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

Abstract

For a fixed prime p, the maximum coefficient (in absolute value) M (p) of the cyclotomic polynomial Φ pqr (x), where r and q are free primes satisfying r > q > p exists. Sister Beiter conjectured in 1968 that M (p) ≤ (p + 1)/2. In 2009 Gallot and Moree showed that M (p) ≥ 2p(1 − )/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).

A Survey on Coefficients of Cyclotomic Polynomials

“…Moreover, they computed or bound M(p, q) for p and q satisfying certain conditions, and they posed several problems regarding M(p, q) [51, Section 11]. Cobeli, Gallot, Moree, and Zaharescu (2013) [34], using techniques from the study of the distribution of modular inverses, in particular bounds on Kloosterman sums, improved the lower bound in (6) to…”

The family of ternary cyclotomic polynomials with one free prime

“…We have B(11) = {4}, B(13) = {5}, B(17) = {7} and B(19) = {8}. In general one can show [12] using Kloosterman sum techniques that…”

Lower Bounds for Maximum Gap in (Inverse) Cyclotomic Polynomials

“…x − e 2πi k n For example, we have Φ 15 (x) = 1 − x + x 3 − x 4 + x 5 − x 7 + x 8 Ψ 15 (x) = −1 − x − x 2 + x 5 + x 6 + x 7 There have been extensive studies on the coefficients of cyclotomic polynomials [21,4,3,11,19,15,14,25,5,16,12,6,24,10,13,7,8], and more recently, on inverse cyclotomic polynomials [22,5,8].…”