For any large \(d\) and an integer \(n\geq 2\) the authors prove an asymptotic formula for the \(\frac1n\)-th power moment of \(S_1(2c,d)\), with \(c\) running over the integers prime to \(d\) in an interval \([1, N]\), where \(N\leq d^{1-\epsilon}\). Here for even integer \(c\) and odd integer \(d\), \(S_1(c,d)=\sum_{j\bmod{d}}(-1)^{[cj/d]}((\frac{j}{d}))\) where \(((x))=x-[x]-1/2\) if \(x\) is not an integer, and let \(((x))=0\) otherwise.
The proof uses an asymptotic formula from [W. Zhang and Y. Yi, J. Math. Anal. Appl. 256, No. 2, 542–555 (2001; Zbl 0972.11089)].