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.

Abstract: where ck(x), ψk(x) are unknown functions of the scalar variable x. They have been called the vector analogs of the Cauchy equation. Note that the classic Cauchy equation is a particular case of (1) corresponding to N = 0, and in this case all solutions of (1) are exponential functions. It was proved that for N = 1 and N = 2 all the solutions of this equation are the Baker-Akhiezer functions corresponding to algebraic curves of genus 1 and 2, respectively. A starting point for the consideration of (1) in [5] was close connections of this equation with the theory of one-dimensional integrable systems of the Calogero-Moser system type. The main goal of this note is to show that the multivariable Baker-Akhiezer functions give solutions of the following multivariable generalization of (1):

Abstract: (T) at infinity is an analogue of thecompletion R of Q at infinity (=the archimedean prime), and hence thestatement is indeed analogous to the classical Riemann hypothesis aboutzeros being on a ‘‘line’’ (even though the second variable y may bearbitrary). Notice that in contrast to the ArtinWeil zeta function which isa simple rational function satisfying the Riemann hypothesis (as is trivialto prove for F

Abstract: We discuss some aspects of the search for identities using computer algebra and symbolic methods. To keep the discussion as concrete as possible, we shall focus on so-called Apery-like formulae for special values of the Riemann Zeta function. Many of these results are apparently new, and much more work needs to be done before they can be formally proved and properly classified. A first step in this direction can be found in [1].

Abstract: In this chapter we give an exposition of the definite integral of a real-valued function defined on a closed bounded interval. We assume familiarity with this concept from a previous study of calculus, but want to develop the theory in a more precise way than is typical for calculus courses, and also take a closer look at what kind of functions can be integrated. The integral to be defined and studied here is now widely known as the Riemann integral; in a later chapter we will study the more general Lebesgue integral.

Abstract: The change of variables formula for the Riemann integral is discussed and a theorem is proved which perhaps compares favorably with its counterpart in Lebesgue theory.

Abstract: For $1 0$ if and only if the Riemann zeta function satisfies $\zeta(s) eq 0$ for $\sigma = Re s > 1/p$.

Abstract: Let X be any Riemann surface. By Koebe’s uniformization theorem we know that the universal covering space of X is conformally equivalent to either Riemann sphere, complex plane, or the unit disc in the complex plane. If X is allowed to vary with parameters we may inquire the parameter dependence of the corresponding family of the universal covering spaces.

Abstract: We generalize the classical Wolff-Denjoy theorem to certain infinitely connected Riemann surfaces. Let X be a non-parabolic Riemann surface with Martin boundary i\. Suppose each Martin function ky, y E i\, extends continuously to Ai \ {y} and vanishes there. We show that if f is an endomorphism of X and the iterates of f converge to the point at infinity, then the iterates converge locally uniformly to a point in Ai. As an application, we extend the Wolff-Denjoy theorem to non-elementary Gromov hyperbolic covering spaces of compact Riemann surfaces. Such covering surfaces are of independent interest. Finally, we use the theory of non-tangential boundary limits to give a version of the Wolff-Denjoy theorem that imposes certain mild restrictions on f but none on X itself. Introduction. The classical theorem on the iteration of endomorphisms of the unit disc ID1, i.e., holomorphic maps from ID1 into itself, is the theorem of Wolff and Denjoy of 1926. It states that if such a map f is not an elliptic automorphism of ID1, then the iterates fn converge locally uniformly to a point in the closed unit disc ID1. The following theorem of Heins [Hei] seems to be the strongest available generalization of the Wolff-Denjoy theorem to arbitrary Riemann surfaces. Theorem (Heins). Let f be an endomorphism of a Riemann surface X covered by ID1. Then one of the following holds: (1) The iterates of f converge locally uniformly to a point in X. This point is the unique fixed point of f. (2) f is an automorphism of X of finite order. (3) X is D, D\\{0} or an annulus, and f is an irrational rotation, so a subsequence (ffnk) converges locally uniformly to the identity. (4) The iterates of f converge locally uniformly to an end of X. Moreover, in case (4), if X is a smoothly bounded domain in a compact Riemann surface, then the iterates of f converge locally uniformly to a boundary point of X. Received by the editors March 3, 1995. 1991 Mathematics Subject Classification. Primary 30F25, 32H50.

Abstract: For the special case of a Riemann surface which arises as the double of a planar domainR, the trisecant identity has a natural interpretation as a relation among reproducing kernels for subspaces of the Hardy spaceH2(R). This relation and Riemann's theorem on the vanishing of the theta function is applied to Nevanlinna-Pick interpolation onR.

