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.

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.

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.

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.

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~$X$.

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.