Note on a Paper By L. G. Sathe

While the results of Sathe's paper [J. Indian Math. Soc. 17 (1953), 63-141; 18 (1954), 27-81] are very beautiful and highly interesting, the way the author has proceeded in order to prove these results is a rather complicated and involved one, and this by necessity since a proof by induction after v starting from the case v = 1, presents overwhelming difficulties in keeping track of the estimates of the remainder terms in their dependence of the two parameters v and x.