key: cord-0047188-4kkihzlb
authors: Farr, Ricky E.; Pauli, Sebastian; Saidak, Filip
title: Evaluating Fractional Derivatives of the Riemann Zeta Function
date: 2020-06-06
journal: Mathematical Software - ICMS 2020
DOI: 10.1007/978-3-030-52200-1_9
sha: e370934ad989e2557e7b776700fd6d3c61f065f9
doc_id: 47188
cord_uid: 4kkihzlb

We present a method for evaluating the reverse Grünwald-Letnikov fractional derivatives of the Riemann Zeta function [Formula: see text] and use it to explore the location of zeros of integral and fractional derivatives on the left half plane.

The Riemann zeta function ζ(s) and its derivatives ζ (k) (s) are defined by

everywhere in the half-plane (s) > 1. By a process of analytic continuation these functions can be extended to meromorphic functions with a single pole at s = 1. Moreover, ζ(s) has the Laurent series expansion:

where γ 0 is the Euler constant and for n ≥ 1 γ n are the Stieltjes constants. Unlike ζ(s) itself, the functions ζ (k) (s) have neither Euler products nor functional equations. Thus their nontrivial zeros do not lie on a line, but appear to be distributed seemingly at random with most zeros located to the right of the critical line σ = 1 2 . Speiser [16] was the first to show, in 1934, that the Riemann Hypothesis is equivalent to the fact that ζ (s) has no zeros with 0 < σ < 1 2 . Spira [17] noticed that the zeros of ζ (s) and ζ (s) seem to come in pairs, where a zero of ζ (s) is located to the right of a zero of ζ (s). More recently, with the help of extensive computations, Skorokhodov [15] observed this behavior for higher derivatives as well.

Our results from [2] support a straightforward one-to-one correspondence between the zeros of ζ (k) (s) and ζ (m) (s) for large k and m on the right half plane. Furthermore in [3] we have observed an interesting behavior of the zeros of ζ (k) (s) on the left half plane, namely they seem to lie on curves which are extensions of chains of zeros of ζ (k) (s) that were observed on the right half plane.

Also some of the zeros of ζ (k) (s) on the negative real axis appeared to be part the chains.

We are investigating this correspondence between the zeros of different derivatives by considering curves of zeros of fractional derivatives ζ (k) (s) that connect the zeros of integral derivatives. We have found that among the multitude of existing definitions of fractional derivatives, the reverse Grünwald-Letnikov fractional derivative works best for situations dealing with ζ(s).

In [4] we have applied it in a proof of a conjecture by Kreminski [10] and in [5] we have been able to apply some of the properties of the fractional Stieltjes constants to prove that the zero free region of ζ(s) of radius one about s = 1 generalizes to fractional derivatives.

In [14] we present generalizations of the zero free regions of integral derivatives of ζ(s) on the right half plane from [2] to fractional derivatives. This yields the existence of curves of zeros of fractional derivatives on the right half plane. Here we conduct numerical investigations of the zeros of fractional derivatives where we concentrate our attention to the left half plane.

The fractional derivative introduced by Grünwald [7] in 1867 was simplified both in approach and notation, by Letnikov in 1869 [11, 12] . For N ∈ N and h > 0, let

be the N -th finite difference of f . Then for all n ∈ N we have:

This can be naturally extended to the fractional case with the generalization of

where α ∈ C and α k =

whenever the limit exists. Defined this way, the fractional derivatives D α s [f (s)] coincides with the integral derivatives for all α ∈ N. Furthermore, they satisfy D 0

For c ∈ C we have that D α s [c] = 0 and for m = 0 we have D α s [e ms ] = m α e ms . So for s ∈ C with (s) > 1 and α > 0 we have as the generalization of (1) that

We have already used the generalization of (2) to the fractional domain in our proof [4] of a conjecture of Kreminski [10] . For 1 = s ∈ C and α > 0 we have

where the γ α are the fractional Stieltjes constants. Because of the branch cut of the complex logarithm there is a discontinuity along (−∞, 1] for α ∈ N. On C \ (−∞, 1] the fractional derivative is analytic. As a direct consequence we obtain the following useful property: Proposition 1. Let α be a positive real number.

