The Riemann hypothesis is justified by three different proof ways
The author attempts to justify Riemann hypothesis in three ways. The first way, already described in his paper [``Showing how to imply proving the Riemann hypothesis'', European J. Math. Sci. 2, No. 2, 126--139 (2013)], consists of showing that the integral NEWLINE\[NEWLINEI(x)=\int_1^{\infty} \frac{\psi(x)-x}{x^{s+1}}dxNEWLINE\]NEWLINE is analytic for \(\Re(s)>\frac{1}{2}\) (here \(\psi(x)=\sum_{p^m\leq x}\log p\) is the second Chebyshev function). As a result, the logarithmic derivative of the Riemann zeta function \(\zeta(s)\) has no pole in the region \(\frac{1}{2}<\Re(s)<1\) and \(\zeta(s)\) has no zero in the same region. According to Luboš Motl blog, there is a loophole (by a circular reasoning) in Lemma 3 (see \url{http://motls.blogspot.fr/2013/05/a-proof-of-riemann-hypothesis-using.html}).NEWLINENEWLINEIn the second proposed proof, the author writes NEWLINE\[NEWLINE\zeta(s)=(s-s_0)^m \phi(s),NEWLINE\]NEWLINE close to a zero of order \(m\) at \(s=s_0=\sigma+it_0\) and attempts to establish (in a quite elusive way) that \(\sigma=\frac{1}{2}\).NEWLINENEWLINEThe third proposed proof claims that the assumption \(s_0\neq \overline{1-s_0}\) for a zero \(s_0\) outside the critical line \(\sigma=\frac{1}{2}\) is topologically incorrect.
scientific article; zbMATH DE number 6241511