On theta type functions associated with the zeros of the Selberg zeta functions

In this paper, we consider a kind of theta type function concerning the zeros of the Selberg zeta function. This is obtained from an application of Cartier-Voros type Selberg trace formula for non co-compact but co-finite volume discrete subgroups ofPSL(2, R).     

On the nontrivial zeros of modified Epstein zeta functions

We study the zeros of modified Epstein zeta functions having functional equations. The result is that for any $\delta >0$, all but finitely many nontrivial zeros of such a function in $\{s\in \mathbb{C}:|s?\frac{1}{2}|<\delta \}$ are simple and on the critical line. As an immediate consequence of this theorem, all but finitely many nontrivial zeros of many modified Epstein zeta functions are simple and on the critical line. To cite this article: H. Ki, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

The zero-free region of the derivative of Selberg zeta functions

In this article, we study the zero-free region of the derivative of Selberg zeta functions associated with compact Riemann surfaces and three-dimensional compact hyperbolic spaces. We obtain the zero-free region with respect to the left of the critical line of each Selberg zeta function. This is an improvement of Wenzhi Luo’s zero-free region theorem for compact Riemann surfaces.     

Asymptotic expansions for a class of zeta-functions

In this paper, we shall reveal the hidden structure in recent results of Katsurada as the Meijer G-function hierarchy. In Sect. 1, we consider the holomorphic Eisenstein series and show that Katsurada’s two new expressions are variants of the classical Chowla–Selberg integral formula (Fourier expansion) with or without the beta-transform of Katsurada being incorporated. In Sect. 2, we treat the Taylor series expansion of the Lipschitz–Lerch transcendent in the perturbation variable. In the proofs, we make an extensive use of the beta-transform (used to be called the Mellin–Barnes formula).     

An Explicit Formula and its Application to the Selberg Trace Formula

We prove that the explicit formula in a symmetric case for a triple (Z, [(Z)\tilde]\tilde{Z} , Φ) in Jorgenson-Lang’s fundamental class of functions holds for a larger class of (not necessarily differentiable or even continuous) test functions. As one of the most important applications, we show that the Selberg trace formula for a strictly hyperbolic cocompact Fuchsian group Γ is valid for a larger class of test functions. Further applications to growth estimates for the logarithmic derivative of the Selberg zeta function are considered.     

Mesoscopic fluctuations of the zeta zeros

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.     

Majoration du Nombre de Zéros D’Une Fonction Méromorphe en Dehors D’Une Droite Verticale et Applications

We study the distribution of the zeros of functions of the form f(s) = h(s) ± h(2a − s), where h(s) is a meromorphic function, real on the real line, a is a real number. One of our results establishes sufficient conditions under which all but finitely many of the zeros of f(s) lie on the line ℜs = a, called the critical line for the function f(s), and that they are simple, provided that all but finitely many of the zeros of h(s) lie on the half-plane ℜs < a. This result can be regarded as a generalization of the necessary condition of stability for the function h(s), in the Hermite-Biehler theorem. We apply our results to the study of translations of the Riemann Zeta Function and L functions, and integrals of Eisenstein Series, among others.     

The zero-free region of the derivative of Selberg zeta functions

In this article, we study the zero-free region of the derivative of Selberg zeta functions associated with compact Riemann surfaces and three-dimensional compact hyperbolic spaces. We obtain the zero-free region with respect to the left of the critical line of each Selberg zeta function. This is an improvement of Wenzhi Luo’s zero-free region theorem for compact Riemann surfaces.     

Explicit formula for the spectral counting function of the Laplace operator on a compact Riemannian surface of genus g > 1

Let the standard Riemannian metric of constant curvature K = −1 be given on a compact Riemannian surface of genus g > 1. Under this condition, for a class of strictly hyperbolic Fuchsian groups, we obtain an explicit expression for the spectral counting function of the Laplace operator in the form of a series over the zeros of the Selberg zeta function.     

The zeta functions of Ruelle and Selberg for hyperbolic manifolds with cusps

In this paper, we study the Ruelle zeta function and the Selberg zeta functions attached to the fundamental representations for real hyperbolic manifolds with cusps. In particular, we show that they have meromorphic extensions to \mathbbC{\mathbb{C}} and satisfy functional equations. We also derive the order of the singularity of the Ruelle zeta function at the origin. To prove these results, we completely analyze the weighted unipotent orbital integrals on the geometric side of the Selberg trace formula when test functions are defined for the fundamental representations.     

An Explicit Formula and its Application to the Selberg Trace Formula

We prove that the explicit formula in a symmetric case for a triple (Z, , Φ) in Jorgenson-Lang’s fundamental class of functions holds for a larger class of (not necessarily differentiable or even continuous) test functions. As one of the most important applications, we show that the Selberg trace formula for a strictly hyperbolic cocompact Fuchsian group Γ is valid for a larger class of test functions. Further applications to growth estimates for the logarithmic derivative of the Selberg zeta function are considered.     