. While this establishes symmetry for the location of the zeros D α σ [ζ(s)] in C, with respect to the real axis, the symmetry is not perfect. It only refers to the location, and not the actual mirroring of properties, or the dynamics surrounding the zeros. Nevertheless it asserts that chains of zeros can be observed on the upper as well as lower half plane.

One of the most effective ways for evaluating (4) and its analytic continuations to the regions where σ < 1 is Euler-Maclaurin summation. We use the following form of the summation formula: We now use this to approximate

converges for (s) > 1. We assume that v is even. We evaluate the first summand of (5) as is, namely as

The second term of the right hand side of (5) 

For the third term we get:

Now we determine a bound for the fourth term of (5). We denote the falling factorial by (α) i = Γ (α+1) Γ (α−i+1) and the Stirling numbers of the first kind by s(j, i). Let

be the the non-central Stirling numbers. The derivatives of g can be written as [8, Theorem 1] :

The error term E α s (m, v) converges for σ + v > 1 and m > 2. For all s ∈ C \ (∞, 1] we can choose m ∈ N and v ∈ N such that |E α s (m, v)| becomes arbitrarily small. We can thus approximate D α s [ζ(s)] as

where the error is |E α s (m, v)|. We have implemented the method described above in the computer algebra system SageMath [18] using the library mpmath [9] . A considerable increase in speed was obtained by caching the values of the non-central Stirling numbers, which we evaluate by their recurrence relation. Figures 1 and 2 were generated with our implementation.

With our implementation of the approximation to ζ (α) (s), see Sect. 3, we have investigated the distribution of the zeros on the left half plane. We observe, see Fig. 1 , that the zeros on the left half plane given in [3] appear to be connected in a similar manner as on the right half plane.

Furthermore they connect to zeros of integral derivatives on the negative real axis. Note that there is a discontinuity of ζ (α) (s) for α ∈ N ∪ {0} on the real axis σ < 1. We find different patterns how zeros of integral derivatives are connected, see Fig. 2 . Some of the curves start and stop at zeros of integral derivatives on the left real axis such as shown in the first plot in Fig. 2 . Further to the left we find curves touching (or crossing) the real line at integral derivative and jumping between those points, as shown in the second, third, and fourth plot in Fig. 2 .

Levinson and Montgomery [13] have shown that ζ (k) (s) for k ∈ N has only finitely many non-real zeros on the left half plane. Taking derivatives of the Laurent series expansion (2) of ζ(s) one immediately sees that the order of the pole of the k-th derivative of ζ(s) is k + 1. Thus the argument of ζ (k) (γ(t)) on a curve γ : [0, 2π) → C around s = 1 whose interior does not contain any zeros cycles through all of [0, 2π) exactly k + 1 times. Each of these cycles "spawns" at most 2 zeros of ζ (k) (s). If those zeros were evenly distributed, there would be at most k+1 2 such zeros in the upper left half plane. Experiments suggest that this is indeed an upper bound for the count of such zeros (see Table 1 ) and that these are the only non-real zeros on the upper left half plane (see Fig. 1 ). This leads us to conjecture: 

Let k ∈ N. The number of pairs of non-real zeros of ζ (k) (s) with σ ≤ 0 is at most k+1 2 . Fig. 2 shows that this is not the case for fractional derivatives. 

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More zeros of the derivatives of the riemann zeta function on the left half plane

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A zero free region for the fractional derivatives of the Riemann zeta function

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Zeros of the derivatives of the Riemann zetafunction

Zeros of fractional derivatives of the Riemann zeta function

Padé approximants and numerical analysis of the Riemann zeta function

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Zero-free regions of ζ (k) (s)

The Sage Developers: SageMath, the Sage Mathematics Software System

Zeros of ζ (s) & ζ (s) in σ 1