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The volume of the unit sphere in every dimension is given an interpretation as a product of special values of the zeta function of ℤ, akin to volume formulas of Minkowski and Siegel in the theory of arithmetic groups. A product formula is found for this zeta function that specializes to Catalan numbers. Moreover, certain closed-form expressions for various other zeta values are deduced, in particular leading to an alternative perspective on Euler’s values for the Riemann zeta function.

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 introduce the Hadamard topology on the Witt ring of rational functions, giving a simultaneous refinement of the weight and point-counting topologies. Zeta functions of algebraic varieties over finite fields are elements of the rational Witt ring, and the Hadamard topology allows for a conjectural unification of results in arithmetic and motivic statistics: The completion of the Witt ring for the Hadamard topology can be identified with a space of meromorphic functions which we call Hadamard functions, and we make the meta-conjecture that any “natural” sequence of zeta functions which converges to a Hadamard function in both the weight and point-counting topologies converges also in the Hadamard topology. For statistics arising from Bertini problems, zero-cycles or the Batyrev-Manin conjecture, this yields an explicit conjectural unification of existing results in motivic and arithmetic statistics that were previously connected only by analogy. As evidence for our conjectures, we show that Hadamard convergence holds for many natural statistics arising from zero-cycles, as well as for the motivic height zeta function associated to the motivic Batyrev-Manin problem for split toric varieties.

An important problem in mathematics is understanding the behavior of functions. If we can show that complicated functions can be simplified, computation becomes more feasible. In this thesis, we describe two major projects: counting arcs in projective space and counting subrings in $\Z^n$. In the case of arcs in $\mb{P}^{k-1}(\Fq)$, we seek to understand whether the counting function can be expressed by a simple formula like a polynomial. In the case of subrings of $\Z^n$, we are interested in the asymptotic growth of the function that counts subrings of index at most $X$. An $n$-arc in $(k-1)$-dimensional projective space is a set of $n$ points so that no $k$ lie on a hyperplane. Let $C_{n,k}(q)$ be the number of $n$-arcs in $\mb{P}^{k-1}(\Fq)$. We discuss new results for $n$-arcs when $k = 3$ and 4. Building off work of Kaplan, Kimport, Lawrence, Peilen, and Weinreich, we show that $C_{10,3}(q)$ is not quasipolynomial in $q$. For almost all $n$, no formulas for $C_{n,k}(q)$ were known when $k \ge 4$. We introduce a new algorithm to count $n$-arcs in $\mb{P}^3(\Fq)$ in terms of a small number of special combinatorial objects and use it to compute $C_{n,4}(q)$ for $n \le 7$. Finally, we discuss generalizations of this algorithm to higher-dimensional projective space.Let $f_n(p^e)$ count the number of subrings of index $p^e$ in $\Z^n$. We study lower bounds for $f_n(p^e)$ using techniques from combinatorics and arithmetic geometry. Using these lower bounds, we give the best known result about the asymptotic growth of subrings in $\Z^n$ when $n \ge 8$ by studying the analytic properties of the subring zeta function for $\Z^n$. These results immediately give new information about the asymptotic growth of orders in a fixed degree $n$ number field. Finally, we consider the behavior of $f_n(p^e)$ as a function of $p$. We give evidence that this function may always be polynomial in $p$.

In this paper we develop new identities in the spirit of Euler. We shall investigate and report on new Euler identities of weight p+ 2, for p an odd integer, but with a non unitary argument of the harmonic numbers. Some examples of these Euler identities will be given in terms of Riemann zeta values, Dirichlet values and other special functions

Producción Científica, The purpose of this note is to give a brief overview on zeta functions of curve singularities and to provide some evidences on how these and global zeta functions associated to singular algebraic curves over perfect fields relate to each other, Ministerio de Economía, Industria y Competitividad ( grants MTM2012-36917-C03-03 / MTM2015-65764- C3-2-P)

We study stacks of truncated Barsotti-Tate groups and the G-zips defined by Pink, Wedhorn & Ziegler. The latter occur naturally when studying truncated Barsotti-Tate groups of height 1 with additional structure. By studying objects over finite fields and their automorphisms we determine the zeta functions of these stacks. These zeta functions can be expressed in terms of the Weyl group of the reductive group G and its action on the root system. The main ingredients are the classification of G-zips over algebraically closed fields and their automorphism groups by Pink, Wedhorn & Ziegler, and the study of truncated Barsotti-Tate groups and their automorphism groups by Gabber & Vasiu.

