Showing papers on "Riemann zeta function published in 2010"

Zeta Functions of Graphs: A Stroll through the Garden

Abstract: List of illustrations Preface Part I. A Quick Look at Various Zeta Functions: 1. Riemann's zeta function and other zetas from number theory 2. Ihara's zeta function 3. Selberg's zeta function 4. Ruelle's zeta function 5. Chaos Part II. Ihara's Zeta Function and the Graph Theory Prime Number Theorem: 6. Ihara zeta function of a weighted graph 7. Regular graphs, location of poles of zeta, functional equations 8. Irregular graphs: what is the RH? 9. Discussion of regular Ramanujan graphs 10. The graph theory prime number theorem Part III. Edge and Path Zeta Functions: 11. The edge zeta function 12. Path zeta functions Part IV. Finite Unramified Galois Coverings of Connected Graphs: 13. Finite unramified coverings and Galois groups 14. Fundamental theorem of Galois theory 15. Behavior of primes in coverings 16. Frobenius automorphisms 17. How to construct intermediate coverings using the Frobenius automorphism 18. Artin L-functions 19. Edge Artin L-functions 20. Path Artin L-functions 21. Non-isomorphic regular graphs without loops or multiedges having the same Ihara zeta function 22. The Chebotarev Density Theorem 23. Siegel poles Part V. Last Look at the Garden: 24. An application to error-correcting codes 25. Explicit formulas 26. Again chaos 27. Final research problems References Index.

The Arakawa–Kaneko zeta function

Abstract: We present a very natural generalization of the Arakawa–Kaneko zeta function introduced ten years ago by T. Arakawa and M. Kaneko. We give in particular a new expression of the special values of this function at integral points in terms of modified Bell polynomials. By rewriting Ohno’s sum formula, we are able to deduce a new class of relations between Euler sums and the values of zeta.

Barnes-type multiple q-zeta functions and q-Euler polynomials

Abstract: The purpose of this paper is to present a systemic study of some families of multiple q-Euler numbers and polynomials and we construct multiple q-zeta functions which interpolate multiple q-Euler numbers at a negative integer. This is a partial answer to the open question in a previous publication (see Kim et al 2001 J. Phys. A: Math. Gen. 34 7633–8).

Functional equations for zeta functions of groups and rings

Abstract: We introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or T -)groups, and the normal zeta functions of T -groups of class 2. In particular we solve the two problems posed in [9, Section 5]. We deduce our theorems from a ‘blueprint result’ on certain p-adic integrals which generalises work of Denef and others on Igusa’s local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to ‘linearise’ the problems of counting subgroups and representations in T -groups, respectively.

Traces of high powers of the Frobenius class in the hyperelliptic ensemble

Abstract: The zeta function of a curve over a finite field may be expressed in terms of the characteristic polynomial of a unitary symplectic matrix, called the Frobenius class of the curve. We compute the expected value of the trace of the n-th power of the Frobenius class for an ensemble of hyperelliptic curves of genus g over a fixed finite field in the limit of large genus, and compare the results to the corresponding averages over the unitary symplectic group USp(2g). We are able to compute the averages for powers n almost up to 4g, finding agreement with the Random Matrix results except for small n and for n=2g. As an application we compute the one-level density of zeros of the zeta function of the curves, including lower-order terms, for test functions whose Fourier transform is supported in (-2,2). The results confirm in part a conjecture of Katz and Sarnak, that to leading order the low-lying zeros for this ensemble have symplectic statistics.

On the zeros of the Riemann Zeta function

Abstract: This paper is divided into two independent parts. The first part presents new integral and series representations of the Riemaan zeta function. An equivalent formulation of the Riemann hypothesis is given and few results on this formulation are briefly outlined. The second part exposes a totally different approach. Using the new series representation of the zeta function of the first part, exact information on its zeros is provided.

More than 41% of the zeros of the zeta function are on the critical line

Abstract: We prove that at least 41.05% of the zeros of the Riemann zeta function are on the critical line.

