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

By Shalosh B. Ekhad and Doron Zeilberger

[Appeared in "Analytic number theory, modular forms and q-hypergeometric series"( in honor of Krishna Alladi, F. Garvan, ed.), 253-265, Springer Proc. Math. Stat., 221, Springer, Cham, 2017.]

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First written: May 20, 2016

Previous version: May 27, 2016

Last update (of webpage, and article): June 7, 2016

Dedicated to Krishnaswami "Krishna" Alladi, the tireless apostle of Srinivasa Ramanujan, yet a great mathematician on his own right.

Krishna Alladi's 60th birthday celebration (perfectly organized by Cindy Garvan, Frank Garvan, and George Andrews) was one of the most stimulating conferences that I have been to. All the talks were great, and because Krishna is interested in so many things, we got to hear stimulating talks in all aspects of number theory, both fancy and plain.

In particular, I really enjoyed Hugh Montgomery's Erdős Colloquium talk about the Rudin-Shapiro polynomials, that lead to the present article, Maple package, and ample output.

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.

Added May 23, 2016: Brad Rodgers, independently, and simultaneously, found a (complete) proof of Saffari's conjecture, that he is writing up now and will soon post in the arxiv. Meanwhile, you can read his proof in a letter that he wrote one of us (DZ).

Added May 30, 2016: Tamas Erdelyi has an interesting article where he examines the Mahler measure of the Rudin-Shapiro polynomials. This turns out to be closely related to the moments of the Rudin-Shapiro polynomials, (see Theorem 2.2), from which an upper bound

Mn(k) < c((2^n)/(\log n))*(2^{kn})

follows immediately. This is far from Saffari's Conjecture but it is quite non-trivial.

Added June 7, 2016: Brad Rodgers' beautiful article has just been posted. He also proves the more general Montgomery conjecture. Highly recommended.

Maple Packages

Sample Input and Output for the Maple package HaroldSilentShapiro.txt

Sample Input and Output for the Maple package ShapiroGeneral.txt

Doron Zeilberger's List of Papers

Doron Zeilberger's Home Page