Fourier-Stieltjes coefficients of the Minkowski question mark function

In this paper we investigate the Fourier-Stieltjes coefficients of the Minkowski question mark function. In 1943, R. Salem asked whether these coefficients vanish at infinity. We propose the conjecture which implies the affirmative answer to Salem's problem. Further, we prove infinite linear identities among these Fourier-Stieltjes coefficients. We also provide a method to numerically calculate special values at integers of the associated zeta function with a high precision (more than 30 digits).