Showing papers on "Riemann zeta function published in 1996"

Area, Lattice Points, and Exponential Sums

Abstract: Introduction Part I Elementary Methods 1 The rational line 2 Polygons and area 3 Integer points close to a curve 4 Rational points close to a curve Part II The Bombieri-Iwaniec Method 5 Analytic methods 7 The simple exponential sum 8 Exponential sums with a difference 9 Exponential sums with a difference 10 Exponential sums with modular form coefficients Part III The First Spacing Problem: Integer Vectors 11 The ruled surface method 12 The Hardy Littlewood method 13 The first spacing problem for the double sum Part IV The Second Spacing Problem: Rational vectors 14 The first and second conditions 15 Consecutive minor arcs Part V Results and Applications 17 Exponential sum theorems 18 Lattice points and area 19 Further results 20 Sums with modular form coefficients m 21 Applications to the Riemann zeta function 22 An application to number theory: prime integer points Part IV Related Work and Further Ideas 23 Related work 24 Further ideas References

Heat-kernels and functional determinants on the generalized cone

Abstract: We consider zeta functions and heat-kernel expansions on the bounded, generalized cone in arbitrary dimensions using an improved calculational technique. The specific case of a global monopole is analysed in detail and some restrictions thereby placed on theA 5/2 coefficient. The computation of functional determinants is also addressed. General formulas are given and known results are incidentally, and rapidly, reproduced.

Hypergeometric series acceleration via the wz method

Abstract: Based on the WZ method, some series acceleration formulas are given. These formulas allow us to write down an inflnite family of parametrized identities from any given identity of WZ type. Further, this family, in the case of the Zeta function, gives rise to many accelerated expressions for ‡(3).

A Note on Bernoulli Numbers and Shintani Generalized Bernoulli Polynomials

Abstract: Generalized Bernoulli polynomials were introduced by Shintani in 1976 in order to express the special values at non-positive integers of Dedekind zeta functions for totally real numbers. The coefficients of such polynomials are finite combinations of products of Bernoulli numbers which are difficult to get hold of. On the other hand, Zagier was able to get the explicit formula for the special values in cases of real quadratic number fields. In this paper, we shall improve Shintani's formula by proving that the special values can be determined by a finite set of polynomials. This provides a convenient way to evaluate the special values of various types of Dedekind functions. Indeed, a much broader class of zeta functions considered by the author [4] admits a similar formula for its special values. As a consequence, we are able to find infinitely many identities among Bernoulli numbers through identities among zeta functions. All these identities are difficult to prove otherwise. 1. IDENTITIES AMONG BERNOULLI NUMBERS The Bernoulli numbers Bn (n = 0,1, 2, ..) are defined by t _00 Bntn et1 E n! ' Itl 1 since the function t t et _1 2 is an even function of t by direct verification. Bernoulli numbers are used to express the special values of Riemann zeta function

Dynamical Zeta Functions, Nielsen theory and Reidemeister torsion

Abstract: The paper consists of four parts Part I presents a brief survey of the Nielsen fixed point theory Part II deals with dynamical zeta functions connected with Nielsen fixed point theory Part III is concerned with congruences for the Reidemeister and Nielsen numbers Part IV deals with the Reidemeister torsion In Chapter 2 we prove that the Reidemeister zeta function of a group endomorphism is a rational function with functional equation in the following cases: the group is finitely generated and an endomorphism is eventually commutative; the group is finite ; the group is a direct sum of a finite group and a finitely generated free abelian group; the group is finitely generated, nilpotent and torsion free In Chapter 3 we show that the Nielsen zeta function has a positive radius of convergence which admits a sharp estimate in terms of the topological entropy of the map For a periodic map of a compact polyhedron we prove that Nielsen zeta function is a radical of a rational function In sect ion 34 and 35 we give sufficient conditions under which the Nielsen zeta function coincides with the Reidemeister zeta function and is a rational function with functional equation In Chapter 5 we prove analog of Dold congruences for Reidemeister and Nielsen numbers In section 62 we establish a connection between the Reidemeister torsion and Reidemeister zeta function In section 63 we establish a connection between the Reidemeister torsion of a mapping torus, the eta-invariant, the Rochlin invariant and the multipliers of the theta function In section 64 we describe with the help of the Reidemeister torsion the connection between the topology of the attraction domain of an attractor and the dynamic of the system on the attractor

Explicit Formulas and Asymptotic Expansions for Certain Mean Square of Hurwitz Zeta-Functions.

