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

For a fixed prime pp, the maximum coefficient (in absolute value) M(p)M(p) of the cyclotomic polynomial Φpqr(x)Φpqr(x), where rr and qq are free primes satisfying r>q>pr>q>p exists. Sister Beiter conjectured in 1968 that M(p)≤(p+1)/2M(p)≤(p+1)/2. In 2009 Gallot and Moree showed that M(p)≥2p(1−ϵ)/3M(p)≥2p(1−ϵ)/3 for every pp 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)M(p).