About generalization of Euler's totient function and some its applications (Q2901763)

The present paper is devoted to the arithmetic function \(\varphi_k(n)\), \(k,n\in {\mathbb N}\), which is a generalization of Euler's totient function. Let \(\varphi_k(n)\) is defined to be the number of positive integers \(j\) less than \(n-k\) that \(\gcd(j,n)=\gcd(j+1,n)=\dots =\gcd(j+k-1,n)=1\). It is proven that this function is multiplicative with \(\varphi_k(p^\alpha) = p^{\alpha-1}\cdot\varphi_k(p)=p^{\alpha-1}\cdot(p-k)\). Moreover, the value of \(\varphi_k(n)\) for \(n > 1\) can be computed using the fundamental theorem of arithmetic: if \(n=p_1^{\alpha_1}\cdot p_2^{\alpha_2}\cdot p_l^{\alpha_l}\), where each \(p_i\) is a distinct prime, \(p_i<p_{i+1}\;\forall i\) and \(k\in\{1,2,\dots ,p_l-1\}\), then \(\varphi_k(n) = n\cdot\left( {1 - \frac{k}{p_1 }} \right) \cdot \left( {1 - \frac{k}{p_2 }} \right) \cdot \dots \cdot \left( {1 - \frac{k}{p_l }} \right)\). Also given the application of this function to calculation the number of non-isomorphic \(2\)-colored chord diagrams of special kind and the number of topologically nonequivalent smooth functions with one (degenerate) saddle critical point, one minimum and one maximum on closed oriented surface of genus \(g\geq 1\).