Let
be the -th cyclotomic polynomial. In 1949, in his first published
paper, Paul Bateman
[1]
proved the inequality
where is the divisor function, and deduced from this
the bound
Bateman’s proof of and takes just over one page, and is a model
of simplicity and elegance. His work was motivated by a 1946 paper of
Paul Erdős
[e1],
in which Erdős gave a lower bound for
among other results and conjectures.
The papers by Erdős and Bateman gave impetus to significant further
work on the coefficients of cyclotomic polynomials. We mention here some
of the key results that have been obtained during the past few
decades. In 1974,
Robert Vaughan
[e2]
showed that the upper
bound is best possible, in the sense that an inequality in the
other direction holds for infinitely many . In 1984, Bateman,
Carl Pomerance,
and Vaughan
[2]
gave upper and lower bounds for
in terms of the number of prime factors of . In 1990,
Helmut Maier
[e3]
proved a long-standing conjecture on
the “typical” size of the maximal coefficient by
showing that, for “almost all” , one has for any given function . In 1993,
Gennady Bachman
[e4]
considered the size of the coefficient
when is fixed and varies, and he obtained an
asymptotic formula for as .
The above papers settle some of the key problems in this area, but many
other open questions remain, and the subject continues to attract the
attention of researchers. Of the fourteen papers that reference the
1984 paper
[2]
of Bateman, Pomerance, and Vaughan, according to
the MathSciNet database, five have been published since 2010.