Mesoscopic fluctuations of the zeta zeros

Abstract: We prove a multidimensional extension of Selberg’s central limit theorem for log ζ, in which non-trivial correlations appear. In particular, this answers a question by Coram and Diaconis about the mesoscopic fluctuations of the zeros of the Riemann zeta function. Similar results are given in the context of random matrices from the unitary group. This shows the correspondence n ↔ log t not only between the dimension of the matrix and the height on the critical line, but also, in a local scale, for small deviations from the critical axis or the unit circle.

From monoids to hyperstructures: in search of an absolute arithmetic

Abstract: We show that the trace formula interpretation of the explicit formulas expresses the counting functionN.q/ of the hypothetical curveC associated to the Riemann zeta function, as an intersection number involving the scaling action on the adele class space. Then, we discuss the algebraic structure of the adele class space both as a monoid and as a hyperring. We construct an extension R convex of the hyperfield S of signs, which is the hyperfield analogue of the semifield R max of tropical geometry, admitting a one parameter group of automorphisms fixing S. Finally, we develop function theory over Spec K and we show how to recover the field of real numbers from a purely algebraic construction, as the function theory over Spec S.

Graph Zeta Function in the Bethe Free Energy and Loopy Belief Propagation

Abstract: We propose a new approach to the analysis of Loopy Belief Propagation (LBP) by establishing a formula that connects the Hessian of the Bethe free energy with the edge zeta function. The formula has a number of theoretical implications on LBP. It is applied to give a sufficient condition that the Hessian of the Bethe free energy is positive definite, which shows non-convexity for graphs with multiple cycles. The formula clarifies the relation between the local stability of a fixed point of LBP and local minima of the Bethe free energy. We also propose a new approach to the uniqueness of LBP fixed point, and show various conditions of uniqueness.

Computing zeta functions of nondegenerate curves

Abstract: In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since in practice all known cases, e.g. hyperelliptic, superelliptic and Cab curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope. For a genus g curve over Fpn , the expected running time is e O(ng+ng), whereas the space complexity amounts to e O(ng), assuming p is fixed.

Schemes over 1 and zeta functions

Abstract: We determine the real counting function N(q) (q∈[1,∞)) for the hypothetical ‘curve’ over 𝔽1, whose corresponding zeta function is the complete Riemann zeta function. We show that such a counting function exists as a distribution, is positive on (1,∞) and takes the value −∞ at q=1 as expected from the infinite genus of C. Then, we develop a theory of functorial 𝔽1-schemes which reconciles the previous attempts by Soule and Deitmar. Our construction fits with the geometry of monoids of Kato, is no longer limited to toric varieties and it covers the case of schemes associated with Chevalley groups. Finally we show, using the monoid of adele classes over an arbitrary global field, how to apply our functorial theory of -schemes to interpret conceptually the spectral realization of zeros of L-functions.

Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction

Abstract: The Harmonic Oscillator.- The Weyl-Hoermander Calculus.- The Spectral Counting Function N(?) and the Behavior of the Eigenvalues: Part 1.- The Heat-Semigroup, Functional Calculus and Kernels.- The Spectral Counting Function N(?) and the Behavior of the Eigenvalues: Part 2.- The Spectral Zeta Function.- Some Properties of the Eigenvalues of .- Some Tools from the Semiclassical Calculus.- On Operators Induced by General Finite-Rank Orthogonal Projections.- Energy-Levels, Dynamics, and the Maslov Index.- Localization and Multiplicity of a Self-Adjoint Elliptic 2x2 Positive NCHO in .

On Witten multiple zeta-functions associated with semisimple Lie algebras II

Abstract: We prove certain general forms of functional relations among Witten multiple zeta-functions in several variables (or zeta-functions of root systems). The structural background of these functional relations is given by the symmetry with respect to Weyl groups. From these relations, we can deduce explicit expressions of values of Witten zeta-functions at positive even integers, which are written in terms of generalized Bernoulli numbers of root systems. Furthermore, we introduce generating functions of Bernoulli numbers of root systems, using which we can give an algorithm of calculating Bernoulli numbers of root systems.

