We show that the conjecture of [27] for the special value at $s=1$ of the zeta function of an arithmetic surface is equivalent to the Birch–Swinnerton–Dyer conjecture for the Jacobian of the generic fibre.

We show that the ergodic integrals for the horocycle flow on the two-torus associated by Giulietti and Liverani with an Anosov diffeomorphism either grow linearly or are bounded; in other words, there are no deviations. For this, we use the topological invariance of the Artin–Mazur zeta function to exclude resonances outside the open unit disc. Transfer operators acting on suitable spaces of anisotropic distributions and their Ruelle determinants are the key tools used in the proof. As a bonus, we show that for any $C^\infty $ Anosov diffeomorphism F on the two-torus, the correlations for the measure of maximal entropy and $C^\infty $ observables decay with a rate strictly smaller than $e^{-h_{\mathrm {top}}(F)}$ . We compare our results with very recent related work of Forni.

The motivic Hilbert zeta function of a variety $X$ is the generating function for classes in the Grothendieck ring of varieties of Hilbert schemes of points on $X$. In this paper, the motivic Hilbert zeta function of a reduced curve is shown to be rational.

The $p$-cohomology of an algebraic variety in characteristic $p$ lies naturally in the category $D_{c}^{b}(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl, Illusie, Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc., and the special values of their zeta functions. These results provide compelling evidence that $D_{c}^{b}(R)$ is the correct target for $p$-cohomology in characteristic $p$.

We construct adelic objects for rank two integral structures on arithmetic surfaces and develop measure and integration theory, as well as elements of harmonic analysis. Using the topological Milnor K2-delic and K1×K1-delic objects associated to an arithmetic surface, an adelic zeta integral is defined. Its unramified version is closely related to the square of the zeta function of the surface. For a proper regular model of an elliptic curve over a global field, a two-dimensional version of the theory of Tate and Iwasawa is derived. Using adelic analytic duality and a two-dimensional theta formula, the study of the zeta integral is reduced to the study of a boundary integral term. The work includes first applications to three fundamental properties of the zeta function: its meromorphic continuation and functional equation and a hypothesis on its mean periodicity; the location of its poles and a hypothesis on the permanence of the sign of the fourth logarithmic derivative of a boundary function; and its pole at the central point where the boundary integral explicitly relates the analytic and arithmetic ranks.

We define a topological space over the p-adic numbers, in which Euler products and Dirichlet series converge. We then show how the classical Riemann zeta function has a (p-adic) Euler product structure at the negative integers. Finally, as a corollary of these results, we derive a new formula for the non-Archimedean Euler–Mascheroni constant.

We show that the zeta function of a regular graph admits a representation as a quotient of a determinant over a L2-determinant of the combinatorial Laplacian.

This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as a consequence, precise asymptotics (previously conjectured via random matrix theory) for moments of zeta functions and quadratic L-series. As an application of the theory, in a third section, we obtain the current best known error term for mean values of cubes of cent ral values of Dirichlet L-series. The methods utilized to derive this result are the convexity principle for functions of several complex-variables combined with a knowledge of groups of functional equations for certain multiple Dirichlet series.