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Variations on a theme by Hurwitz. (English) Zbl 0712.11017

The Hurwitz lemma states that if a: [0,1]\(\Rightarrow {\mathbb{R}}\) is a function and \(g(n)=\sum_{1\leq k\leq n}a(k/n)\), \(f(n)=\sum_{k\in A(n)}a(k/n)\), then \(f(n)=\sum_{d| n}g(d)\mu (n/d)\) where \(\mu\) denotes the Möbius function, and \(A(n)=\{k : k\in {\mathbb{N}}^*,1\leq k<n,(k,n)=1\}.\) This result has many applications and we have obtained methods for calculations of trigonometric or algebraic sums, and products e.g. \(\prod_{k\in A(n)}\cos k\pi /n=(-1/4)^{\phi (n)/2)},\) for \(n\geq 3\), odd. For the coefficients of the cyclotomic polynomial \(\Phi_ n(x)=x^{\phi (n)}+a_ 1x^{\phi (n)-1}+a_ 2x^{\phi (n)- 2}+...+a_{\phi (n)}\) a general formula (using determinants) is given. As an application, e.g. for \((n,6)=1\) we have \(a_ 3=-1/6[\mu (n)^ 3- 3\mu (n)^ 2+2\mu (n)].\) Denoting \(\alpha_ k(n)=\phi (n)[\mu (n/(n,k))/\phi (n/(n,k))],\) and \(M_ i=\max \{i-1,\alpha_ 1,\alpha_ 2,...,\alpha_ i\}\), then for the coefficients the inequality \(| a_ i| \leq (M_ i)^ i(i^{i/2})/i!\) is proved. The last result is the following divisibility property: For \(a>1\), \(n\in {\mathbb{N}}^*\) we have: n divides \(\phi (\Phi_ n(a))\). As a corollary we mention that for \(a>1\), \(m| n\), \(m<n\) we have \(n| \phi [(a^ n-1)/(a^ m- 1)]| \phi (a^ n-1)\).
Reviewer: J.Sandor

MSC:

11B83 Special sequences and polynomials
11A25 Arithmetic functions; related numbers; inversion formulas
15A15 Determinants, permanents, traces, other special matrix functions
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