A sum analogous to Dedekind sums and its mean value formula (Q2706880)

Let \(c\) be an even integer and \(d\) be a positive odd integer. Define the sum NEWLINE\[NEWLINE s_{1}(c,d)=\sum_{j \bmod d}(-1)^{[\frac{cj}{d}]} \Biggl(\biggl(\frac{j}{d}\biggr)\Biggr),NEWLINE\]NEWLINE where \(((x))=x-[x]-1/2\) if \(x\) is not an integer, and \(((x))=0\) if \(x\) is integer. An asymptotic formula for the sum NEWLINE\[NEWLINE\sum_{k=1, (k,d)=1}^{\frac{d-1}{2}}|s_{1}(2k,d)|^{2}NEWLINE\]NEWLINE is found in this paper.