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

Cristian Cobeli, Yves Gallot, Pieter Moree, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review

Abstract

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).

Original languageEnglish (US)
Pages (from-to)915-929
Number of pages15
JournalIndagationes Mathematicae
Volume24
Issue number4
DOIs
StatePublished - Nov 15 2013

Keywords

  • Cyclotomic coefficients
  • Kloosterman sums
  • Modular inverses
  • Sister Beiter conjecture

ASJC Scopus subject areas

  • Mathematics(all)

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