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

[摘要] 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.

[发布日期] 1994-10-01