Evaluating Fractional Derivatives of the Riemann Zeta Function

Abstract

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

Introduction

The Riemann zeta function and its derivatives are defined by

1

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

2

where is the Euler constant and for are the Stieltjes constants.

Unlike itself, the functions 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 . Speiser [16] was the first to show, in 1934, that the Riemann Hypothesis is equivalent to the fact that has no zeros with . Spira [17] noticed that the zeros of and seem to come in pairs, where a zero of is located to the right of a zero of . 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 and for large k and m on the right half plane. Furthermore in [3] we have observed an interesting behavior of the zeros of on the left half plane, namely they seem to lie on curves which are extensions of chains of zeros of that were observed on the right half plane. Also some of the zeros of 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 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 .

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 of radius one about generalizes to fractional derivatives.

In [14] we present generalizations of the zero free regions of integral derivatives of 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.

Grünwald-Letnikov Fractional Derivatives of

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

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

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

where and . The reverse Grünwald-Letnikov derivative of a function f(z) is now defined as:

3

whenever the limit exists.

Defined this way, the fractional derivatives coincides with the integral derivatives for all . Furthermore, they satisfy and , for all . For we have that and for we have . So for with and we have as the generalization of (1) that

4

We have already used the generalization of (2) to the fractional domain in our proof [4] of a conjecture of Kreminski [10]. For and we have

where the are the fractional Stieltjes constants. Because of the branch cut of the complex logarithm there is a discontinuity along for . On the fractional derivative is analytic. As a direct consequence we obtain the following useful property:

Proposition 1

Let be a positive real number.

1. If and then is non-real.

2. For we have

While this establishes symmetry for the location of the zeros in , 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.

Evaluating

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

where , , denotes the k-th Bernoulli number, and is the periodic Bernoulli polynomial. If g(x) decreases rapidly enough for , then

5

We now use this to approximate where with . Let . Then converges for . 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) can be written in terms of the Upper Incomplete Gamma function (compare [6, p. 346] and [1, 6.5.3]):

For the third term we get:

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

6

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

Writing and

we obtain

The error term converges for and .

For all we can choose and such that becomes arbitrarily small. We can thus approximate as

where the error is .

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.

Fig. 1.

Zeros with of the fractional derivatives of on the left half plane. For zeros of are labeled with k. Not all zeros on the real axis are shown. The values for are 1/100 apart. For details about see Fig. 2.

Fig. 2.

Selected zeros of the fractional derivatives of on the upper left half plane. For zeros of are labeled with k. The values for are 1/100 apart.

Exploring the Left Half Plane

With our implementation of the approximation to , 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 for on the real axis . 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 for has only finitely many non-real zeros on the left half plane. Taking derivatives of the Laurent series expansion (2) of one immediately sees that the order of the pole of the k-th derivative of is . Thus the argument of on a curve around whose interior does not contain any zeros cycles through all of exactly times. Each of these cycles “spawns” at most 2 zeros of . If those zeros were evenly distributed, there would be at most 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:

Table 1.

The number N of pairs of non-real zeros of for . Levinson and Montgomery [13, Theorem 9], Yıldırım [19, Theorems 2 and 3]. The values for are experimental.

k	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
	0	1	1	2	2	3	3	4	4	5	5	6	6	7	7	8	8
N	0				2	3	3	3	4	4	4	4	4	5	5	5	6

Conjecture 1

Let . The number of pairs of non-real zeros of with is at most .

Fig. 2 shows that this is not the case for fractional derivatives.