Abstract: The main object of this paper is the mean square Ih(s) of higher derivatives of Hurwitz zeta functions ζ(s, α). We shall prove asymptotic formulas for Ih(1/2 + it) as t → +∞ with the coefficients in closed expressions (Theorem 1). We also prove a certain explicit formula for Ih(1/2 + it) (Theorem 2), in which the coefficients are, in a sense, not explicit. However, one merit of this formula is that it contains sufficient information for obtaining the complete asymptotic expansion for Ih(1/2 + it) when h is small. Another merit is that Theorem 1 can be strengthened with the aid of Theorem 2 (see Theorem 3). The fundamental method for the proofs is Atkinson's dissection argument applied to the product ζ(u, α)ζ(v, α) with the independent complex variables u and v.

Gaussian unitary ensemble eigenvalues and Riemann zeta function zeros: A nonlinear equation for a new statistic

Abstract: For infinite Gaussian unitary ensemble random matrices the probability density function ${S}_{\mathrm{nn}}(t)$ for the nearest neighbor eignenvalue spacing (as distinct from the spacing between consecutive eigenvalues) is computed in terms of the solution of a certain nonlinear equation, which generalizes the \ensuremath{\sigma} form of the Painlev\'e $V$ equation. Comparison is made with the empirical value of ${S}_{\mathrm{nn}}(t)$ for the zeros of the Riemann $\ensuremath{\zeta}$ function on the critical line, including data from ${10}^{6}$ consecutive zeros near zero number ${10}^{20}$.

The lap-counting function for linear mod one transformations I: explicit formulas and renormalizability

Abstract: Linear mod one transformations are the maps of the unit interval given by fβα(x) = βx + α (mod 1), with β > 1 and 0 ≤ α < 1. The lap-counting function is the function where the lap number Ln essentially counts the number of monotonic pieces of the nth iterate . We derive an explicit factorization formula for Lβα(z) which directly shows that Lβα(z) is a function meromorphic in the open unit disk {z: |z| < 1} and analytic in the open disk {z: |z| < 1/β}, with a simple pole at z = 1/β.Comparison with a known formula for the Artin—Mazur—Ruelle zeta function ζβ,α(z) of fβα shows that Lβα(z) and ζβ,α(z) have identical sets of singularities in the disk {z: |z| < 1}. We derive two more factorization formulae for Lβ,α(z) valid for certain parameter ranges of (β, α). When 1 < α + β ≤ 2, there is sometimes a ‘renormalization’ structure of such maps present, which has previously been studied in connection with simplified models for the Lorenz attractor. In the case that fβα is non-trivially renormalizable, we obtain a factorization formula for Lβα(z). For (β, α) in a region contained in 2 < α + β ≤ 3 we obtain a factorization formula which relates Lβα(z) to a ‘rescaled’ lap-counting function from the region 1 < α + β ≤ 2. The various factorizations exhibit certain singularities of Lβα(z) on the circle |z| = 1/β. These singularities are related to topological dynamical properties of fβ,α. In parts II and III we show that these comprise the complete set of such singularities on the circle |z| = 1/β.

On special divisors and the two variable zeta function of algebraic curves over finite fields

Abstract: The gonality sequence of a plane curve is computed. A two variable zeta function for curves over a finite field is defined and the rationality and a functional equation are proved.

On the quantum zeta function

Abstract: It is remarkable that the quantum zeta function, defined as a sum over energy eigenvalues E: admits of exact evaluation in some situations for which not a single E be known. Herein we show how to evaluate instances of Z(s), and of an associated parity zeta function Y(s), for various quantum systems. For some systems both Z(n), Y(n) can be evaluated for infinitely many integers n. Such Z,Y values can be used, for example, to effect sharp numerical estimates of a system's ground energy. The difficult problem of evaluating the analytic continuation Z(s) for arbitrary complex s is discussed within the contexts of perturbation expansions, path integration, and quantum chaos.

A probabilistic generalization of the Riemann zeta function

Abstract: There is an old problem that asks for the probability that two integers chosen at random are relatively prime. The informal solution goes as follows.

