TY - JOUR
T1 - Sister Beiter and Kloosterman
T2 - A tale of cyclotomic coefficients and modular inverses
AU - Cobeli, Cristian
AU - Gallot, Yves
AU - Moree, Pieter
AU - Zaharescu, Alexandru
PY - 2013/11/15
Y1 - 2013/11/15
N2 - For a fixed prime p, the maximum coefficient (in absolute value) M(p) of the cyclotomic polynomial Φpqr(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-ε)/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).
AB - For a fixed prime p, the maximum coefficient (in absolute value) M(p) of the cyclotomic polynomial Φpqr(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-ε)/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).
KW - Cyclotomic coefficients
KW - Kloosterman sums
KW - Modular inverses
KW - Sister Beiter conjecture
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U2 - 10.1016/j.indag.2013.01.002
DO - 10.1016/j.indag.2013.01.002
M3 - Article
AN - SCOPUS:84887237909
VL - 24
SP - 915
EP - 929
JO - Indagationes Mathematicae
JF - Indagationes Mathematicae
SN - 0019-3577
IS - 4
ER -