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

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.


Introduction
A classical problem in mathematics which is of common interest in physics and engineering is finding the set of zeros of a function f : Ω ⊂ R n → R n , that is, where · : R n → R denotes any vector norm. Although finding the zeros of a function may seem like a simple problem, many times it involves solving an algebraic equation system given by the following expression . . .
where [f ] k : R n → R denotes the k-th component of the function f . It should be mentioned that system (2) may represent a linear system or a nonlinear system. The function that has the set of zeros of most interest in mathematics, being possibly the most famous set of zeros, corresponds to the Riemann zeta function 310 An Approximation to Zeros of the Riemann Zeta Function Using Fractional Calculus The Riemann hypothesis, under the assumption that it is true, is responsible for giving a general expression to the zeros of the function ζ, and may be written compactly as when ξ = −2x it is known as a nontrivial zero of the function ζ. In general, it is necessary to use numerical methods of the iterative type to construct a sequence {x i } ∞ i=0 , that under certain conditions, allows to approximate to values ξ ∈ ker (ζ) with ξ = −2x, that is, If we want to find N values ξ ∈ ker (ζ), it is necessary to give N initial conditions x 0 . This is an intrinsic problem of iterative methods, because time must first be spent determining to initial condition x 0 before beginning to search the values ξ. Furthermore, it is sometimes necessary that the initial conditions are near to the searched values to guarantee convergence, that is, These problems are partially solved using fractional iterative methods, because they can determine N values ξ using a single initial condition x 0 , and the initial condition does not need to be near the searched values. In this paper a fractional iterative method is used, which has the peculiarity that it uses the fractional derivative of constants to find the zeros of functions, and due to this peculiarity when implementing the fractional derivative operator it is possible to avoid expressions involving hypergeometric functions, Mittag-Leffler functions or infinite series. Then, it is an ideal iterative method to work with functions that can be expressed in terms of a series, as is the case of the Riemann zeta function, or to solve nonlinear systems in several variables.

Fixed Point Method
Let Φ : R n → R n be a function. It is possible to build a sequence {x i } ∞ i=0 defining the following iterative method if it is true that the sequence {x i } ∞ i=0 converges to a value ξ ∈ R n , and if the functions Φ is continuous around ξ, it holds that the above result is the reason by which the method (5) is known as fixed point method. Furthermore, the function Φ is called an iteration function. The following theorem allows characterizing the order of convergence of an iteration function Φ with its derivatives [1,2,3,4]. Before continuing, we need to consider the following multi-index notation. Let N 0 be the set N ∪ {0}, if γ ∈ N n 0 , then Theorem 2.1. Let Φ : Ω ⊂ R n → R n be an iteration function with a fixed point ξ ∈ Ω. Assuming that Φ is p-times differentiable in ξ for some p ∈ N, and furthermore where Φ (1) denotes the Jacobian matrix of the function Φ, then Φ is (locally) convergent of (at least) order p.
Proof. The proof may be found in the reference [4].
The next corollary follows from the previous theorem.
such that x i → ξ, and if the following condition is fulfilled then Φ has an order of convergence (at least) linear in B(ξ; δ).

Riemann-Liouville Fractional Operators
One of the fundamental operators of fractional calculus is the operator Riemann-Liouville fractional integral, which is defined as follows [5,6] which is a fundamental piece to build the operator Riemann-Liouville fractional derivative, which is defined as follows [5,7] where n = α and a I 0 . Applying the operator (11) with a = 0 to the function x µ , with µ > −1, we obtain the following result [4]:

Fractional Pseudo-Newton Method
Although the interest in fractional calculus has mainly focused on the study and development of techniques to solve differential equation systems of order non-integer [5,6,7,8,9,10,11]. Over the years, iterative methods have also been developed that use the properties of fractional derivatives to solve algebraic equation systems [4,12,13,14,15,16,17,18,19,20]. These methods may be called fractional iterative methods, which under certain conditions, may accelerate their speed of convergence with the implementation of the Aitken's method [2,4], recently these methods have been useful in the search for solutions to algebraic equation systems related to hybrid solar receivers [3,19]. It should be noted that depending on the definition of fractional derivative used, fractional iterative methods have the particularity that they may be used of local form [12] or of global form [18]. Furthermore, for some definitions of fractional derivative, it is fulfilled that the derivative of the order α of a constant is different from zero, that is, 312 An Approximation to Zeros of the Riemann Zeta Function Using Fractional Calculus where ∂ α k denotes any fractional derivative applied only in the component k, that does not cancel the constants (for example: Riesz, Grünwald-Letnikov, Riemann-Liouville, etc. [5,6,7,8,9,10,11]), and that fulfills the following continuity relation with respect to the order α of the derivative Considering a function Φ : (R \ Z) × C n → C n , recently a fractional iterative method has been proposed that has already been implemented in an engineering application, which is called the fractional pseudo-Newton method and is defined by the following expression [3]: where P ,β (x i ) is a matrix evaluated in the value x i , which is given by the following expression where with δ jk the Kronecker delta, a positive constant 1, and β(α, [x i ] k ) a function defined as follows Due to the part of the integral operator that fractional derivatives usually have, we consider in the matrix (16) that each fractional derivative is obtained for a real variable [x] k , and if the result allows it, this variable is subsequently substituted by a complex variable [x i ] k , that is, It should be mentioned that the value α = 1 in (18), is taken to avoid the discontinuity that is generated when using the fractional derivative of constants in the value x = 0. Furthermore, since in the previous method generated by the iterative method (15) has an order of convergence (at least) linear.

