22 research outputs found
Jordan totient quotients
The Jordan totient can be defined by . In this paper, we study the average behavior of fractions
of two products and of Jordan totients, which we call Jordan totient
quotients. To this end, we describe two general and ready-to-use methods that
allow one to deal with a larger class of totient functions. The first one is
elementary and the second one uses an advanced method due to Balakrishnan and
P\'etermann. As an application, we determine the average behavior of the Jordan
totient quotient, the normalized derivative of the cyclotomic
polynomial at , the second normalized derivative of the
cyclotomic polynomial at , and the average order of
the Schwarzian derivative of at .Comment: 16 page
Constrained ternary integers
An integer is said to be ternary if it is composed of three distinct odd
primes. In this paper, we asymptotically count the number of ternary integers
with the constituent primes satisfying various constraints. We apply
our results to the study of the simplest class of (inverse) cyclotomic
polynomials that can have coefficients that are greater than 1 in absolute
value, namely to the (inverse) cyclotomic polynomials with ternary
. We show, for example, that the corrected Sister Beiter conjecture is true
for a fraction of ternary integers.Comment: 21 pages, to appear in International Journal of Number Theor