Properties and Inequalities of Generalized k-Gamma, Beta and Zeta Functions

Abstract: Recently R.Diaz and E.Pariguan introduced [2] the k-generalized Gamma function Γk(x), Beta function Bk(x, y) and Zeta function ζk(x, s) and gave some identities which they satisfy. We give some more properties and inequalities for the above k-generalized functions. Mathematics Subject Classification: 33B15, 33D05, 26D07, 24A48

Zeros of the Riemann zeta function on the critical line

Abstract: it is proved that at least 41.28% zeros of the Riemann zeta function are on the critical line

Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators

Abstract: In a previous paper by the first two authors, a tube formula for fractal sprays was obtained which also applies to a certain class of self-similar fractals. The proof of this formula uses distributional techniques and requires fairly strong conditions on the geometry of the tiling (specifically, the inner tube formula for each generator of the fractal spray is required to be polynomial). Now we extend and strengthen the tube formula by removing the conditions on the geometry of the generators, and also by giving a proof which holds pointwise, rather than distributionally. Hence, our results for fractal sprays extend to higher dimensions the pointwise tube formula for (1-dimensional) fractal strings obtained earlier by Lapidus and van Frankenhuijsen. Our pointwise tube formulas are expressed as a sum of the residues of the "tubular zeta function" of the fractal spray in $\mathbb{R}^d$. This sum ranges over the complex dimensions of the spray, that is, over the poles of the geometric zeta function of the underlying fractal string and the integers $0,1,...,d$. The resulting "fractal tube formulas" are applied to the important special case of self-similar tilings, but are also illustrated in other geometrically natural situations. Our tube formulas may also be seen as fractal analogues of the classical Steiner formula.

Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function

Abstract: Let s 0 0 be the abscissa of absolute convergence of the dynamical zeta function Z ( s ) for several disjoint strictly convex compact obstacles K i ⊂ R N , i = 1 , … , κ 0 , κ 0 ⩾ 3 and let R χ ( z ) = χ ( − Δ D − z 2 ) −1 χ , χ ∈ C 0 ∞ ( R N ) , be the cut-off resolvent of the Dirichlet Laplacian − Δ D in Ω = R N ∖ ⋃ i = 1 κ 0 K i ¯ . We prove that there exists σ 2 s 0 such that Z ( s ) is analytic for R ( s ) ⩾ σ 2 and the cut-off resolvent R χ ( z ) has an analytic continuation for ℑ ( z ) − i σ 2 , | R ( z ) | ⩾ C . To cite this article: V. Petkov, L. Stoyanov, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Quantum statistical mechanics, l-series and anabelian geometry

Abstract: It is known that two number fields with the same Dedekind zeta f unction are not neces- sarily isomorphic. The zeta function of a number field can be i nterpreted as the partition function of an associated quantum statistical mechanical system, which is a C � -algebra with a one parameter group of automorphisms, built from Artin reciprocity. In the first part of this paper, we prove that isomorphism of number fields is the same as isomorphism of the se associated systems. Considering the systems as noncommutative analogues of topological spaces, this result can be seen as another version of Grothendieck's "anabelian" program, much like t he Neukirch-Uchida theorem character- izes isomorphism of number fields by topological isomorphis m of their associated absolute Galois groups. In the second part of the paper, we use these systems to prove the following. If there is a con- tinuous bijection : y G ab � ! y G ab between the character groups (viz., Pontrjagin duals) of the abelianized Galois groups of the two number fields that induces an equalit y of all corresponding L- series LK(�,s) = LL( (�),s) (not just the zeta function), then the number fields are isomo rphic.

An integral representation of multiple hurwitz–lerch zeta functions and generalized multiple bernoulli numbers

Abstract: A surface integral representation of a multiple generalization of the Hurwitz–Lerch zeta function is given, which is a direct analogue of the well-known contour integral representation of the Riemann zeta function of Hankel's type. From this integral representation, we derive a detailed description of the set of its possible singularities. In addition, we present two formulae for special values of the zeta function at non-positive integers in terms of generalizations of Bernoulli numbers. These results are refinements of previously known ones.

Phase Plots of Complex Functions: a Journey in Illustration

Abstract: We propose to visualize complex (meromorphic) functions $f$ by their phase $P_f:=f/|f|$. Color--coding the points on the unit circle converts the function $P_f$ to an image (the phase plot of $f$), which represents the function directly on its domain. We discuss how special properties of $f$ are reflected by their phase plots and indicate several applications. In particular we reformulate a universality theorem for Riemann's Zeta function in the language of phase plots.