Some Examples
Instructions for implementing the method (15) along with information to provide values α ∈ [−2, 2] \ Z are found in the reference [18]. On the other hand, it should be mentioned that the method (15) may be implemented through recursive programming in a way analogous to that presented in the reference [21]. For rounding reasons, only for the examples the following function is defined Combining the function (20) with the method (15), the following iterative method is defined where γ is the Euler-Mascheroni constant [22]. Then considering the value k = 50, the initial condition x 0 = 0.018 is chosen to use the iterative method given by (21) along with fractional derivative given by (12). Consequently, we obtain the results of the Then considering the value k = 50, the initial condition x 0 = 1.85 is chosen to use the iterative method given by (21) along with fractional derivative given by (12). Consequently, we obtain the results of the Then the initial condition x 0 = (0.86, 0.86) T is chosen to use the iterative method given by (21) along with fractional derivative given by (12). Consequently, we obtain the results of the  Table 3. Results obtained using the iterative method (21) with = e − 3.

Approximation to zeros of the Riemann zeta function
A systematic study of the Riemann zeta function is beyond the intent of this paper. However, the basic information necessary to approximate their zeros using the iterative method (21) will be presented. Let ζ : Ω ⊂ C \ {1} → C be the Riemann zeta function with Ω = {x ∈ C : Re (x) > 1}. The function ζ is defined as follows the previous expression may be extended for all x ∈ C \ {1} by analytic continuation. With which it is possible to obtain the following functional equation On the other hand, there exists a series version of the Riemann zeta function, which has the characteristic of being globally convergent for all x ∈ C \ {1}. This version of the function ζ was conjectured by Konrad Knopp and proved by Helmut Hasse in 1930 [23], and it is given by the following expression It is necessary to mention that the expression (24) is very useful, because it allows us to make numerical approximations of the nontrivial zeros of the Riemann zeta function.
Then considering the value k = 50, the initial condition x 0 = 0.5 + 31.51i is chosen to use the iterative method given by (21) along with fractional derivative given by (12). Consequently, we obtain the results of the The comments below are made under the assumption that the Riemann hypothesis given in equation (3) is true. Table 4 shows certain x n values with |Re (x n ) − 0.5| ≥ 0.1, this is partly a consequence of the approximation of the function ζ by means of the function f k , because in general the following condition is true 316 An Approximation to Zeros of the Riemann Zeta Function Using Fractional Calculus ker (f k ) = ker (ζ) ⇔ k < ∞.
In addition, it is necessary to mention that the use of iterative methods does not guarantee that we can get as near as we want to the value ξ, normally it is only possible to determine a value x n near to the value ξ given by the following expression considering (25), it is necessary to give a definition that allows us to characterize the behavior of a function with respect to the values x n in B(ξ; δ ξ ).
Definition 5.2. Let f : Ω ⊂ R n → R n be a function with a value ξ ∈ Ω such that f (ξ) = 0. If when doing ξ → ξ + δ ξ , with δ ξ < 1, it holds that then the function f is (locally) stable with respect to the value ξ in B(ξ, δ ξ ).
The condition (26), implies that for a function f to be (locally) stable, it is necessary that a slight perturbation δ ξ in its zeros does not generate a great perturbation δ f in its images. To try to observe the stability of function ζ, we may consider δ = 10 −12 and the following values By performing multiple examples, it is possible to show for the trivial zeros of function ζ the validity of the following affirmation: The previous affirmation implies that it is complicated to make numerical approximations to trivial zeros of ζ by iterative methods using an initial condition x 0 = −2x with x ∈ N when x → ∞. Furthermore, since the behavior of the nontrivial zeros of function ζ is not completely determined, which is one of the many reasons for which the Riemann hypothesis remains an unsolved problem [23,24,25], it is possible to generate the following question: If ξ = 0.5 + xi with x ∈ R ⇒ Is ζ (locally) unstable in ξ if |x| → ∞?

Conclusions
Due to the fact that the derivatives of constant functions are identically zero functions in conventional calculus, it is difficult to imagine them as tools that may be used to approximate the zeros of more complicated functions. However, in the fractional calculus, the derivatives of constant functions may be used to approximate the zeros of more complicated functions and even nondifferentiables. It is necessary to mention that constant functions are the easiest functions to work with in fractional calculus, since the fractional derivatives of more complicated functions are expressed in many occasions in terms of Mittag-Leffler functions, hypergeometric functions or infinite series. In this paper, an approximation to the zeros of the Riemann zeta function has been obtained for the first time using fractional derivatives of constant functions.
Since the fractional derivative of constants are not always zero. 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. Then, it is an ideal iterative method to work with functions that can be expressed in terms of a series, as is the case of the Riemann zeta function. The method does not explicitly depend on the fractional derivative of the function for which zeros are sought, then it is an ideal iterative method for solving nonlinear systems in several variables.
The fractional iterative methods are efficient at finding multiple zeros of a function using a single initial condition, in addition to having the particularity of finding complex zeros of polynomials using real initial conditions. For this reason, they are iterative methods suitable for working with functions that have a large number of zeros, as is the case of the Riemann zeta function.