* Presented to the Society, December 30, 1931; received by the editors August 24, 1932. t National Research Fellow. î Comptes Rendus, vol. 149 (1909), pp. 18-22. In this note Riesz considered only real positive orders r. § Comptes Rendus, vol. 152 (1911), pp. 1651-1654. Here again Riesz considered only the case r>0. || The terminology used in this paper is that given by W. A. Hurwitz, Bulletin of the American Mathematical Society, vol. 28 (1922), pp. 17-36. H We use the symbols [t] and [t~] to denote respectively the greatest integer á I and the greatest integer«. ** Hardy-Riesz, General Theory of Dirichlet's series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 18.