Harbingers of Artin's Reciprocity Law. III. Gauss's Lemma and Artin's Transfer

We briefly review Artin's reciprocity law in the classical ideal theoretic language, and then study connections between Artin's reciprocity law and the proofs of the quadratic reciprocity law using Gauss's Lemma

Elementary proofs of Berndt's reciprocity laws

Using analytic functional equations, Berndt derived three reciprocity laws connecting five arithmetical sums analogous to Dedekind sums. This paper gives elementary proofs of all three reciprocity laws and obtains them all from a common source, a polynomial reciprocity formula of L. Carlitz

Harbingers of Artin's Reciprocity Law. II. Irreducibility of Cyclotomic Polynomials

In the first article of this series we have presented the history of auxiliary primes from Legendre's proof of the quadratic reciprocity law up to Artin's reciprocity law. We have also seen that the proof of Artin's reciprocity law consists of several steps, the first of which is the verification of the reciprocity law for cyclotomic extensions. In this article we will show that this step can be identified with one of Dedekind's proofs of the irreducibility of the cyclotomic polynomial

Upstream reciprocity in heterogeneous networks

Many mechanisms for the emergence and maintenance of altruistic behavior in social dilemma situations have been proposed. Indirect reciprocity is one such mechanism, where other-regarding actions of a player are eventually rewarded by other players with whom the original player has not interacted. The upstream reciprocity (also called generalized indirect reciprocity) is a type of indirect reciprocity and represents the concept that those helped by somebody will help other unspecified players. In spite of the evidence for the enhancement of helping behavior by upstream reciprocity in rats and humans, theoretical support for this mechanism is not strong. In the present study, we numerically investigate upstream reciprocity in heterogeneous contact networks, in which the players generally have different number of neighbors. We show that heterogeneous networks considerably enhance cooperation in a game of upstream reciprocity. In heterogeneous networks, the most generous strategy, by which a player helps a neighbor on being helped and in addition initiates helping behavior, first occupies hubs in a network and then disseminates to other players. The scenario to achieve enhanced altruism resembles that seen in the case of the Prisoner's Dilemma game in heterogeneous networks.Comment: 10 figures, Journal of Theoretical Biology, in press (2010

Elliptic Reciprocity

The paper introduces the notions of an elliptic pair, an elliptic cycle and an elliptic list over a square free positive integer d. These concepts are related to the notions of amicable pairs of primes and aliquot cycles that were introduced by Silverman and Stange. Settling a matter left open by Silverman and Stange it is shown that for d=3 there are elliptic cycles of length 6. For d not equal to 3 the question of the existence of proper elliptic lists of length n over d is reduced to the the theory of prime producing quadratic polynomials. For d=163 a proper elliptic list of length 40 is exhibited. It is shown that for each d there is an upper bound on the length of a proper elliptic list over d. The final section of the paper contains heuristic arguments supporting conjectured asymptotics for the number of elliptic pairs below integer X. Finally, for d congruent to 3 modulo 8 the existence of infinitely many anomalous prime numbers is derived from Bunyakowski's Conjecture for quadratic polynomials.Comment: 17 pages, including one figure and two table