Abstract: Let s = σ + it be a complex variable, and let, as usual, ζ(s) denote the Riemann zeta-function. It is well known that probabilistic methods can be applied for the examination of value-distribution of the function ζ(s), and the obtained results are usually stated as probabilistic limit theorems in the sense of weak convergence of probability measures. In recent years much attention was devoted to functional limit theorems for the Riemann zeta-function. B. Bagchi [1] proved such a limit theorem in the space of meromorphic functions with the topology of uniform convergence on compacta, and the first author of this article obtained a limit theorem in the space of continuous functions [4]. The latter paper also contains a survey on limit theorems for the Riemann zeta-function. In [7] we have presented a limit theorem with weight for ζ(s) in the space of analytic functions with the topology of uniform convergence on compacta. Let w(t) be a positive function with bounded variation on [T0,∞), T0 > 0, such that its variance V b aw on [a, b] satisfies the inequality V b aw ≤ cw(a) with some c > 0 for all b ≥ a ≥ T0. Moreover, let

Abstract: In this paper we present an example of an anticonformal automorphism whose square has prime order and is not embeddable. We prove that every embeddable automorphism of odd order of a compact Riemann surface is the square of an orientation-reversing self-homeomorphism. Finally we study whether a conformal involution is the square of an orientation-reversing automorphism. A smooth surface embedded in R inherits a conformal structure from R. A Riemann surface is embeddable if it is conformally equivalent to a smooth surface which is embedded in R. It has been shown that every Riemann surface is embeddable (see [G] and [R1]). If X is a Riemann surface and f is a conformal automorphism of X , we say that f is embeddable when there is a conformal embedding d : X → R such that dfd−1 is the restriction of a rotation. R. A. Rüedy [R2] has given necessary and sufficient conditions for an automorphism to be embeddable. In [Z1], using [R2], it is claimed that an automorphism of prime order of a compact Riemann surface is embeddable if it is the square of an anticonformal automorphism. In this paper we present an example of an anticonformal automorphism whose square has prime order and is not embeddable. Therefore Theorem 1 of [Z1] is not valid. Theorem 1 has been used in [Z2] and [Z3], leading to some mistakes which were later corrected in [BC] and [Y]. We also give an alternative proof of Theorem 2 of [Z1] asserting that every embeddable automorphism of odd order of a compact Riemann surface is the square of an orientation-reversing selfhomeomorphism. Finally we study whether a conformal involution is the square of an orientation-reversing automorphism. Let f be a conformal automorphism of prime order of a Riemann surface X and Fix(f) be the fixed point set of f . Given a point p ∈ Fix(f), there exists a chart (U, φ) on X such that φ(p) = 0 and φfφ−1(z) = z exp iα. Now α = α(f, p) is unique up to a multiple of 2π, and it is independent of the choice of the chart. We normalize α by requiring −π < α ≤ π. The following remark will be very useful. Remark 1. If the genus n of X satisfies n > 1, then there is a surface fuchsian group Λ uniformizing X and a fuchsian group ∆ containing Λ such that X/f can Received by the editors December 30, 1993 and, in revised form, April 29, 1994 and September 16, 1994. 1991 Mathematics Subject Classification. Primary 30F99. The author was partially supported by DGICYT PB 92-0716 and EU project CHRX-CT93-408. c ©1996 American Mathematical Society

Abstract: A new family of Dirichlet series having interesting combinatorial properties is introduced. Although they have no functional equation or Euler product, under the Riemann Hypothesis it is shown that these functions have no zeros in Re(s) > 1/2. Some identities in the ring of formal power series involving rook theory and continued fractions are developed.

Abstract: We obtain exact expressions for the moments of single order statistics from a standard Cauchy distribution. These are expressed as linear combinations of Riemann zeta functions. Using these and numerical integration methods, means of order statistics from samples of sizes upto 25 have been tabulated. Second order moments and variances are then obtained by applying the recurrence relation given by Barnett (1966). They are also tabulated. Finally, we obtain expressions for product moments in terms of means of order statistics and Riemann zeta functions.

Abstract: Let D 1 = {s ∈ C: 1/2 1}. We define the probability measures $${P_{j,T}}(A) = V_T^\tau (\zeta (s + i\tau ) \in A)$$ on (H(D j ), Β(H(D j ))), j = 1, 2. The aim of this chapter is to prove that the measures P j,T , converge weakly to some measure as T → ∞. Let D = {s ∈ C: σ > 1/2}.

Abstract: Weinberger in 1973 has shown that under the Generalized Riemann Hypothesis for Dedekind zeta functions, an algebraic number field with infinite unit group is Euclidean if and only if it is a principal ideal domain. Using a method recently introduced by us, we give two examples of cubic fields which are Euclidean but not norm-Euclidean.

Abstract: In this chapter the limit theorems on the modulus of the function ζ(s) in the half-plane σ ≥ 1/2 will be proved. The attention will be devoted mainly to the cases when σ → 1/2+0 or σ = 1/2.