Let $\alpha>0$ be a real number and $(s_\alpha(n))_{n\geq1}$ be the lexicographically greatest Sturmian word of slope $\alpha$. We investigate Dirichlet series of the form $\sum_{n=1}^\infty s_\alpha(n) n^{-s}$. 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 $01$, this paper establishes an identity $\sum_{n=1}^\infty s_\alpha(n) n^{-s}=1+\sum_{n=1}^\infty \varphi_\alpha(n) (\zeta(s)-\zeta(s,1+n^{-1})) n^{-s}$.