Abstract
Let Ψ n ( x ) be the monic polynomial having precisely all non-primitive nth roots of unity as its simple zeros. One has Ψ n ( x ) = ( x n − 1 ) / Φ n ( x ) , with Φ n ( x ) the nth cyclotomic polynomial. The coefficients of Ψ n ( x ) are integers that like the coefficients of Φ n ( x ) tend to be surprisingly small in absolute value, e.g. for n < 561 all coefficients of Ψ n ( x ) are ⩽1 in absolute value. We establish various properties of the coefficients of Ψ n ( x ) , especially focusing on the easiest non-trivial case where n is composed of 3 distinct odd primes.
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