An approximation to zeros of the Riemann zeta function using fractional calculus

Torres Hernandez, Anthony; Brambila Paz, Fernando

An approximation to zeros of the Riemann zeta function using fractional calculus

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Torres-Hernandez A, Brambila-Paz F. An approximation to zeros of the Riemann zeta function using fractional calculus. Mathematics and statistics. 2021;9(3):309-18. DOI: 10.13189/ms.2021.090312

Abstract

In this paper an approximation to the zeros of the Riemann zeta function has been obtained for the first time using a fractional iterative method which originates from a unique feature of the fractional calculus. This iterative method, valid for one and several variables, uses the property that the fractional derivative of constants are not always zero. This allows us to construct a fractional iterative method to find the zeros of functions in which it is possible to avoid expressions that involve hypergeometric functions, Mittag-Leffler functions or infinite series. Furthermore, we can find multiple zeros of a function using a singe initial condition. This partially solves the intrinsic problem of iterative methods, which in general is necessary to provide N initial conditions to find N solutions. 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. Some examples of its implementation are presented, and finally 53 different values near to the zeros of the Riemann zeta function are shown.