On monotone almost (P, mu)-u. d. mod 1 sequence

Using the result of almost u.d. mod 1 and G. HOHEISEL [6] and ERDOS'S resul t, AKIYAMA [1] gave a different proof of the ERDOS and TURAN result [4], i. e., Delta(2) log p(n) changes its sign infinitely many times, where p(n) is the n-th prime number and Delta(2)f(n) = Delta(Delta f(n)), Delta f(n) = f (n + 1) - f(n). In this paper, we show the necessary condition for monotone almost uniform distribution mod 1. We give Fejer's type theorem for almost uniform distrib ution mod 1 and give a similar result of NIEDERREITER [8]. Also we obtain t he converse of its result with some conditions. As an application, we give a simple proof [1] of ERDOS and TURAN'S result using ERDOS'S result [cf. 3, 9, 10, 11, and 2]. Also we obtain the condition of P(n) that Delta(2) log P(n) changes its sign infinitely many times.