The Riemann hypothesis illuminated by the Newton flow of ζ^☆

The flow lines of the continuous Newton method [5] for a given function are lines where the phase of this function is constant. Curves where ζ assumes real values are a special class of these flow lines and have been studied in great detail in the Doktorarbeit by Albert Arnold Utzinger [6] in 1934, and more recently by Hugh Lowell Montgomery and John Griggs Thompson [7]. They are at the very heart of an equivalent formulation of the Riemann hypothesis put forward by Andreas Speiser [8]: all non-trivial zeros of the first derivative are to the right of the critical line.

Curves in the complex plane where ζ is purely real or purely imaginary have been analyzed over a large range of arguments of ζ in [9]. Since the emerging picture resembles the x-ray photograph of the skeleton of a body this work carries the suggestive title 'X-ray of Riemann zeta function'. In our article we go further and find all flow lines of ζ.

We emphasize that our approach is also different from [10] which discusses the holomorphic rather than the Newton flow of ζ. Indeed, the continuous Newton method distinguishes [5] between the zeros of the function which are sinks and the poles which are sources of the flow. In contrast, in the holomorphic flow the zeros can be sinks or sources.

It is also worth mentioning that [11] introduces a family of parametrized zeta functions and studies the flow of the corresponding zeros on the Riemann sphere as the functions tend towards ζ. In this parameter flow the zeros approach the equator. In contrast, we focus exclusively on the Riemann zeta function and the generated Newton flow is associated with every point of the complex plane. In the present article the zeros are stationary and the flow lines terminate in them.

The important role of the Riemann zeta function in number theory [12] and, in particular, in the distribution of primes is well known. For this reason ζ has been the topic of extensive research [13–15] exemplified by the celebrated asymptotic formulae of Carl-Ludwig Siegel [16], and Michael Victor Berry and John Peter Keating [17]. They suggest that the Riemann hypothesis is indeed correct. The strongest support of this assertion is the work of Xavier Gourdon and Patrick Demichel [18] who have numerically verified that the first 10¹³ zeros of ζ are on the critical line.

We emphasize that ζ appears not only at the interface of number theory and quantum physics such as the factorization of numbers with Bose–Einstein condensates [19] but is prevalent in physics [10, 20]. To illustrate this point we list only a few physical phenomena where ζ manifests itself.

For example, $\zeta (4)={\pi }^{4}/90$ is hidden in the constant of the Stefan–Boltzmann law [21] of black-body radiation connecting the energy density and the fourth power of the temperature. Indeed, the numerical factor emerging from the integration of the Planck law over frequency contains this value.

We emphasize that there also exist interesting connections between the distribution of non-trivial zeros of ζ and random matrix theory [22, 23], or the eigenvalue distribution of Hermitian matrices [24, 25]. Moreover, the entanglement [26] of quantum systems [27, 28] is a necessary resource to construct [29] a quantum simulator for ζ. We also note the appearance of ζ in the partition function of a Bose–Einstein condensate [30–32], and in radiation patterns [33] as first observed by Balthasar van der Pol [34]. This way of creating ζ is also closely related to the quantum Mellin transform [35].

Our article is organized as follows: in section 2 we briefly summarize the essential ingredients of the continuous Newton method and apply it in section 3 to the Riemann zeta function in the complex plane. Here we employ the flow lines corresponding to phases which are integer multiples of π to rederive the Riemann–von Mangoldt formula, equation (1). Section 4 contains our discussion of the Newton flow of ζ on the Riemann sphere. We conclude in section 5 by summarizing our results, indicating a different approach towards the Riemann hypothesis and analyzing the Newton flow of a function which resembles ζ but does not satisfy the Riemann conjecture.

In the appendix we compare and contrast the Newton flows of ζ and the sine function. In particular, we use the asymptotic properties of the flow lines to show that the zeros of sine are on the real axis. This comparison serves us as our motivation that indeed the Newton flow of ζ can provide insight into the Riemann hypothesis.

Figure 1 represents the Newton flow of ζ defined by equations (6)–(8) in the complex plane, and discussed in more detail in [4]. We recognize the role of the pole at s = 1, indicated by a black dot, as a source and the first trivial zero at s = −2, depicted by a red dot, as a sink. Moreover, the flow converges towards the trivial zeros at s = −2k, with $k=1,2,...$ on the real axis, or the non-trivial zeros along the critical axis $\sigma =1/2,$ represented by an orange line.