This paper solves the general Erdös-Szemeredi conjecture for some classes of increasing families of finite subsets of self-similar subsets of the integers. It does so by applying zeta function methods for discrete self similar sets.

We initiate the study of spectral zeta functions zeta(X) for finite and infinite graphs X, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function zeta(s) is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of the logarithmic derivative of zeta(s). We relate zeta(Z) to Euler's beta integral and show how to complete it giving the functional equation xi(Z)(1 - s) = xi(Z)(s). This function appears in the theory of Eisenstein series although presumably with this spectral intepretation unrecognized. In higher dimensions d we provide a meromorphic continuation of zeta(Zd) (s) to the whole plane and identify the poles. From our aymptotics several known special values of zeta(s) are derived as well as its non-vanishing on the line Re(s) = 1. We determine the spectral zeta functions of regular trees and show it to be equal to a specialization of Appell's hypergeometric function F-1 via an Euler-type integral formula due to Picard.

Eine wichtige Invariante von Modulkurven ist die arithmetische Selbstschnittzahl der relativ dualisierenden Garbe. Auf dem minimalen regulären Modell von X(N) ist diese Selbstschnittzahl durch den gewöhnlichen Schnitt einiger ausgezeichneter vertikaler Divisoren (dem geometrischen Beitrag) und durch die Auswertung der kanonischen Greenschen Funktion an einigen Spitzen (dem analytischen Beitrag) vollständig festgelegt. Das Ziel dieser Arbeit ist es, jeden dieser Beiträge in Abhängigkeit von der Stufe N zu bestimmen und das asymptotische Verhalten der Selbstschnittzahl zu studieren, wenn die Stufe N gegen unendlich geht., An important invariant of modular curves is the arithmetic self-intersection of the relative dualizing sheaf. On the minimal regular model of X(N) this self-intersection is completely described by the usual intersection of some distinguished vertical divisors (geometric contribution) and the evaluation of the canonical Green’s function at certain cusps (analytic contribution). The aim of this thesis is to determine each of these contributions in terms of the level N and study the asymptotic behaviour of the self-intersection as N tends to infinity.

This is the first semester of a one year graduate course in number theory covering standard topics in algebraic and analytic number theory. At various points in the course, we will make reference to material from other branches of mathematics, including topology, complex analysis, representation theory, and algebraic geometry.

We initiate the study of spectral zeta functions ζX for finite and infinite graphs X, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function ζ(s) is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of the logarithmic derivative of ζ(s). We relate ζZ to Euler's beta integral and show how to complete it giving the functional equation ξZ(1−s)=ξZ(s). This function appears in the theory of Eisenstein series although presumably with this spectral intepretation unrecognized. In higher dimensions d we provide a meromorphic continuation of ζZd(s) to the whole plane and identify the poles. From our aymptotics several known special values of ζ(s) are derived as well as its non-vanishing on the line Re(s)=1. We determine the spectral zeta functions of regular trees and show it to be equal to a specialization of Appell's hypergeometric function F1 via an Euler-type integral formula due to Picard.

