It took decades for mathematicians to verify his results, and to. The conditions are given in terms of the lefschetz numbers of the iterates of the map andor its lefschetz zeta function. Pdf on the lefschetz zeta function for quasiunipotent maps. Riemanns zeta function if s 1 is a real number, then the series s x n 1 1 ns converges. It is an algebraic version of the artinmazur 1 zeta function which in dynamical systems theory is studied frequently 27. Values at infinities, specific values 60 formulas zeta. It is an algebraic version of the artinmazur zeta function, which in dynamical systems theory is studied frequently.
This zeta function is an in nite product over prime ideals pin o k, the ring of algebraic integers of k. Topological entropy of continuous selfmaps on a graph. It is implemented in terms of path integrals with the statistics physics interpretation in mind. Despite not resembling a holomorphic function, the special case for the poset of integer divisibility is related as a formal dirichlet series to the riemann zeta function. Artinmazur zeta function, lefschetz zeta function, ruelle zeta function for discrete dynamical systems, ruelle zeta function for. Having a lefschetz number, it is custom to define a lefschetz zeta function which encodes the lefschetz numbers of iterates t n of the graph automorphism t. Hyperbolic geometry and closed geodesics perhaps one of the easier routes into understanding dynamical zeta functions for ows is via. A brouwer fixedpoint theorem for graph endomorphisms. Multiple dirichlet series, automorphic forms, and analytic number theory.
The zeta function and its relation to the prime number theorem ben rifferreinert abstract. By a global zeta function we mean an object that takes into account all primes. Why is the zeta function of a variety over a finite field not a polynomial. This thesis is an exposition of the riemann zeta function. Pdf on jun 20, 1992, ross geoghegan and others published lefschetz trace formulae, zeta functions and torsion in dynamics find, read and cite all the research you need on researchgate. Related content quiver theories from d6branes via mirror symmetry amihay hanany and amer iqbaltoric duality as seiberg duality and brane diamonds bo feng, amihay hanany, yanghui he et al. There are various methods to derive this functional equation, see e. One of the main tools in delignes proof of the riemann hypothesis is the generalized lefschetz trace formula for adic sheaves on varieties over nite elds. A particularly important zeta function in both analysis and geometry is the selberg zeta function. Indeed, it is shown in 7, 8 that, for a quasiunipotent self map f. The present book is concerned with these zeta functions from a cohomological point of view. In this paper, i will demonstrate an important fact about the zeros of the zeta function, and how it relates to the prime number theorem. Zeta functions in algebraic geometry mircea mustat. The riemann zeta function david jekel june 6, 20 in 1859, bernhard riemann published an eightpage paper, in which he estimated \the number of prime numbers less than a given magnitude using a certain meromorphic function on c.
Lefschetz zeta function lerch zeta function local zeta. Bo feng et al jhep022003056 view the article online for updates and enhancements. The fact that homeomorphisms with nonzero lefschetz numbers have fixed open invariant sets, can be seen as a kakutani fixed point theorem for finite simple graphs. As a particular example the theory of the generalized selberg zeta function is. Pdf lefschetz trace formulae, zeta functions and torsion. Dynamical zeta functions, nielsen theory alexander felshtyn. We consider the cases where the manifold is a compact orientable and nonorientable surface, the ndimensional torus, the product of n spheres of dimension. The prime number theorem, hardys theorem on the zeros of. T exp x1 m1 n m m tm the n mturn out to be exactly the number of xed points of the qmfrobenius morphism acting on x x spec f q spec f q. Generalised selberg zeta functions and a conjectural lefschetz formula anton deitmar s. In the case of quasiunipotent maps of tori the lefschetz zeta function can be computed completely explicitly. In this paper we will discuss characterizations and properties of each, and how the two are connected.
