Continuous crop circles drawn by Riemann's zeta function (Q1981593)

In the paper under review, the author describes an empirical method to produce a very good approximation to the Riemann zeta function \[\zeta(s)=\sum_{n=1}^\infty n^{-s} \] and to the Dirichlet eta function \[\eta(s)=\sum_{n=1}^\infty (-1)^{n+1} n^{-s}=(1-2^{1-s})\zeta(s).\] Actually, there are different ways to approximate the infinite Dirichlet series of \(\zeta(s)\) and \(\eta(s)\) using finite Dirichlet series \[\kappa_N(s)=\sum_{n=1}^N a_{N, n} n^{-s}.\] One way is to define for \(N=2L+1\), the coefficients \(a_{N, n}\) by \(a_{N,1}=1\) and \(\kappa_N(1/2 \mp i\gamma_k)=0\), for \(k=1,\cdots ,L\) and \(\gamma_1<\gamma_2<\cdots<\gamma_L\) are the positive imaginary parts of the zeroes of \(\zeta(s)\) assuming the Riemann hypothesis together with the simplicity of the zeroes. By doing extensive numerical calculations, it is conjectured that for fixed \(n\) and \(N\to \infty\), \(a_{N,n}\to (-1)^{n+1}\) and \(\kappa_N(s)\) is a very good approximation of \(\eta(s)\) for large \(N\). Finally, the author explains the second empirical method using the crop circles to find \(\kappa_N(s)\) which satisfies an approximate functional equation \[g(1-s)\kappa_N(1-s)\approx g(s)\kappa_N(s),\] where \(g(s)=\pi^{-s/2}(s-1)\Gamma(1+\frac{s}{2}).\) By Hamburger's theorem which asserts that \(\zeta(s)\) is uniquely determined by its functional equation, \(\kappa_N(s)\) is expected to be a very good approximation of \(\zeta(s)\) for large \(N\).