In order to shed light on Orlov’s conjecture that derived equivalent smooth, projective varieties have isomorphic Chow motives, we examine the zeta functions of derived equivalent varieties over finite fields; in this setting Orlov’s conjecture predicts equality of zeta functions. It is demonstrated that derived equivalent smooth, projective varieties over finite fields that are abelian or satisfy a certain condition on their cohomology. This condition is satisfied, for example, by a surface or Calabi–Yau 3–fold.One of our approaches to comparing the zeta functions of derived equivalent varieties over finite fields comes from using the Lefschetz Fixed Point Theorem to turn the question into one of comparing the -adic ́etale cohomology of varieties. Cohomology groups are not in general preserved under the action of Fourier–Mukai equivalences on cohomology, but cohomological structures we call even and odd Mukai–Hodge structures, which are a realization of the Mukai motive, are preserved. Investigation into when isomorphism of these cohomological structures implies equality of zeta functions gives us our cohomological condition for equality of zeta functions.We also develop a relative version of the map Fourier–Mukai transforms induce on cohomology and define a relative notion of even and odd Mukai–Hodge structures, and show these structures are preserved in a situation arising from the derived equivalence of smooth, projective varieties with semiample (anti-)canonical bundles. Using this result, it is demonstrated that when derived equivalent smooth, projective varieties have semiample (anti-)canonical bundles, the fibers over any fixed geometric point in their shared (anti-)canonical variety must also have isomorphic even and odd Mukai–Hodge structures. Hence, for any such varieties over finite fields, if their geometric fibers satisfy any of the conditions identified for isomorphism of Mukai–Hodge structures to imply equality of zeta functions, then the varieties t

Text. In this paper, we prove a generalization of Mertens' theorem to Beurling primes, namely that lim(x ->infinity) 1/Inx Pi(p <= x)(1 - p(-1))(-1) = Ae(gamma), where gamma is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit M = lim(x ->infinity) (Sigma(p <= x) p(-1) - In(Inx)) exists. We also show that this limit coincides with lim(alpha -> 0+)(Sigma(p) p(-1)(In p)(-alpha) - 1/alpha); for ordinary primes this claim is called Meissel's theorem. Finally, we will discuss a problem posed by Beurling, namely how small vertical bar N(x)-[4]vertical bar can be made for a Beurling prime number system Q not equal P. where P is the rational primes. We prove that for each c > 0 there exists a Q such that vertical bar N(x) - [x]vertical bar < cInx and conjecture that this is the best possible bound. Video. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=Kw3iNo3fAbk/. (C) 2010 Elsevier Inc. All rights reserved.

We introduce orbital functionals $\int \boldsymbol{\beta} $ simultaneously for each commensurability class of orbital surfaces. They are realized on infinitely dimensional \emph{orbital} divisor spaces spanned by (arithmetic-geodesic real $2$-dimensional) orbital curves on any orbital surface. We discover infinitely many of them on each commensurability class of orbital Picard surfaces, which are real $4$-spaces with cusps and negative constant Kähler-Einstein metric degenerated along an orbital cycle. For a suitable (Heegner) sequence $\int \mathbf{h}_N$, $N \in \mathbb{N}$, of them we investigate the corresponding formal orbital $q$-series $\mathop{\sum}\limits_{N=0}^\infty (\int \mathbf{\mathbf{h}}_N)q^N$. We show that after substitution $q = \re^{2\pi\ri\tau}$ and application to arithmetic orbital curves $\mathbf{\hat{C}}$ on a fixed Picard surface class the series $\mathop{\sum}\limits_{N=0}^{\infty} (\int_{\mathbf{\hat{C}}} \mathbf{\mathbf{h}}_N)\re^{2\pi\ri\tau}$ define modular forms of well-determined fixed weight, level and Nebentypus. The proof needs a new orbital understan-ding of orbital hights introduced in \cite{Ho1} and Mumford-Fulton's rational intersection theory on singular surfaces in Riemann-Roch-Hirzebruch style. It has to be connected with Zeta and Theta functions of hermitian lines, indefinit quaternionic fields and of a matrix algebra along a research marathon over $75$ years represented by Cogdell, Kudla, Hirzebruch, Zagier, Shimura, Schoeneberg and Hecke. Our aim is to open a door to an effective enumerative geometry for complex geodesics on orbital varieties with nice metrics.

A Poisson structure (anti-symmetric bilinear local operator on functionals, obeying the Jacobi identify) is established on certain function spaces (spaces of symbols of pseudodifferential operators on R to the nth power). The spaces of functionals thus become (infinite-dimensional) Lie Algebras. This type of Lie algebra structure has been established previously for functionals of functions of a single variable (n = 1) only. For n = 1, the theorem of Gardner, as generalized by Gel'fand, Dikii, and others, is proved: that is, the residues of the zeta-function of the elliptic symbol are in involution with respect to the appropriate Poisson bracket. In contrast, it is shown by explicit example that the residues of the zeta functions of higher-dimensional elliptic symbols are generally not in involution.

