On rational reciprocity

Rational Quartic Reciprocity Ii

for every prime p ≡ 1 mod 4 such that (p/pj) = +1 for all 1 ≤ j ≤ r. This is ’the extension to composite values of m’ that was referred to in [3], to which this paper is an addition. Here I will fill in the details of a proof, on the one hand because I was requested to do so, and on the other hand because this general law can be used to derive general versions of Burde’s and Scholz’s reciprocit...

Rational Quartic Reciprocity

In 1985, K. S. Williams, K. Hardy and C. Friesen [11] published a reciprocity formula that comprised all known rational quartic reciprocity laws. Their proof consisted in a long and complicated manipulation of Jacobi symbols and was subsequently simplified (and generalized) by R. Evans [3]. In this note we give a proof of their reciprocity law which is not only considerably shorter but also she...

Rational Reciprocity Laws

The purpose of this note is to provide an overview of Rational Reciprocity (and in particular, of Scholz’s reciprocity law) for the non-number theorist. In the first part, we will describe the background in number theory that will be necessary for a complete understanding of the material to be discussed in the second part. The second part focuses on a proof of Scholz’s reciprocity law using the...

On Stanley’s Reciprocity Theorem for Rational Cones

We give a short, self-contained proof of Stanley’s reciprocity theorem for a rational cone K ⊂ R. Namely, let σK(x) = ∑ m∈K∩Zd x m. Then σK(x) and σK◦(x) are rational functions which satisfy the identity σK(1/x) = (−1) σK◦(x). A corollary of Stanley’s theorem is the Ehrhart-Macdonald reciprocity theorem for the lattice-point enumerator of rational polytopes. A distinguishing feature of our proo...

Quadratic Reciprocity for the Rational Integers and the Gaussian Integers