The next two sections develop the mathematical theory of and its zeros starting with basic results such as the intermediate value theorem from real analysis and the argument princi. This means that the artinmazur zeta function is a formal function, but the lefschetz zeta function is always a function. Pdf leafwise cohomological expression of dynamical zeta. Selberg introduced, together with the trace formula, the selberg zeta function for compact quotients of the hyperbolic plane. For these gradients the novikov complex is defined over the ring of rational functions and the lefschetz zeta function is also rational. By the lefschetz xed point theorem we can thus express them as an alternating sum of traces.
Beta pdf cauchy pdf chisquared pdf exponential pdf f pdf gamma pdf laplace pdf lognormal pdf normal pdf. Zeta functions have generally been associated with. Algebraic properties of the lefschetz zeta function, periodic. The gamma function and the riemann zeta function are two special functions that are critical to the study of many di erent elds of mathematics. Basic zeta functions and some applications in physics. The selberg zeta function is an example of a zeta function which can be interpreted in this way. Quiver theories, soliton spectra and picard lefschetz. On topological entropy, lefschetz numbers and lefschetz. A functional equation for the lefschetz zeta arxiv. The connection between lefschetz formulae and zeta function is explained. Pdf cohomological theory of dynamical zeta functions. Chapter 9 the functional equation for the riemann zeta function we will eventually deduce a functional equation, relating s to 1 s. Weil on the zeta functions of curves and abelian varieties, weil in 1949 made.
In the present paper we prove that for every c 0generic fgradient there is a homotopy equivalence between these two chain complexes, such that its torsion equals the lefschetz zeta function of the gradient flow. Some generalizations of the riemann zeta function are outlined, as well as the calculation of zeta constants and the development of some identities. As a particular example the theory of the generalized selberg zeta. Using the lefschetz zeta function of f, we provide a sufficient condition in order that f has positive topological entropy. A brouwer fixedpoint theorem for graph endomorphisms topic. News about this project harvard department of mathematics.
Notes on the riemann zeta function january 25, 2007 1 the zeta function 1. The riemann zeta function is the dirac zeta function of the circle. As a particular example the theory of the generalized selberg zeta function is presented. It is an algebraic version of the artinmazur 1 zeta function which in dynamical systems theory is studied frequently 24. The corresponding lefschetz formula can be generalized to a multidimensional lefschetz formula for higher rank spaces. In mathematics, the lefschetz fixedpoint theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space to itself by means of traces of the induced mappings on the homology groups of. The latter can be applied to give meromorphic extension of generalized selberg zeta functions or to prove higher rank prime. Arithmetic zeta function artinmazur zeta function barnes zeta function bessel zeta function beurling zeta function cotangent zeta function dedekind zeta function epstein zeta function fibonacci zeta function goss zeta function hasseweil zeta function height zeta function igusa zeta function ihara zeta function lefschetz zeta function lerch. These are analytic objects complex functions that are intimately related to the global elds. On topological entropy, lefschetz numbers and lefschetz zeta. Chapter 9 the functional equation for the riemann zeta. David roe harvard university university of calgary zeta functions with padic cohomology 9 28. A generalisation of the selberg zeta function, or rather its log.
This talk aims at explaining the statement and proof of this generalized lefschetz trace formula. Included are techniques of analytic continuation and relationships to special functions. Actually, as we will see in this article, the lefschetz formula implies that the lefschetz zeta function of a graph automorphism is the product of two zeta functions defined in dynamical systems theory. Recover the zeta function or the cardinality of xfq from the characteristic polynomial of m0and the weil conjectures. Titchmarsh, the theory of the riemann zeta function. Algebraic properties of the lefschetz zeta function. In this case one expects to have a product formula in terms of local factors. The first term, i presented several classical results on zeta functions in characteristic p weils calculation of the zeta. In particular we compute the lefschetz zeta function for quasiunipotent maps, such that the characteristic polynomial of the induced map on the first homology group is the inline formula th. The riemann zeta function is defined as the analytic continuation of the function defined for. It is named after solomon lefschetz, who first stated it in 1926. Anosov flows and dynamical zeta functions annals of mathematics.
