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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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