On a method of estimating from below the number of sign changes of arithmetic functions.
Let \(V(\Delta,X)\) denote the number of sign-changes in \((0,X]\), \((X>0)\), of the remainder term \(\Delta(x)=\psi(x)-x\) in the prime number theorem, where \(\psi(x)=\sum_{p^ r\leq x} \log p\) stands for the Chebyshev's function. Assume that \((A_ H)\) for all non-trivial zeros \(\rho=\beta+i\gamma\) of the Riemann zeta function \(\zeta\) with \(|\gamma|<H\) we have \(\beta=1/2\). Then the earlier author's result [Math. Ann. 283, 151-163 (1989; Zbl 0636.10035); see also ibid. 283, 139-149 (1989; Zbl 0636.10036)] is basically as follows: \[ V(\Delta,X)\geq(1-b/H)(\gamma_ 0/\pi)\log X \qquad (\log X\geq BH^ c) \tag{1} \] with \(b=3\) and \(B=0.09\), if \(H\geq 4400\), where \(\gamma_ 0=14.13\dots\) is the imaginary part of the first non-trivial zero of \(\zeta\). It is now natural to ask about efficiency of the method used by the author in the proof of (1). More specifically, one may ask about the best constants \(b\) and \(B\) in (1) obtainable when using the method. The author succeeds in answering this question and gives the lower limits for \(b\) and \(B\). In particular, he remarks that there is an explicitly given constant \(b_ 0>0\) \(b_ 0=1.242\dots\)) so that the inequality (1) with \(b<b_ 0\) is not attainable by means of the present method under the assumption \((A_ H)\) for any \(H\). It is to mention that if one limits oneself to the basic problem of determination of \(\ell=\liminf_{X\to\infty} V(\Delta,X)/\log X\), then the better result \(\ell\geq\gamma_ 0/\pi+2\times 10^{-248}\) was obtained by the reviewer [Acta Arith. 59, 37-58 (1991; Zbl 0736.11048)] (without any hypothesis).
scientific article; zbMATH DE number 62237