The novelty of the Jean Pierre Badiali last scientific works stems to a quantum approach based on both (i) a return to the notion of trajectories (Feynman paths) and (ii) an irreversibility of the quantum transitions. These iconoclastic choices find again the Hilbertian and the von Neumann algebraic point of view by dealing statistics over loops. This approach confers an external thermodynamic origin to the notion of a quantum unit of time (Rovelli Connes' thermal time). This notion, basis for quantization, appears herein as a mere criterion of parting between the quantum regime and the thermodynamic regime. The purpose of this note is to unfold the content of the last five years of scientific exchanges aiming to link in a coherent scheme the Jean Pierre's choices and works, and the works of the authors of this note based on hyperbolic geodesics and the associated role of Riemann zeta functions. While these options do not unveil any contradictions, nevertheless they give birth to an intrinsic arrow of time different from the thermal time. The question of the physical meaning of Riemann hypothesis as the basis of quantum mechanics, which was at the heart of our last exchanges, is the backbone of this note.

This thesis is concerned with the analytic properties of arithmetic zeta functions, which remain largely conjectural at the time of writing. We will focus primarily on the most basic amongst them - meromorphic continuation and functional equation. Our weapon of choice is the so-called “mean-periodicity correspondence”, which provides a passage between nicely behaved arithmetic schemes and mean-periodic functions in certain functional spaces. In what follows, there are two major themes. 1. The comparison of the mean-periodicity properties of zeta functions with the much better known, but nonetheless conjectural, automorphicity properties of Hasse–Weil L functions. The latter of the two is a widely believed aspect of the Langlands program. In somewhat vague language, the two notions are dual to each other. One route to this result is broadly comparable to the Rankin-Selberg method, in which Fesenko’s “boundary function” plays the role of an Eisenstein series. 2. The use of a form of “lifted” harmonic analysis on the non-locally compact adele groups of arithmetic surfaces to develop integral representations of zeta functions. We also provide a more general discussion of a prospective theory of GL1(A(S)) zeta-integrals, where S is an arithmetic surface. When combined with adelic duality, we see that mean-periodicity may be accessible through further developments in higher dimensional adelic analysis. The results of the first flavour have some bearing on questions asked first by Langlands, and those of the second kind are an extension of the ideas of Tate for Hecke L-functions. The theorems proved here directly extend those of Fesenko and Suzuki on two-dimensional adelic analysis and the interplay between mean-periodicity and automorphicity.

This thesis is concerned with the analytic properties of arithmetic zeta functions, which remain largely conjectural at the time of writing. We will focus primarily on the most basic amongst them - meromorphic continuation and functional equation. Our weapon of choice is the so-called “mean-periodicity correspondence”, which provides a passage between nicely behaved arithmetic schemes and mean-periodic functions in certain functional spaces. In what follows, there are two major themes. 1. The comparison of the mean-periodicity properties of zeta functions with the much better known, but nonetheless conjectural, automorphicity properties of Hasse–Weil L functions. The latter of the two is a widely believed aspect of the Langlands program. In somewhat vague language, the two notions are dual to each other. One route to this result is broadly comparable to the Rankin-Selberg method, in which Fesenko’s “boundary function” plays the role of an Eisenstein series. 2. The use of a form of “lifted” harmonic analysis on the non-locally compact adele groups of arithmetic surfaces to develop integral representations of zeta functions. We also provide a more general discussion of a prospective theory of GL1(A(S)) zeta-integrals, where S is an arithmetic surface. When combined with adelic duality, we see that mean-periodicity may be accessible through further developments in higher dimensional adelic analysis. The results of the first flavour have some bearing on questions asked first by Langlands, and those of the second kind are an extension of the ideas of Tate for Hecke L-functions. The theorems proved here directly extend those of Fesenko and Suzuki on two-dimensional adelic analysis and the interplay between mean-periodicity and automorphicity.