2015/11/10 by Sai Teja Somu, Somu, Sai Teja
Mathematics · #Analytic Number Theory Research #Meromorphic and Entire Functions #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.1511.03226
Let a(r,n) be rth coefficient of nth cyclotomic polynomial. Suzuki proved that \a(r,n)|r≥ 1,n≥ 1\=ℤ. If m and n are two natural numbers we prove an analogue of Suzuki's theorem for divisors of xn-1 with exactly m irreducible factors. We prove that for every finite sequence of integers n1,…,nr there exists a divisor f(x)=∑i=0deg(f)cixi of xn-1 for some n∈ ℕ such that ci=ni for 1≤ i ≤ r. Let H(r,n) denote the maximum absolute value of rth coefficient of divisors of xn-1. In the last section of the paper we give tight bounds for H(r,n).