We improve upon the traditional error term in the truncated Perron formula for the logarithm of an L-function. All our constants are explicit.

We develop multisummability, in the positive real direction, for generalized power series with natural support, and we prove o-minimality of the expansion of the real field by all multisums of these series. This resulting structure expands both $\mathbb {R}_{\mathcal {G}}$ and the reduct of $\mathbb {R}_{\text {an}^*}$ generated by all convergent generalized power series with natural support; in particular, its expansion by the exponential function defines both the gamma function on $(0,\infty )$ and the zeta function on $(1,\infty )$.

In the first part of the paper, we introduce notions of asymptotic continuous orbit equivalence and asymptotic conjugacy in Smale spaces and characterize them in terms of their asymptotic Ruelle algebras with their dual actions. In the second part, we introduce a groupoid $C^{\ast }$-algebra that is an extended version of the asymptotic Ruelle algebra from a Smale space and study the extended Ruelle algebras from the view points of Cuntz–Krieger algebras. As a result, the asymptotic Ruelle algebra is realized as a fixed point algebra of the extended Ruelle algebra under certain circle action.

Let $K$ be a number field with a ring of integers ${\mathcal{O}}$. We follow Ferraguti and Micheli [‘On the Mertens–Cèsaro theorem for number fields’, Bull. Aust. Math. Soc.93(2) (2016), 199–210] to define a density for subsets of ${\mathcal{O}}$ and use it to find the density of the set of $j$-wise relatively $r$-prime $m$-tuples of algebraic integers. This provides a generalisation and analogue for several results on natural densities of integers and ideals of algebraic integers.

Let $K$ be a number field with ring of integers ${\mathcal{O}}$. After introducing a suitable notion of density for subsets of ${\mathcal{O}}$, generalising the natural density for subsets of $\mathbb{Z}$, we show that the density of the set of coprime $m$-tuples of algebraic integers is $1/{\it\zeta}_{K}(m)$, where ${\it\zeta}_{K}$ is the Dedekind zeta function of $K$. This generalises a result found independently by Mertens [‘Ueber einige asymptotische Gesetze der Zahlentheorie’, J. reine angew. Math. 77 (1874), 289–338] and Cesàro [‘Question 75 (solution)’, Mathesis 3 (1883), 224–225] concerning the density of coprime pairs of integers in $\mathbb{Z}$.

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$({{C}_{k}}{{\left| s \right|}^{\delta }})$ in strips $|\,\text{Re}\,s\,|\,\le \,K$, 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 $\left\{ \left| \,\operatorname{Re}s \right|\,\le \,{{\left| \,\operatorname{Im}s \right|}^{\alpha }} \right\}$ is given, followed by weaker lower bound estimates in strips $\{\text{Re}\,s\,>\,-C,\,|\,\text{IM}\,s|\,\le \,r\}$, and logarithmic neighbourhoods $\{|\text{Re}s|\le {{}_{\rho }}\log |\operatorname{Im}s|\}$. Recent numerical work of Strain-Zworski suggests the upper bounds in strips are optimal.

Over a perfect field $k$ of characteristic $p > 0$, we construct a ‘Witt vector cohomology with compact supports’ for separated $k$-schemes of finite type, extending (after tensorization with ${\mathbb Q}$) the classical theory for proper $k$-schemes. We define a canonical morphism from rigid cohomology with compact supports to Witt vector cohomology with compact supports, and we prove that it provides an identification between the latter and the slope ${<}1$ part of the former. Over a finite field, this allows one to compute congruences for the number of rational points in special examples. In particular, the congruence modulo the cardinality of the finite field of the number of rational points of a theta divisor on an abelian variety does not depend on the choice of the theta divisor. This answers positively a question by J.-P. Serre.

We construct multiple zeta functions as absolute tensor products of usual zeta functions. The Euler product expression is established for the most basic case $\zeta(s,\mathbf{F}_p)\otimes\zeta(s,\mathbf{F}_q)$ by using the signed double Poisson summation formula and the theory of the double sine function.

We prove that the zeta function of an irreducible hypersurface quasi-ordinary singularity f equals the zeta function of a plane curve singularity g. If the local coordinates $(x_1,\dots,x_{d+1})$ of f are ‘nice’, then $g=f(x_1,0,\dots,0,x_{d+1})$. Moreover, the Puiseux pairs of g can also be recovered from (any set of) distinguished tuples of f.

