Expand all Collapse all | Results 1 - 7 of 7 |
1. CJM 2013 (vol 65 pp. 1320)
Orbital $L$-functions for the Space of Binary Cubic Forms We introduce the notion of orbital $L$-functions
for the space of binary cubic forms
and investigate their analytic properties.
We study their functional equations and residue formulas in some detail.
Aside from their intrinsic interest,
the results from this paper are used to
prove the existence of secondary terms in counting
functions for cubic fields.
This is worked out in a companion paper.
Keywords:binary cubic forms, prehomogeneous vector spaces, Shintani zeta functions, $L$-functions, cubic rings and fields Categories:11M41, 11E76 |
2. CJM 2007 (vol 59 pp. 311)
Growth and Zeros of the Zeta Function for Hyperbolic Rational Maps This paper describes new results on the growth and zeros of the Ruelle
zeta function for the Julia set of a hyperbolic rational map. It is
shown that the zeta function is bounded by $\exp(C_K |s|^{\delta})$ in
strips $|\Real s| \leq K$, where $\delta$ is the dimension of the
Julia set. This leads to bounds on the number of zeros in strips
(interpreted as the Pollicott--Ruelle resonances of this dynamical
system). An upper bound on the number of zeros in polynomial regions
$\{|\Real s | \leq |\Imag s|^\alpha\}$ is given, followed by weaker
lower bound estimates in strips $\{\Real s > -C, |\Imag s|\leq r\}$,
and logarithmic neighbourhoods
$\{ |\Real s | \leq \rho \log |\Imag s| \}$.
Recent numerical work of Strain--Zworski suggests the upper
bounds in strips are optimal.
Keywords:zeta function, transfer operator, complex dynamics Category:37C30 |
3. CJM 2003 (vol 55 pp. 292)
Infinitely Divisible Laws Associated with Hyperbolic Functions The infinitely divisible distributions on $\mathbb{R}^+$ of random
variables $C_t$, $S_t$ and $T_t$ with Laplace transforms
$$
\left( \frac{1}{\cosh \sqrt{2\lambda}} \right)^t, \quad \left(
\frac{\sqrt{2\lambda}}{\sinh \sqrt{2\lambda}} \right)^t, \quad \text{and}
\quad \left( \frac{\tanh \sqrt{2\lambda}}{\sqrt{2\lambda}} \right)^t
$$
respectively are characterized for various $t>0$ in a number of
different ways: by simple relations between their moments and
cumulants, by corresponding relations between the distributions and
their L\'evy measures, by recursions for their Mellin transforms, and
by differential equations satisfied by their Laplace transforms. Some
of these results are interpreted probabilistically via known
appearances of these distributions for $t=1$ or $2$ in the description
of the laws of various functionals of Brownian motion and Bessel
processes, such as the heights and lengths of excursions of a
one-dimensional Brownian motion. The distributions of $C_1$ and $S_2$
are also known to appear in the Mellin representations of two
important functions in analytic number theory, the Riemann zeta
function and the Dirichlet $L$-function associated with the quadratic
character modulo~4. Related families of infinitely divisible laws,
including the gamma, logistic and generalized hyperbolic secant
distributions, are derived from $S_t$ and $C_t$ by operations such as
Brownian subordination, exponential tilting, and weak limits, and
characterized in various ways.
Keywords:Riemann zeta function, Mellin transform, characterization of distributions, Brownian motion, Bessel process, LÃ©vy process, gamma process, Meixner process Categories:11M06, 60J65, 60E07 |
4. CJM 2002 (vol 54 pp. 916)
ConvexitÃ©, complÃ¨te monotonie et inÃ©galitÃ©s sur les fonctions zÃªta et gamma sur les fonctions des opÃ©rateurs de Baskakov et sur des fonctions arithmÃ©tiques |
ConvexitÃ©, complÃ¨te monotonie et inÃ©galitÃ©s sur les fonctions zÃªta et gamma sur les fonctions des opÃ©rateurs de Baskakov et sur des fonctions arithmÃ©tiques We give optimal upper and lower bounds for the function
$H(x,s)=\sum_{n\geq 1}\frac{1}{(x+n)^s}$ for $x\geq 0$ and $s>1$. These
bounds improve the standard inequalities with integrals. We deduce from them
inequalities about Riemann's $\zeta$ function, and we give a conjecture
about the monotonicity of the function
$s\mapsto[(s-1)\zeta(s)]^{\frac{1}{s-1}}$. Some applications concern the
convexity of functions related to Euler's $\Gamma$ function and optimal
majorization of elementary functions of Baskakov's operators. Then, the
result proved for the function $x\mapsto x^{-s}$ is extended to completely
monotonic functions. This leads to easy evaluation of the order of the
generating series of some arithmetical functions when $z$ tends to 1. The
last part is concerned with the class of non negative decreasing convex
functions on $]0,+\infty[$, integrable at infinity.