The Determinant of the Dirichlet-to-Neumann Map for Surfaces with Boundary

Abstract: For any orientable compact surface with boundary, we compute the regularized determinant of the Dirichlet-to-Neumann (DN) map in terms of main value at 0 of a Ruelle zeta function using uniformization of Mazzeo-Taylor. We apply it to compact hyperbolic surfaces with totally geodesic boundary and obtain various interpretations of the determinant of the Dirichlet Laplacian in terms of dynamical zeta functions. We also relate in any dimension the DN map for the Yamabe operator to the scattering operator for a conformally compact related problem by using uniformization.

The twisted fourth moment of the riemann zeta function

Abstract: We compute the asymptotics of the fourth moment of the Riemann zeta func- tion times an arbitrary Dirichlet polynomial of length T 1 11 " . The study of the moments of the Riemann zeta function has a long and distinguished history, starting with the work of Hardy and Littlewood in 1918 and continuing to the present day. One motivation for understanding moments is that they yield information about the maximum size of the zeta function (the Lindelof Hypothesis); another application is to zero density estimates which in turn have consequences for primes in short intervals. However they have become an interesting topic in their own right. Very few rigorous results are known, just the second and fourth power moments. Indeed, it is only recently that a believable conjecture for higher powers has been made. The twisted moments (that is, moments of the Riemann zeta function times an arbitrary Dirichlet polynomial) are important too, for example Levinson's method of detecting zeros of the zeta function lying on the critical line requires knowing the asymptotics of the mollified second moment. In a series of papers, Duke, Friedlander, and Iwaniec used estimates for amplified moments of a family of L-functions in order to deduce a subconvexity bound for an individual member of the family. Of course, there are far easier methods to give a subconvexity bound for zeta, but there are close analogies between different families and it is desirable to understand the structure of these amplified moments in general. In this paper, we prove an asymptotic formula for the twisted fourth moment of the Riemann zeta function, where we may take a Dirichlet polynomial of length up to T 1 11 −e .

Some Remarks on the Algebra of Bounded Dirichlet Series

Abstract: We consider here the algebra of functions which are analytic and bounded in the right half-plane and can moreover be expanded as an ordinary Dirichlet series. We first give a new proof of a theorem of Bohr saying that this expansion converges uniformly in each smaller half-plane; then, as a consequence of the alternative definition of this algebra as an algebra of functions analytic in the infinite-dimensional polydisk, we first observe that it does not verify the corona theorem of Carleson; and then, we give in a deterministic way a new quantitative proof of the Bohnenblust-Hille optimality theorem, through the construction of a generalized Rudin-Shapiro sequence of polynomials. Finally, we compare this proof with probabilistic ones.

Stochastic Complexity and Generalization Error of a Restricted Boltzmann Machine in Bayesian Estimation

Abstract: In this paper, we consider the asymptotic form of the generalization error for the restricted Boltzmann machine in Bayesian estimation. It has been shown that obtaining the maximum pole of zeta functions is related to the asymptotic form of the generalization error for hierarchical learning models (Watanabe, 2001a,b). The zeta function is defined by using a Kullback function. We use two methods to obtain the maximum pole: a new eigenvalue analysis method and a recursive blowing up process. We show that these methods are effective for obtaining the asymptotic form of the generalization error of hierarchical learning models.

Sharpening the Becker-Stark Inequalities

Abstract: In this paper, we establish a general refinement of the Becker-Stark inequalities by using the power series expansion of the tangent function via Bernoulli numbers and the property of a function involving Riemann's zeta one.

Riemann Zeros and Random Matrix Theory

Abstract: In the past dozen years random matrix theory has become a useful tool for conjecturing answers to old and important questions in number theory. It was through the Riemann zeta function that the connection with random matrix theory was first made in the 1970s, and although there has also been much recent work concerning other varieties of L-functions, this article will concentrate on the zeta function as the simplest example illustrating the role of random matrix theory.