Oscillatory properties of arithmetical functions. II (Q1094446)

In the first paper in this series [ibid. 48, 173--185 (1986; Zbl 0613.10036)] the authors proved that a real-valued function \(f(x)\) has at least \(c(f) \log Y\) sign-changes in the interval \((0,Y]\) provided that the analytic continuation of its Mellin transform has all of its singularities of a certain form. The purpose of this paper is to generalize this result for a much wider class of functions. The theorem has immediate applications to the theory of distribution of prime numbers, the oscillatory properties of the difference \(\pi(x)-li(x)\), and of the similar difference for primes in an arithmetic progression.