Nous prouvons un encadrement optimal pour la quantit\'e
$H(x,s)=\sum_{n\geq 1}\frac{1}{(x+n)^s}$ pour $x\geq 0$ et $s>1$, qui
am\'eliore l'encadrement standard par des int\'egrales. Cet encadrement
entra{\^\i}ne des in\'egalit\'es sur la fonction $\zeta$ de Riemann, et
am\`ene \`a conjecturer la monotonie de la fonction
$s\mapsto[(s-1)\zeta(s)]^{\frac{1}{s-1}}$. On donne des applications \`a
l'\'etude de la convexit\'e de fonctions li\'ees \`a la fonction $\Gamma$
d'Euler et \`a la majoration optimale des fonctions \'el\'ementaires
intervenant dans les op\'erateurs de Baskakov. Puis, nous \'etendons aux
fonctions compl\`etement monotones sur $]0,+\infty[$ les r\'esultats \'etablis
pour la fonction $x\mapsto x^{-s}$, et nous en d\'eduisons des preuves
\'el\'ementaires du comportement, quand $z$ tend vers $1$, des s\'eries
g\'en\'eratrices de certaines fonctions arithm\'etiques. Enfin, nous
prouvons qu'une partie du r\'esultat se g\'en\'eralise \`a une classe de
fonctions convexes positives d\'ecroissantes.
Keywords:arithmetical functions, Baskakov's operators, completely monotonic functions, convex functions, inequalities, gamma function, zeta function Categories:26A51, 26D15 |
5. CJM 2001 (vol 53 pp. 1194)
Explicit Upper Bounds for Residues of Dedekind Zeta Functions and Values of $L$-Functions at $s=1$, and Explicit Lower Bounds for Relative Class Numbers of $\CM$-Fields |
Explicit Upper Bounds for Residues of Dedekind Zeta Functions and Values of $L$-Functions at $s=1$, and Explicit Lower Bounds for Relative Class Numbers of $\CM$-Fields We provide the reader with a uniform approach for obtaining various
useful explicit upper bounds on residues of Dedekind zeta functions of
numbers fields and on absolute values of values at $s=1$ of $L$-series
associated with primitive characters on ray class groups of number
fields. To make it quite clear to the reader how useful such bounds
are when dealing with class number problems for $\CM$-fields, we
deduce an upper bound for the root discriminants of the normal
$\CM$-fields with (relative) class number one.
Keywords:Dedekind zeta functions, $L$-functions, relative class numbers, $\CM$-fields Categories:11R42, 11R29 |
6. CJM 2001 (vol 53 pp. 834)
Zeta Functions and `Kontsevich Invariants' on Singular Varieties Let $X$ be a nonsingular algebraic variety in characteristic zero. To
an effective divisor on $X$ Kontsevich has associated a certain
motivic integral, living in a completion of the Grothendieck ring of
algebraic varieties. He used this invariant to show that birational
(smooth, projective) Calabi-Yau varieties have the same Hodge
numbers. Then Denef and Loeser introduced the invariant {\it motivic
(Igusa) zeta function}, associated to a regular function on $X$, which
specializes to both the classical $p$-adic Igusa zeta function and the
topological zeta function, and also to Kontsevich's invariant.
This paper treats a generalization to singular varieties. Batyrev
already considered such a `Kontsevich invariant' for log terminal
varieties (on the level of Hodge polynomials of varieties instead of
in the Grothendieck ring), and previously we introduced a motivic zeta
function on normal surface germs. Here on any $\bbQ$-Gorenstein
variety $X$ we associate a motivic zeta function and a `Kontsevich
invariant' to effective $\bbQ$-Cartier divisors on $X$ whose support
contains the singular locus of~$X$.
Keywords:singularity invariant, topological zeta function, motivic zeta function Categories:14B05, 14E15, 32S50, 32S45 |
7. CJM 1998 (vol 50 pp. 794)
Upper bounds on $|L(1,\chi)|$ and applications We give upper bounds on the modulus of the values at $s=1$ of
Artin $L$-functions of abelian extensions unramified at all
the infinite places. We also explain how we can compute better
upper bounds and explain how useful such computed bounds are
when dealing with class number problems for $\CM$-fields. For
example, we will reduce the determination of all the
non-abelian normal $\CM$-fields of degree $24$ with Galois
group $\SL_2(F_3)$ (the special linear group over the finite
field with three elements) which have class number one to the
computation of the class numbers of $23$ such $\CM$-fields.
Keywords:Dedekind zeta function, Dirichlet series, $\CM$-field, relative class number Categories:11M20, 11R42, 11Y35, 11R29 |