Bounds for the R ‐th Coefficients of Cyclotomic Polynomials

We consider the cyclotomic polynomials ~m(z) _ Fl (z-e(m/n)), m=1 (m, n)= 1 where e(a) = e2nia and write (Dn in the form n (1) 0(n) (D n(z) _ y_ a,(n)zr, (2) r=o where a) is Euler's function. Bounds for a,(n) in terms of n have been obtained by a number of l ar(n)I < exp (n 'llog log n) and Erdős [7, 8] has shown that this is best possible. Mirsky has mentioned in conversation that it is possible to obtain a bound for a,(n) which is independent of n. Moreover, Möller [15 ; (9) and Satz 3] has shown that l a,(n)I < p(r)-p(r-2), (3) where p(m) is the number of partitions of m, and also that max l a,(n)I > r m (r > ro(in)). (4) There is clearly a close connection between the size of a,(n) and the values (D n (z) takes as lzl-). 1-. Thus we first of all prove THEOREM 1. For each z with lzl < 1 we have I4)n(z)l < exp(c(1-Izl)-'+C,(1-lzl)-3/4), where (5) T = IZ (1-P(P+1)) (6) Although this cannot be far from the truth, we suspect that the right hand side of (5) should be as Izl-+ 1- .