On the even power mean of a sum analogous to Dedekind sums
The authors investigate sums which were introduced by \textit{B. C. Berndt} [J. Reine Angew. Math. 303/304, 332--365 (1978; Zbl 0384.10011)] and which are defined by \[ S_1(h,k)=\sum^k_{a=1}(-1)^{[\frac{ah}k]}\left(\left(\frac ak\right)\right), \] where \(((x))=x-[x]-\frac 12\) if \(x\) is not an integer, \(((x))=0\) if \(x\) is an integer. If one drops the signs in this formula then one obtains the classical Dedekind sum \(S(h,k)\). The authors prove an asymptotic formula for the mean value powers \(S^{2m}_1(h,k)\). For a fixed integer \(m\geq 2\) and for all sufficiently large odd \(k\), it reads \[ \sum_hS^{2m}_1(h,k)=g_m(k)\cdot \left(\frac k2\right)^{2m}+O(k^{2m-1}). \] The summation is on all even \(h\), \(1\leq h\leq k\), which are relatively prime to \(k\), and \(g_m(k)\) is defined to be the \(k\)-th coefficient in the Dirichlet series expansion of \[ \frac{(2^{s+4m}-2)(2^s-1)}{(2^{s+2m}-1)^2(2^{2m}+1)}\cdot \frac{\zeta^2(2m)}{\zeta(4m)}\cdot \frac{\zeta(s+4m-1)\zeta(s)}{\zeta^2(s+2m)}. \] A similar asymptotic formula for the mean value of \(S^{2m}(h,k)\) was proved by \textit{J. B. Conrey, E. Fransen, R. Klein} and \textit{C. Scott} [J. Number Theory 56, 214--226 (1996; Zbl 0851.11028)], and improved by \textit{C. Jia} [J. Number Theory 87, 173--188 (2001; Zbl 0976.11044)].
scientific article; zbMATH DE number 5001796