Sister Beiter and Kloosterman: a tale of cyclotomic coefficients and modular inverses

TL;DR

This paper investigates the maximum coefficients of certain cyclotomic polynomials, testing Sister Beiter's conjecture, and uses modular inverse distribution and Kloosterman sums to identify counterexamples and refine bounds.

Contribution

It applies Kloosterman sums and modular inverse distribution techniques to analyze cyclotomic coefficients, providing new bounds and counterexamples to Sister Beiter's conjecture.

Findings

Counterexamples to Sister Beiter's conjecture identified.

Refined lower bounds for maximum cyclotomic coefficients established.

Distribution of modular inverses linked to cyclotomic coefficient behavior.

Abstract

For a fixed prime $p$, the maximum coefficient (in absolute value) $M(p)$ of the cyclotomic polynomial $\Phi_{pqr}(x)$, where $r$ and $q$ are free primes satisfying $r>q>p$ exists. Sister Beiter conjectured in 1968 that $M(p)\le(p+1)/2$. In 2009 Gallot and Moree showed that $M(p)\ge 2p(1-\epsilon)/3$ for every $p$ sufficiently large. In this article Kloosterman sums (`cloister man sums') and other tools from the distribution of modular inverses are applied to quantify the abundancy of counter-examples to Sister Beiter's conjecture and sharpen the above lower bound for $M(p)$.