Abstract. In this paper, we provide tight estimates for the divisor class number of hyperelliptic function fields. We extend the existing methods to any hyperelliptic function field and improve the previous bounds by a factor proportional to g with the help of new results. We thus obtain a faster method of computing regulators and class numbers. Furthermore, we provide experimental data and heuristics on the distribution of the class number within the bounds on the class number. These heuristics are based on recent results by Katz and Sarnak. Our numerical results and the heuristics imply that our approximation is in general far better than the bounds suggest. 1.

Abstract. Let p be an odd prime. Assuming the Extended Riemann Hypothesis, we show how to construct O((log p) 4 (log log p) −3) residues modulo p, one of which must be a primitive root, in deterministic polynomial time. Granting some well-known character sum bounds, the proof is elementary, leading to an explicit algorithm. 1.

Abstract. We provide a number of results that can be used to derive approximations for the Euler product representation of the zeta function of an arbitrary algebraic function field. Three such approximations are given here. Our results have two main applications. They lead to a computationally suitable algorithm for computing the class number of an arbitrary function field. The ideas underlying the class number algorithms in turn can be used to analyze the distribution of the zeros of its zeta function. 1.

. We develop a framework for computing with rational approximations to real numbers in number theoretic computations. We use that framework to analyze the approximate evaluation of the value L(1; \Delta ) where L(s; \Delta ) is the L-function of the quadratic order of discriminant \Delta. 1. Introduction In many computations in number theory approximations to real numbers are used. Important examples are the computation of values of L-series, in particular the approximation of the product of the regulator and the class number of a number field by means of the analytic class number formula (see for example [Sha72], [Len82], [BW89], [Coh95], [JLW95]) and the determination of a system of fundamental units and the regulator from a generating set of units (see for example [Buc89],[PZ89], [Coh95]). Those computations are carried out with rational approximations of a certain precision. Therefore, roundoff errors occur and those errors have to be taken into account. Unfortunately, most alg...

? ) E. Teske 1 and H.C. Williams ??2 1 Technische Universitat Darmstadt Institut fur Theoretische Informatik Alexanderstrae 10, 64283 Darmstadt Germany 2 University of Manitoba Dept. of Computer Science Winnipeg, MB Canada R3T 2N2 Abstract. Let p be a prime congruent to 1 modulo 4, n p the Legendre symbol and S(k) = P p 1 n=1 n k n p . The problem of nding a prime p such that S(3) > 0 was one of the motivating forces behind the development of several of Shanks' ideas for computing in algebraic number elds, although neither he nor D. H. and Emma Lehmer were ever successful in nding such a p. In this extended abstract we summarize some techniques which were successful in producing, for each k such that 3 k 2000, a value for p such that S(k) > 0. 1 Introduction Let d denote a fundamental discriminant of an imaginary quadratic eld IK = Q( p d ) and let h(d) denote the class number of IK. Let p be a prime ( 3(mod 4)), n p the Legendre symbol and S...