On the Coefficients of Cyclotomic PolynomialsThis book studies the coefficients of cyclotomic polynomials. Let $a(m, n)$ be the $m$th coefficient of the $n$th cyclotomic polynomial $\Phi_n(z)$, and let $a(m)={\rm max _n \vert a(m, n)\vert$. The principal result is an asymptotic formula for ${\rm log a(m)$ that improves a recent estimate of Montgomery and Vaughan. Bachman also gives similar formulae for the logarithms of the one-sided extrema $a (m)={\rm max _na(m, n)$ and $a_*(m)={\rm min _na(m, n)$. In the course of the proof, estimates are obtained for certain exponential sums which are of independent inter |
Contents
| 1 | |
1 Statement of results | 4 |
2 Proof of Theorem 0 the upper bound | 11 |
3 Preliminaries | 13 |
4 Proof of Theorem 1 the minor arcs estimate | 28 |
5 Proof of Theorem 1 the major arcs estimate | 33 |
6 Proof of Theorem 2 preliminaries | 55 |
7 Proof of Theorem 2 completion | 64 |
8 Proof of Propositions 1 and 2 | 68 |
9 Proof of Theorem 3 | 70 |
Appendix | 74 |
| 79 | |
Other editions - View all
Common terms and phrases
algebras American Mathematical Society analytic continuation asymptotic formula Ax is defined Cauchy’s coefficients Combining these estimates completes the proof coprime integers cos(y/K cyclotomic polynomial defined by 1.3 denotes Department of Mathematics differential Dirichlet Dirichlet series dźz e-mail Editors equation error term exist integers exponential sums follows fundamental discriminant g satisfying g given Hence hypothesis implies inequality inner sum integer composed k(t log Kronecker symbol large in terms last sum Lemma log a(m log a(rn log log lower bound M(gp Mathematics Subject Classification modulo q Montgomery and Vaughan multiplicative function nía nk+1 non-principal character paper partial summation pê Q pe?x pe(t positive integer prime factors prime number theorem primitive character modulo principal character modulo Proof of Theorem Proposition prove real numbers real primitive character result right-hand side set of primes So(z squarefree Suppose upper bound Vlog write XL g(p)(log