We prove that if q is a power of an odd prime, then there is no genus-2 curve over $\mathbf{F}_q$ whose Jacobian has characteristic polynomial of Frobenius equal to $x^4 + (2 - 2q)x^2 + q^2$. Our proof uses the Brauer relations in a biquadratic extension of $\mathbb{Q}$ to show that every principally polarized abelian surface over $\mathbf{F}_q$ with the given characteristic polynomial splits over $\mathbf{F}_{q^2}$ as a product of polarized elliptic curves.

We present a new approach to the problem of computing the zeta function of a hypersurface over a finite field. For a hypersurface defined by a polynomial of degree $d$ in $n$ variables over the field of $q$ elements, one desires an algorithm whose running time is a polynomial function of $d^n \log(q)$. (Here we assume $d \geq 2$, for otherwise the problem is easy.) The case $n = 1$ is related to univariate polynomial factorisation and is comparatively straightforward. When $n = 2$ one is counting points on curves, and the method of Schoof and Pila yields a complexity of $\log(q)^{C_d}$, where the function $C_d$ depends exponentially on $d$. For arbitrary $n$, the theorem of the author and Wan gives a complexity which is a polynomial function of $(pd^n \log(q))^n$, where $p$ is the characteristic of the field. A complexity estimate of this form can also be achieved for smooth hypersurfaces using the method of Kedlaya, although this has only been worked out in full for curves. The new approach we present should yield a complexity which is a small polynomial function of $pd^n\log(q)$. In this paper, we work this out in full for Artin–Schreier hypersurfaces defined by equations of the form $Z^p - Z = f$, where the polynomial $f$ has a diagonal leading form. The method utilises a relative $p$-adic cohomology theory for families of hypersurfaces, due in essence to Dwork. As a corollary of our main theorem, we obtain the following curious result. Let $f$ be a multivariate polynomial with integer coefficients whose leading form is diagonal. There exists an explicit deterministic algorithm which takes as input a prime $p$, outputs the number of solutions to the congruence equation $f = 0 \bmod{p}$, and runs in ${\mathcal O}(p^{2 + \varepsilon})$ bit operations, for any $\varepsilon > 0$. This improves upon the elementary estimate of ${\mathcal O}(p^{n-1 + \varepsilon})$ bit operations, where $n$ is the number of variables, which can be achieved using Berlekamp's root counting algorithm.

Nous prouvons un encadrement optimal pour la quantité $H\left( x,\,s \right)\,=\,\sum{_{n\ge 1}\,\frac{1}{{{\left( x+n \right)}^{s}}}}$ pour $x\,\ge \,0$ et $s\,>\,1$, qui améliore l'encadrement standard par des intégrales. Cet encadrement entraîne des inégalités sur la fonction $\zeta $ de Riemann, et amène à conjecturer la monotonie de la fonction $s\,\mapsto \,{{[(s\,-\,1)\text{ }\!\!\zeta\!\!\text{ (}s\text{)}]}^{\frac{1}{S-1}}}$. On donne des applications à l'étude de la convexité de fonctions liées à la fonction $\Gamma $ d'Euler et à la majoration optimale des fonctions élémentaires intervenant dans les opérateurs de Baskakov. Puis, nous étendons aux fonctions complètement monotones sur ]$0,\,+\infty $[ les résultats établis pour la fonction $x\,\mapsto \,{{x}^{-s}}$, et nous en déduisons des preuves élémentaires du comportement, quand $z$ tend vers 1, des séries génératrices de certaines fonctions arithmétiques. Enfin, nous prouvons qu'une partie du résultat se généralise à une classe de fonctions convexes positives décroissantes.

We show that for all finite fields Fq, there exists a curve C over Fq of genus 3 such that the number of rational points on C is within 3 of the Serre–Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over Fq.

The well-behaved Sylow theory for soluble groups is exploited to prove an Euler product for zeta functions counting certain subgroups in pro-soluble groups. This generalizes a result of Grunewald, Segal and Smith for nilpotent groups.

