Sign Changes of E(T), Δ(x), and P(x) Original Research Article

Let Δ(x), P(x), and E(T) denote the usual error terms in the divisor problem, the circle problem, and the mean value theorem for ζ(image + 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 √K]. We show, in contrast, that a positive proportion of intervals [K, K + C √K (log K)−5] contain no such sign change. In the case of E(T) this disproves a conjecture of A. Iviimage and H. J. J. te Riele (Math. Comp.56, 1991, 303-328).