultat, en dÃ©montrant que le
rang de $\operatorname{Cl}_2(k)$ est toujours supÃ©rieur ou Ã©gal Ã $2$. Nous
dÃ©montrons aussi, que la valeur $2$ est optimale pour une famille
infinie de corps $k$.
Keywords:class group, units, multiquadratic number fields Categories:11R29, 11R11 |
55. CMB 2010 (vol 53 pp. 654)
Variations on a Paper of ErdÅs and Heilbronn It is shown that an old direct argument of ErdÅs and Heilbronn may be elaborated to yield a result of the current inverse
type.
Categories:11L07, 11P70 |
56. CMB 2010 (vol 54 pp. 39)
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 |
57. CMB 2010 (vol 53 pp. 661)
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 |
58. CMB 2010 (vol 53 pp. 385)
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 |
59. CMB 2010 (vol 53 pp. 571)
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 |
60. CMB 2009 (vol 53 pp. 140)
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 Â- one with $\{0,1\}$-coefficients Â- 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 |
61. CMB 2009 (vol 53 pp. 187)
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 |
62. CMB 2009 (vol 53 pp. 102)
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 |
63. CMB 2009 (vol 53 pp. 58)
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 |
64. CMB 2009 (vol 53 pp. 95)
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 |
65. CMB 2009 (vol 53 pp. 204)
Corrigendum for "Consecutive large gaps in sequences defined by multiplicative constraints" No abstract.
Categories:11N25, 11B05 |
66. CMB 2009 (vol 53 pp. 87)
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 |
67. CMB 2009 (vol 52 pp. 481)
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 |
68. CMB 2009 (vol 52 pp. 583)
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 |
69. CMB 2009 (vol 52 pp. 511)
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 |
70. CMB 2009 (vol 52 pp. 195)
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 |
71. CMB 2009 (vol 52 pp. 186)
Extension of the Riemann $\xi$-Function's Logarithmic Derivative Positivity Region to Near the Critical Strip |
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 |
72. CMB 2009 (vol 52 pp. 237)
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 |
73. CMB 2009 (vol 52 pp. 66)
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 |
74. CMB 2009 (vol 52 pp. 3)
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 |
75. CMB 2009 (vol 52 pp. 53)
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 |