A green's function for the annulus

Abstract: In this paper we find an expression for Green's junction for the operator Δ2 in a planar circular annulus with Dirichlet boundary conditions (clamped elastic plate). We likewise determine the corresponding Poisson type kernels and the harmonic Bergman kernel. These results come in terms of certain new transcendental functions which in a natural way generalize the Weierstrass zeta function. They are analogous to the result of R.Courant D.Hubert (Methoden der Mathematischen Physik I (3. Aufl.), Springer-Verlag, Berlin, Heidelberg, New York (1968), pp. 335-337)and H.Villat (Rend. Circ. Mat. Palermo,33 (1912), pp. 134–175)respectively. As an application we show that, regardless of the size of the ratio of the radii of the bounding circles, the Green's function always assumes negative values, which constitutes another rather striking counter-example to the wellknown Boggio-Hadamard conjecture.

The semiclassical resonance spectrum of Hydrogen in a constant magnetic field

Abstract: We present the first purely semiclassical calculation of the resonance spectrum in the Diamagnetic Kepler problem (DKP), a hydrogen atom in a constant magnetic field with $L_z =0$. The classical system is unbound and completely chaotic for a scaled energy $\epsilon \sim E B^{-2/3}$ larger than a critical value $\epsilon_c >0$. The quantum mechanical resonances can in semiclassical approximation be expressed as the zeros of the semiclassical zeta function, a product over all the periodic orbits of the underlying classical dynamics. Intermittency originating from the asymptotically separable limit of the potential at large electron--nucleus distance causes divergences in the periodic orbit formula. Using a regularisation technique introduced in [G.\ Tanner and D.\ Wintgen, Phys.~Rev.~Lett.~{\bf 75}, 2928 (1995)] together with a modified cycle expansion, we calculate semiclassical resonances, both position and width, which are in good agreement with quantum mechanical results obtained by the method of complex rotation. The method also provides good estimates for the bound state spectrum obtained here from the classical dynamics of a scattering system. A quasi Einstein--Brillouin--Keller (QEBK) quantisation is derived that allows for a description of the spectrum in terms of approximate quantum numbers and yields the correct asymptotic behaviour of the Rydberg--like series converging towards the different Landau thresholds.

Lyapunov exponents and anomalous diffusion of a Lorentz gas with infinite horizon using approximate zeta functions

Abstract: We compute the Lyapunov exponent, the generalized Lyapunov exponents, and the diffusion constant for a Lorentz gas on a square lattice, thus having infinite horizon. Approximate zeta functions, written in terms of probabilities rather than periodic orbits, are used in order to avoid the convergence problems of cycle expansions. The emphasis is on the relation between the analytic structure of the zeta function, where a branch cut plays an important role, and the asymptotic dynamics of the system. The Lyapunov exponent for the corresponding map agrees with the conjectured limit λmap = -2 log(R) + C + O(R) and we derive an approximate value for the constantC in good agreement with numerical simulations. We also find a diverging diffusion constantD(t)∼logt and a phase transition for the generalized Lyapunov exponents.

Limit theorems for the Matsumoto zeta-function

Abstract: © Université Bordeaux 1, 1996, tous droits réservés. L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

On the zeta function of a complete intersection

Abstract: In this article, we compute the p-adic Dwork cohomology of a smooth complete intersection in T^ x A\" or P over a finite field (where T^ is the m-torus). As an application, we prove the \"Katz Conjecture\" (i.e., the assertion that the Newton polygon lies over the Hodge polygon) for such varieties. This result is new in the case of T x A\". (The case of P is due to Mazur [14].)

The zeta function constructed from the zeros of the Bessel function

Abstract: This paper studies the function built from the zeros of the Bessel function . The known first eight terms of the McMahon expansion with are used to construct an accurate approximation to . The quality of this approximation is investigated numerically by comparison with a known but (at least numerically) little-studied integral formula for . Excellent numerical agreement is found for fixed and variable (real) s, and for fixed s and variable . Both formulae for therefore seem to work well. Our approximation also accurately reproduces known special values of . Important properties of are investigated for the first time, including several of its zeros. In addition, some general theory is presented in two areas: (i) perturbed spectra and (ii) the interrelationship between functions like representable as infinite products, and the functions constructed from their infinite spectrum of zeros.