Vernon Armitage (University of Durham)
J.V.Armitage@durham.ac.uk
The Landsberg-Schaar formula enables one to evaluate the quadratic Gauss sum and so to prove the law of quadratic reciprocity. The standard proof is obtained from Jacobi's theta function identity, which connects Θ(τ) with (1/√ τ) Θ (1/τ), by writing τ=2iq/p + ε, ε>0, and then letting ε tend to 0. The method is an example of Hecke's observation that 'exact knowledge of the behaviour of an analytic function in the neighbourhood of its singular points is a source of arithmetic theorems'.
The idea behind this talk (which is based on joint work with Alice Rogers) is to 'investigate the singularity directly', without recourse to Jacobi's identity and the limiting process (except as a source of inspiration), by adapting one of the proofs of Jacobi's identity (which uses the fact that the theta function is a solution of the heat equation) to the discrete, finite case (so avoiding limits) by using Feynman's 'sum over paths' method from quantum mechanics.
The results have applications in quantum computing and may be extended to quadratic Gauss sums and quadratic reciprocity in algebraic number fields and to related questions.