Moreover, for some classes of graphs we improve this condition making it easier to check. The lefschetz zeta function associated to a continuous selfmap f of a compact manifold is a rational function pq. Not all artinmazur zeta functions have a strictly positive radius of convergence, but lefschetz zeta functions always have. Generalised selberg zeta functions and a conjectural. Zeta functions are one of those objects which are interesting in any mathematical field, whether it is number theory, complex analysis, topology, dynamical systems or algebraic geometry. They are useful in studying the statistical properties of the objects in question. The zeta function is an important function in mathematics. The zeta function of circular graphs arxiv local pdf. A brouwer fixedpoint theorem for graph endomorphisms fixed. However, the formula 2 cannot be applied anymore if the real part. In mathematics, the lefschetz zetafunction is a tool used in topological periodic and fixed point theory, and dynamical systems. These notes are essentially the lecture notes for that course. The term applies to such a wide range of functions that no single source contains all of them. Q 1 is a real number, then the series s x n 1 1 ns converges.
Riemann and his zeta function 5 the second form for this integral is called the mellin transform 70 of 1 ex. The exposition is selfcontained, and required a preliminary knowledge of only the ele. The basic example is the wellknown factorization of the riemann zeta function. Having a lefschetz number, it is custom to define a lefschetz zeta function which encodes the lefschetz numbers of iterates tn of the graph automorphism t. Let g be a graph and f be a continuous selfmap on g.
Dynamical zeta functions mark pollicott contents 1 introduction 2. Introduction it is the aim of these lectures to introduce some basic zeta functions and their uses in the areas of. The gamma function and the zeta function paul duncan abstract. Applications are given to the theory of anosov flows and prime geodesic theorems. Special functions are ubiquitous in mathematics and applications of mathematics. A window into zeta and modular physics msri publications volume 57, 2010 basic zeta functions and some applications in physics klaus kirsten 1. We give a proof based on a functional equation for the jacobi theta function z p 1. Dynamical zeta functions and transfer operators david ruelle c ertain generating functionsencodingproperties of objects like prime numbers, periodic orbits, have received the name of zeta functions. According to the parity of the degrees of the polinomials p and q, we analize when the set of periodic points of f is infinite and when the topological entropy is positive. An analogue of the lefschetz formula in abstract algebraic geometry has been established for adic.
Pages 687773 from volume 178 20, issue 2 by paolo giulietti, carlangelo liverani, mark pollicott. While the artinmazur zeta function counts the periodic points geometrically, the lefschetz zeta function counts them homologically. Deninger has suggested that to the riemann zeta function resp. The fact that this function is analytic in this region of the complex plane is a consequence of the following basic fact. The integral 1 can also be viewed as a laplace transform. Nt 19 apr 2007 lefschetz formulae and zeta functions anton deitmar abstract. Moreover, in 1859 riemann gave a formula for a unique the socalled holomorphic extension of the function onto the entire complex plane c except s 1.
As we remarked before the lefschetz zeta function, of a continuous selfmap on t n, is equal 1 if and only if 1 is an eigenvalue of f. The desire to prove them motivated the development of etale cohomology, a process begun by grothendieck and nished by deligne. This is a rational function of a single variable t which depends only on the homotopy class of and can be expressed either in terms of homological invariants of or in terms of the sum of local indices of fixed points of for all n 0. The relation with riemann zeta function is explained, shedding. The weil conjectures are a statement about the zeta function of varieties over nite elds. The hardest part is an analogue of the riemann hypothesis. Zeta functions are one of those objects which are interesting in any mathematical field, whether it is number theory, complex analysis, topology, dynamical systems. Please let me know of any typos, mistakes, incomplete arguments, etc. Pdf on the lefschetz zeta function for quasiunipotent. Cameron franc special values of riemanns zeta function. Originally, the selberg zeta function appeared in the spectral theory of automorphic forms and were suggested by an analogy between weils explicit formula for the riemann zeta function and selbergs trace formula 261. Proceedings of symposia in pure mathematics, volume. The lefschetz zeta function has some better properties than the artinmazur zeta function. Zeta function of an incidence algebra, a function that maps every interval of a poset to the constant value 1.
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