On an exponential sum related to the Möbius function

Let μ ( n ) \mu (n) be the Möbius function and e ( α ) = e 2 π i α e(\alpha )=e^{2\pi i\alpha } . In this paper, we study upper bounds of the classical sum \[ S ( x , α ) ≔ ∑ 1 ≤ n ≤ x μ ( n ) e ( α n ) . S(x,\alpha )≔\sum _{1\leq n\leq x}\mu (n)e(\alpha n). \] We can improve some classical results of Baker and Harman [J. London Math. Soc. (2) 43 (1991), pp. 193–198].