Sister Beiter and Kloosterman: a tale of cyclotomic coefficients and modular inverses (original) (raw)

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 counterexamples to Sister Beiter's conjecture and sharpen the above lower bound for M (p).