SIGN CHANGES OF E(T), DELTA(X), AND P(X)

Let delta(x), P(x), and E(T) denote the usual error terms in the divisor problem, the circle problem, and the mean value theorem for \zeta(1/2 + it)\2. It is not hard to prove that, for a suitable constant C, each of these error terms changes sign on every interval [K, K + C square root K]. We show, in contrast, that a positive proportion of intervals [K, K + C square root K (log K)-5] contain no such sign change. In the case of E(T) this disproves a conjecture of A. Ivic and H. J. J. te Riele (Math. Comp. 56, 1991, 303-328). (C) 1994 Academic Press, Inc.