An asymptotic formula for a type of singular oscillatory integrals

This paper offers a general expansion formula for oscillatory integrals of the form $\smallint _0^1{x^{ - \alpha }}f(x,\{ Nx\} )\,dx$ in which N is a large parameter, Nx denotes the fractional part of Nx, and $\alpha$ is a fixed real number in $0 < \alpha < 1$. Our formula is expressed in terms of some ordinary integrals with integrands containing periodic Bernoulli functions and the generalized Riemann zeta function.