Bemerkungen zu einer Arbeit von Ingham über die Verteilung der Primzahlen. (Remarks on a paper of Ingham on the distribution of primes) (Q1066938)

Using standard notation, let V(y) denote the number of sign changes of \(\pi\) (x)-li x in the interval [2, y]. Ingham proved that if the zeta function has a zero whose real part equals the supremum of the real parts of all zeros, then \(\pi\) (x)-li x has at least one sign change in all intervals of the form [y, Dy] for \(y>Y_ 0\) for some absolute constant D and \(V(y)>(\log y)/(2 \log D)\) for \(y>Y_ 1.\) The purpose of this paper is to prove effective versions of these results, namely that under the assumptions of the Riemann Hypothesis, \(V(y)>10^{-957} \log y\) and \(\pi\) (x)-li x has at least one sign change in the interval \([y^{10^{-957}}, y]\) for \(y>\exp_ 3 7,707.\) The authors also include several remarks about the number of steps which may be required to calculate effective values of D and Y in Ingham's result.