AMS 2000 Mathematics subject classification: Primary 20F16; 11M99

A question of Igusa (from 1978) inquires about the singular behavior of the singular series, determined by a polynomial mapping ${\mathbf P}$:K$^n$→K$^m$, m[les ]n, where K is a local field of characteristic zero. This paper describes in geometric terms the singularities of the singular series for two classes of polynomial maps ${\mathbf P}$=(P$_1$, P$_2$):K$^n$→K$^2$. The main result, which makes possible this description, is a type of uniformization of ${\mathbf P}$ by finitely many monomial maps μ(${\mathbf x}$)=(${\mathbf x}$ $^$$_1$, ${\mathbf x}$$^ $$_2$), such that rank ($^$$_1$ $_$$_2$)= 2. This is proved using resolution of singularities. Using this result, nontrivial estimates of oscillatory integrals with phase λ$_1$P$_1$+ λ$_2$P$_2$ are possible. These will be described elsewhere.

We prove that the zeta function and normal zeta function of a virtually abelian group have meromorphic continuation to the whole complex plane. We do this by relating the functions to classical L-functions of arithmetic orders considered by Hey, Solomon, Bushnell and Reiner. We calculate the zeta functions and normal zeta functions of the plane crystallographic groups. As a corollary of these calculations we produce \begin{enumerate} \item[(1)] examples of two isospectral residually finite groups with non-isomorphic profinite completions and even distinct lattices of subgroups; and \item[(2)] examples of non-nilpotent residually finite groups whose zeta functions enjoy an Euler product.

1991 Mathematics Subject Classification: 11M41, 20H15.

First, we prove that a non-abelian normal CM-field of degree 16 has odd relative class number if and only if it is dihedral, or is a compositum of two normal octic CM-fields with the same maximal real subfield, or has Galois group $G_9 =\langle a,b,z: a^2=b^2=z^4=1,b^{-1}ab=az^2,az=za,bz=zb\rangle$. Then, we solve several relative class number one problems. (1) We solve the relative class number one problem for the dihedral CM-fields of 2-power degrees. First, we remind the reader of the characterization of the dihedral CM-fields of 2-power degrees with odd relative class numbers. Second, we give lower bounds on relative class numbers of dihedral CM-fields of 2-power degrees with odd relative class numbers. We thus obtain an upper bound on the discriminants of the dihedral CM-fields of 2-power degrees with relative class number equal to 1. Third, we compute the relative class numbers of all the dihedral CM-fields of 2-power degrees with odd relative class numbers and discriminants less than or equal to this latter bound. We end up with a list of twenty-four dihedral CM-fields of 2-power degrees with relativeclass numbers equal to 1, and show that exactly twenty-one of them have class number 1. (2) We determine all the non-abelian normal CM-fields of degree 16 with Galois group $G_9$ which have relative class number 1 (there is only one such number field), and then those which have class number 1 (there is only one such number field). (3) We determine some of the non-abelian normal CM-fields $N =N_1N_2$ of degree 16 which are composita of two normal octic CM-fields $N_1$ and $N_2$ with the same maximal real subfield which have relative class number 1, and then those which have class number 1. Indeed, we focus on the case where one of the $N_i$ is a quaternion octic CM-field and prove that there is only one such compositum with relative class number 1 and that this compositum has class number 1.

1991 Mathematics Subject Classification: primary 11R29; secondary 11R21, 11R42, 11M20, 11Y40.

In this paper, we prove that the modular curve $X(11)$ over a field of characteristic 3 admits the Mathieu group $M_{11}$ as an automorphism group. We also examine some aspects of the geometry of the curve $X(11)$ in characteristic 3. In particular, we show that every point of the curve is a point of inflection, the curve has 110 hyperflexes and there are no inflectional triangles and 11232 inflectional pentagons, of which 144 are self-conjugate. The hyperflexes correspond to the supersingular elliptic curves. We comment on the relationship of Ward's quadrilinear invariant for $M_{12}$ to our work and announce for the first time the equations for Klein's A-curve of level 11. We also comment on the relation of our work to some unpublished work of Bott and Tate.

1991 Mathematics Subject Classification: 11F32, 11G20, 14G10, 14H10, 14N10, 20B25, 20C34.