Corresponding to the lattice point problem for a random sphere Kendall and Rankin [8], Nakajima [9] considered the summatory function of the coefficients of the product of two Hurwitz zeta-functions and obtained the Bessel series expression. In this note we treat the case of the product of $\varkappa$ Hurwitz zeta-functions for an arbitrary integer $\varkappa\ge 2$ and obtain the expression in terms of the Voronoi-Steen function. This amounts to a refinement of corrected Nakajima’s formula with streamlining of the ambiguous argument.