Integrals Involving Rudin-Shapiro Polynomials and Sketch of a Proof of Saffari's Conjecture

TL;DR

This paper develops Maple algorithms to compute integrals of Rudin-Shapiro polynomials, confirms Saffari's conjecture and related conjectures for small powers, and outlines a potential proof of Saffari's full conjecture.

Contribution

It introduces new computer algebra methods for analyzing Rudin-Shapiro polynomial integrals and provides evidence towards proving Saffari's conjecture.

Findings

Confirmed Saffari's conjecture for small powers

Validated a related conjecture of Hugh Montgomery

Outlined a proof approach for Saffari's full conjecture

Abstract

Continuing pioneering work of Christophe Doche and Laurent Habsieger from 2004, we develop computer algebra algorithms, implemented in Maple, for finding the (necessarily rational) generating function for any integral of products, and in particular, moments, of Rudin-Shapiro polynomials. We generate a lot of output, and confirm again a conjecture of Saffari for the asymptotics for small (and not so small) powers. We also confirm, for small powers, a related, more general, conjecture, of Hugh Montgomery. Finally, we outline a proof of Saffari's full conjecture, that we believe can be turned into a full proof. [In this version we report that Brad Rodgers has independently found a (complete!) proof of Saffari's conjecture here http://arxiv.org/abs/1606.01637] .