An approximation to zeros of the Riemann zeta function using fractional calculus
A. Torres-Hernandez, F. Brambila-Paz
TL;DR
This work introduces a fractional calculus-based approach to approximating zeros of the Riemann zeta function using a fractional pseudo-Newton method that employs a fractional-derivative–based Jacobian-like operator to generate multiple zeros from a single initial guess. It builds on the Riemann-Liouville fractional derivative and a matrix $P_{\epsilon,\beta}(x)$ to define the iteration $x_{i+1}=x_i- P_{\epsilon,\beta}(x_i) f(x_i)$, aiming to locate nontrivial zeros in the regime of large $\lvert \Im(\xi)\rvert$. The method is applied to a globally convergent Knopp-Hasse representation of $\zeta$ with truncation at $k=50$, producing dozens of near-zeros (including complex ones) and enabling exploration with a single scalar initial condition. The results illustrate the potential of fractional iterative methods for global-like zero search in functions with many zeros, such as $\zeta$, and offer a new numerical tool for zeta-zero analysis and RH-related investigations.
Abstract
In this document, as far as the authors know, an approximation to the zeros of the Riemann zeta function has been obtained for the first time using only derivatives of constant functions, which was possible only because a fractional iterative method was used. This iterative method, valid for one and several variables, uses the properties of fractional calculus, in particular the fact that the fractional derivatives of constants are not always zero, to find multiple zeros of a function using a single initial condition. This partly solves the intrinsic problem of iterative methods that if we want to find N zeros it is necessary to give N initial conditions. Consequently, the method is suitable for approximating nontrivial zeros of the Riemann zeta function when the absolute value of its imaginary part tends to infinity. The deduction of the iterative method is presented, some examples of its implementation, and finally 53 different values near to the zeros of the Riemann zeta function are shown.