On sign-changes in the remainder-term of the prime-number formula, IV
Let T 1 and T 2 be topologies defined on the same set X and let us say that (X, T 1) and (X, T 2) are similar if the families of sets which have nonempty interior with respect to T 1 and T 2 coincide. The aim of the paper is to study how similar topologies are related with each other.
For a positive integer and a real number , let denote the supremum of the real numbers such that there are arbitrarily large positive integers such that are all less than . Here, denotes the distance to the nearest integer. We study the set of values taken by the function and, more generally, we are concerned with the joint spectrum of . We further address several open problems.