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A multivariate arithmetic function of combinatorial and topological significance. (English) Zbl 1230.11007
The author investigates the multivariable arithmetic function \[ E(m_1,m_2,\ldots,m_r) = (1/m) \sum_{k=1}^m \prod_{i=1}^r \Phi(k,m_i) \] where \(m= (m_1,m_2, \ldots,m_r)\) and \[ \Phi(k,n) = \frac{\varphi(n)}{\varphi(\gcd(k,n))} \mu( \frac{n}{\gcd(k,n)} ). \] The main result of the paper is a simple formula for the function \(E\) in terms of the prime factors of \(m\). Besides a variety of properties of \(E\) and a variety of corollaries of the main result, the author presents combinatorial and topological motivations of the function \(E\). Applications of this function include cyclic groups of Riemann surfaces, the Riemann-Hurwitz equation, and the map enumeration.

MSC:
11A25 Arithmetic functions; related numbers; inversion formulas
05A15 Exact enumeration problems, generating functions
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