Towards non-iterative calculation of the zeros of the Riemann zeta function

Abstract

We introduce a family of rational functions $R_N(a,d_0,d_1,\dots ,d_N)$ with the following property. Let $d_0,d_1,\dots ,d_N$ be equal respectively to the value of some function $f\left(s\right)$ and the values of its first N derivatives calculated at a certain complex number a lying not too far from a zero ρ of this function. It is expected that the value of $R_N(a,d_0,d_1,\dots ,d_N)$ is very close to ρ.

We demonstrate this phenomenon on several numerical instances with the Riemann zeta function in the role of $f\left(s\right)$. For example, for $N=10$ and $a=0.6+14i$ we have $\left|R_{10}\right(a,d_0,d_1,\dots ,d_{10})-{\rho }_1|<{10}^{-18}$ where ${\rho }_1=0.5+14.13...i$ is the first non-trivial zeta zero.

Also we define rational functions $R_{N,n}(a,d_0,d_1,\dots ,d_N)$ which (under the same assumptions) have values which are very close to $n^{-\rho }$, that is, to the terms from the Dirichlet series for the zeta function calculated at its zero.

In the case when a is between two consecutive zeros, say ${\rho }_l$ and ${\rho }_{l+1}$, functions $R_{N,n}(a,d_0,d_1,\dots ,d_N)$ approximate neither of $n^{-{\rho }_l}$ no $n^{-{\rho }_{l+1}}$; nevertheless, they allow us to approximate the sum $n^{-{\rho }_l}+n^{-{\rho }_{l+1}}$ and the product $n^{-{\rho }_l}{n}^{-{\rho }_{l+1}}$ and hence to calculate both $n^{-{\rho }_l}$ and $n^{-{\rho }_{l+1}}$ by solving corresponding quadratic equation.