Hecke-Siegel's pull-back formula for the Epstein zeta function with a harmonic polynomial

In this paper, we discuss the generalization of the Hecke's integration formula for the Epstein zeta functions. We treat the Epstein zeta function as an Eisenstein series come from a degenerate principal series. For the Epstein zeta function of degree two, Siegel considered the Hecke's formula as the constant term of a certain Fourier expansion of the Epstein zeta function and obtained the other Fourier coefficients as the Dedekind zeta functions with Grössencharacters of a real quadratic field. We generalize this Siegel's Fourier expansion to more general Eisenstein series with harmonic polynomials. Then we obtain the Dedekind zeta functions with Grössencharacters for arbitrary number fields.     

On the Poles of Maximal Order of the Topological Zeta Function

The global and local topological zeta functions are singularityinvariants associated to a polynomial f and its germ at 0, respectively.By definition, these zeta functions are rational functions inone variable, and their poles are negative rational numbers.In this paper we study their poles of maximal possible order.When f is non-degenerate with respect to its Newton polyhedron,we prove that its local topological zeta function has at mostone such pole, in which case it is also the largest pole; wegive a similar result concerning the global zeta function. Moreover,for any f we show that poles of maximal possible order are alwaysof the form –1/N with N a positive integer. 1991 MathematicsSubject Classification 14B05, 14E15, 32S50.     

On Smooth Maps with Finitely Many Critical Points: Addendum

The paper is an addendum to D. Andrica and L. Funar, ‘Onsmooth maps with finitely many critical points’, J. LondonMath. Soc. (2) 69 (2004) 783–800.     

The Riemann Hypothesis and Inverse Spectral Problems for Fractal Strings

Motivated in part by the first author's work [23] on the Weyl-Berryconjecture for the vibrations of ‘fractal drums’(that is, ‘drums with fractal boundary’), M. L.Lapidus and C. Pomerance [31] have studied a direct spectralproblem for the vibrations of ‘fractal strings’(that is, one-dimensional ‘fractal drums’) and establishedin the process some unexpected connections with the Riemannzeta-function = (s) in the ‘critical interval’0 < s < 1. In this paper we show, in particular, thatthe converse of their theorem (suitably interpreted as a naturalinverse spectral problem for fractal strings, with boundaryof Minkowski fractal dimension D (0,1)) is not true in the‘midfractal’ case when D = , but that it is true for all other D in the criticalinterval (0,1) if and only if the Riemann hypothesis is true.We thus obtain a new characterization of the Riemann hypothesisby means of an inverse spectral problem. (Actually, we provethe following stronger result: for a given D (0,1), the aboveinverse spectral problem is equivalent to the ‘partialRiemann hypothesis’ for D, according to which = (s)does not have any zero on the vertical line Re s = D.) Therefore,in some very precise sense, our work shows that the question(à la Marc Kac) "Can one hear the shape of a fractalstring?" – now interpreted as a suitable converse (namely,the above inverse problem) – is intimately connected withthe existence of zeros of = (s) in the critical strip 0 <Res < 1, and hence to the Riemann hypothesis.     

A short proof of Levinson’s theorem

We give a short proof of Levinson’s result that over 1/3 of the zeros of the Riemann zeta function are on the critical line.     

The a-values of the Selberg zeta-function

We study the roots (a-values) of Z(s) = a, where Z(s) is the Selberg zeta-function attached to a compact Riemann surface. We obtain an asymptotic formula for the number of nontrivial a-values. If a ≠ 0, we show that the analogue of the Riemann hypothesis fails for nontrivial a-values; on other hand, almost all nontrivial a-values are arbitrarily close to the critical line. We also compare distributions of a-values for the Selberg and the Riemann zetafunctions.     

Duality involving the mock theta function f(q)

We show that the coefficients of Ramanujan's mock theta functionf(q) are the first non-trivial coefficients of a canonical sequenceof modular forms. This fact follows from a duality which equatescoefficients of the holomorphic projections of certain weight1/2 Maass forms with coefficients of certain weight 3/2 modularforms. This work depends on the theory of Poincaré series,and a modification of an argument of Goldfeld and Sarnak onKloosterman–Selberg zeta functions.     

Selberg and Ruelle zeta functions for non-unitary twists

In this paper we study the dynamical zeta functions of Ruelle and Selberg associated with the geodesic flow of a compact hyperbolic odd-dimensional manifold. These are functions of a complex variable s in some right half-plane of \(\mathbb {C}\). Using the Selberg trace formula for arbitrary finite dimensional representations of the fundamental group of the manifold, we establish the meromorphic continuation of the dynamical zeta functions to the whole complex plane. We explicitly describe the singularities of the Selberg zeta function in terms of the spectrum of certain twisted Laplace and Dirac operators.     

Zeta Functions for Curves and Log Canonical Models

The topological zeta function and Igusa's local zeta functionare respectively a geometrical invariant associated to a complexpolynomial f and an arithmetical invariant associated to a polynomialf over a p-adic field. When f is a polynomial in two variables we prove a formula forboth zeta functions in terms of the so-called log canonicalmodel of f^-1{0} in A². This result yields moreover a conceptualexplanation for a known cancellation property of candidate polesfor these zeta functions. Also in the formula for Igusa's localzeta function appears a remarkable non-symmetric ‘q-deformation’of the intersection matrix of the minimal resolution of a Hirzebruch-Jungsingularity. 1991 Mathematics Subject Classification: 32S5011S80 14E30 (14G20)