Abstract: The Selberg trace formula on a Riemann surface X connects the discrete spectrum of the Laplacian with the length spectrum of the surface, that is, the set of lengths of the closed geodesics of on X. The connection is most strikingly expressed in terms of the Selberg zeta function, which is a meromorphic function of a complex variable s that is defined for <(s) > 1 in terms of the length spectrum and that has zeros at all s ∈ C for which s(1 − s) is an eigenvalue of the Laplacian in L(X). We will be interested in the case when X is the quotient of the upper half-plane H by either the modular group Γ1 = SL(2,Z) or the extended modular group Γ = GL(2,Z), where γ = ( a b c d ) ∈ Γ acts on H by z 7→ (az + b)/(cz + d) if det(γ) = +1 and z 7→ (az̄ + b)/(cz̄ + d) if det(γ) = −1. In this case the length spectrum of X is given in terms of class numbers and units of orders in real quadratic fields, while the eigenfunctions of the Laplace operator are the non-holomorphic modular functions usually called Maass wave forms. (Good expositions of this subject can be found in [6] and [7]).

Abstract: The topological zeta function and Igusa's local zeta function are respectively a geometrical invariant associated to a complex polynomial $f$ and an arithmetical invariant associated to a polynomial $f$ over a $p$-adic field. When $f$ is a polynomial in two variables we prove a formula for both zeta functions in terms of the so-called log canonical model of $f^{-1} \{ 0 \}$ in $\Bbb A^2$. This result yields moreover a conceptual explanation for a known cancellation property of candidate poles for these zeta functions. Also in the formula for Igusa's local zeta function appears a remarkable non-symmetric ‘$q$-deformation’ of the intersection matrix of the minimal resolution of a Hirzebruch-Jung singularity. 1991 Mathematics Subject Classification: 32S50 11S80 14E30 (14G20)

Abstract: A method for computing provably accurate values of partial zeta functions is used to numerically confirm the rank one abelian Stark Conjecture for some totally real cubic fields of discriminant less than 50000. The results of these computations are used to provide explicit Hilbert class fields and some ray class fields for the cubic extensions.

Abstract: Two types (binomial and exponential types) of power series, together with a related sum, associated with the Riemann zeta-function (s) will be investigated by using Mellin-Barnes type integrals. As for generalizations of these sums we shall introduce hypergeometric type generating functions of (s) and derive their basic properties.

Abstract: Geometric zeta functions of Ihara and Hashimoto are generalized to higher rank. The $p$-adic version of the Patterson conjecture is proven.

Abstract: In this paper, we consider the most non-split parabolic D_4 type prehomogeneous vector space. The vector space is an analogue of the space of Hermitian forms. We determine the principal part of the zeta function.

Abstract: It appears that the only known representations for the Riemann zeta function ((z) in terms of continued fractions are those for z = 2 and 3. Here we give a rapidly converging continued-fraction expansion of ((n) for any integer n > 2. This is a special case of a more general expansion which we have derived for the polylogarithms of order n, n > 1, by using the classical Stieltjes technique. Our result is a generalisation of the Lambert-Lagrange continued fraction, since for n = 1 we arrive at their well-known expansion for log(1 + z). Computation demonstrates rapid convergence. For example, the 11th approximants for all ((n), n > 2, give values with an error of less than 10-9.

Abstract: In [I-S2], we gave an explicit form of zeta functions associated to the space of symmetric matrices. In this paper, the case of L-functions is treated. In the case of definite symmetric matrices, we show the ratinality of special values of these L-functions.

Abstract: For a hyperbolic rational map R of the Riemann sphere of degree d ≥ 2, restricted to its Julia set J(R), we define a zeta function ζR(s), which counts the prepenodic orbib of R, according to the weight function |R'| : J(R) C. An analysis of the analytic domain of ζR(s), using techniques from symbolic dynamics, yields weighted asymptotic formulae for the preperiodic orbits of R. We describe an application to diophantine number theory.

Abstract: We generalize the Sommerfeld theory for a metal where the low temperature and high density expansions are known as the Sommerfeld expansions to that for a degenerate g-on gas — an ideal gas with fractional exclusion (i.e. Haldane–Wu) statistics of 0≤g≤1 — in D dimensions, using the quantum statistical mechanics formulation in the D-dimensional momentum representation. Using the generalized Sommerfeld expansions, the specific heat of the g-on gas is obtained herein, in terms of a generalized Riemann zeta function that is a natural extension of the Riemann zeta function. When g>0, the specific heat of a g-on gas shows a linear T dependence at low temperature as well as that of a metal, but the coefficient depends on statistics g and dimensionality D of the system. Therefore, we claim that this effect of statistics g is testable by an experiment.

Abstract: The aim of this note is to introduce the notion of random sequences of reals and to prove that the answer to the question in the title is negative, as anticipated by the informal discussion of Longpr e and Kreinovich [15].

Abstract: Euler's constant 7 is defined to be the limit of the sequence as Ntends to infinity. This sequence converges rather slowly. We show how to accelerate the convergence by expanding the function f(x) = 1/xin a power series about each integral value x = nthen integrating term by term, and hence expressing the difference between 7 and γ(N) as a power series in 1/N, Evaluating the terms shown at N= 10 yields γ = 0.577 215 664 9... accurate to ten decimal places. Applying the same technique to f(x)= loge xwe establish Stirling's series for the factorial function. Approaching the sequence of functions f(x) =1/x k =2, 3,... in the same way yields similar series for the Riemann zeta function ζ(k)evaluated at k= 2,3,....

Abstract: In this paper, an absolute minimum is found for the specialized theta function defined by a positive-definite ternery quadratic form with real coefficients. The result obtained yields an absolute minimum for the Epstein zeta function of the corresponding form. Bibliography: 15 titles.

Abstract: 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).