THE FRACTIONAL TOTIENT FUNCTION AND STURMIAN DIRICHLET SERIES

Let ${\alpha}$ > 0 be a real number and $(s_{\alpha}(n))_{n{\geq}1}$ be the lexicographically greatest Sturmian word of slope ${\alpha}$. We investigate Dirichlet series of the form ${\sum}^{\infty}_{n=1}s_{\alpha}(n)n^{-s}$. To do this, a generalization of Euler's totient function is required. For a real ${\alpha}$ > 0 and a positive integer n, an arithmetic function ${\varphi}{\alpha}(n)$ is defined to be the number of positive integers m for which gcd(m, n) = 1 and 0 < m/n < ${\alpha}$. Under a condition Re(s) > 1, this paper establishes an identity ${\sum}^{\infty}_{n=1}s_{\alpha}(n)n^{-S}=1+{\sum}^{\infty}_{n=1}{\varphi}_{\alpha}(n)({\zeta}(s)-{\zeta}(s,1+n^{-1}))n^{-s}$.