Some Remarks on the Coefficients of Cyclotomic Polynomials

Cyclotomic polynomials are defined using the primitive roots of the unity, and have numerous applications in various areas of mathematics. Numerous properties of the coefficients of cyclotomic polynomials have been discovered, but practical algorithms for their calculation are hard to find. In this chapter we present recent advances related to the calculation of coefficients of cyclotomic polynomials, including a new recurrence formula involving Ramanujan sums. The chapter concludes with some numerical simulations.