On certain Hardy sums and their \(2m\)-th power mean
The sums in the title are defined by \[ S(d,c)= \sum^{c-1}_{j=1} (-1)^{j+1+[dj/c]}. \] For fixed \(m\geq 2\) and sufficiently large \(k\) it is shown that \[ \sum^k_{h=1} S^{2m}(h, k)= g_m(k) k^{2m}+ O(k^{2m-1}), \] where \(g_m(k)\) is a Dirichlet series that can be evaluated in closed form in terms of the Riemann zeta function. This improves a result of the second author in which \(k\) was restricted to be a prime.
scientific article; zbMATH DE number 2147082