Curve fitting, differential equations and the Riemann hypothesis (Q2490445)

The Riemann \(\xi\)-function can be written as \[ \xi(s)=\int^\infty_0 x^{s/2-1} \varphi(x)\,dx, \] where \[ \varphi(x)=-\frac{1}{2\sqrt x}+ \sum^\infty_{n=1}e^{-\pi n^2x}. \] The author considers three approximations to \(\varphi\), the simplest of which is \[ \varphi^*(x)=-\frac{1}{2(1+x^4)^{1/8}}. \] For each of these it is shown that the corresponding function \(\xi^*(s)\) is real on the critical line. It is also claimed that the function can be real nowhere else, however the reviewer believes the proof to be faulty. Indeed it is clear that \(\xi^*(s)\) is real whenever \(s\) is real. It is shown that each of the approximations \(\xi^* (s)\) is non-vanishing on the critical line, so that the analogy with \(\xi(s)\) is not very good.