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However, the main players of figure 1 are the separatrices which direct the flow into the individual basins of attractions, that is the zeros of ζ. We identify three classes of separatrices: (i) the first one contains those that go through a trivial zero of ζ', that is a hyperbolic point on the negative real axis, and terminate at a trivial zero of ζ with $\sigma \leqslant -4.$ (ii) The second class consists of those that run from the left to the right of figure 1 and then return from there to terminate at a non-trivial zero. We will see that the separatrix emerging from the pole at s = 1 and continuing through the first trivial zero of ζ' at $s^{\prime} \cong -2.72$ to the two trivial zeros of ζ at −2 and −4 is a mixture of both classes. (iii) The third class is defined by the separatrices that start on the left and go through a non-trivial zero of ζ' to conclude in a non-trivial zero of ζ.

Since all separatrices are connected to hyperbolic points we suspect from the separatrices of type (ii) that ζ' vanishes for $\sigma \to +\infty .$ Indeed, the two lowest terms

$$\begin{eqnarray}&&\zeta (s)\cong 1+{{{\rm{e}}}^{}}^{-s\;\mathrm{ln}2}\end{eqnarray} \tag{ 9 }$$

of the Dirichlet series, equation (6), yield

$$\begin{eqnarray}&&{\zeta }^{\prime }\cong -\mathrm{ln}2\;{{{\rm{e}}}^{}}^{-s\;\mathrm{ln}2},\end{eqnarray} \tag{ 10 }$$

which vanishes in the limit of $\sigma \to +\infty$ at the imaginary parts

$$\begin{eqnarray}&&{\tau }_{k}^{\prime }\equiv k\;\displaystyle \frac{\pi }{\mathrm{ln}2},\end{eqnarray} \tag{ 11 }$$

with $k=\ldots ,-2,-1,0,1,2,....$

Moreover, from the asymptotic expansion, equation (9), we find

$$\begin{eqnarray}&&\zeta (\sigma +{\rm{i}}{\tau }_{k}^{\prime })\cong 1+{(-1)}^{k}\;{{{\rm{e}}}^{}}^{-\sigma \;\mathrm{ln}2},\end{eqnarray} \tag{ 12 }$$

and as σ approaches infinity, ζ tends to unity. Hence, at the hyperbolic points at infinity ζ is real and identical to unity.

Since the flow lines are lines of constant phase, ζ is real and positive along these separatrices. In particular, the part of the separatrix which is associated with the imaginary part ${\tau }_{0}^{\prime }\equiv 0$ is the real axis to the right of the pole at s = 1.

The consecutive curves corresponding to even values of k are the parts of the separatrices of class (ii) which run from the left to the right of figure 1. Along all of these lines ζ is larger than unity.

However, for odd values of k the separatrices 'start' at $\sigma \to +\infty$ and approach a non-trivial zero of ζ. Along these lines the value of ζ decays from unity to zero.

Moreover, we find that $k=\pm 1$ is special. Indeed, the lines emerging from ${\tau }_{\pm 1}^{\prime }=\pm \pi /\mathrm{ln}2$ meet each other in the first trivial zero of ζ' at $s^{\prime} \cong -2.72.$

Therefore, all flow lines starting at the pole s = 1 are directed to the first trivial zero of ζ at s = −2, whereas the source of all other flow lines is $\sigma \to -\infty .$

In order to bring out these features of the Newton flow most clearly we display in figure 2 the Riemann zeta function in a rather unusual way in the complex plane. Indeed, we show the function

$$\begin{eqnarray}{\mathcal{Z}}(s)\equiv \left\{\begin{array}{lll}| \zeta (s)| & \mathrm{for} & 0\leqslant \mathrm{Re}\;\zeta \\ -| \zeta (s)| & \mathrm{for} & \mathrm{Re}\;\zeta \leqslant 0,\end{array}\right.\end{eqnarray} \tag{ 13 }$$

that is we depict $| \zeta (s)|$ for all arguments s where $\mathrm{Re}\;\zeta$ is positive, but plot $-| \zeta (s)|$ for all s where $\mathrm{Re}\;\zeta$ is negative.

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Along the green flow lines ζ is real. In the foreground of figure 2 they run most of the time in the plane. An exception is the line along the real axis which descends from the pole at s = 1.

However, for larger values of τ the green lines undergo an infinite change in altitude. They come from the left where ζ is positive infinite, run down the hills to the plains and turn around at values of s where σ is positive infinite. They then continue their descent till they reach a non-trivial zero.

The green lines of figure 2 which represent separatrices play also a prominent role in the analysis of [7] where they are called ${{\mathcal{R}}}_{k}$ and defined for even k as the 'connected component on which $\zeta (s)$ is positive real with asymptote $t=k\pi /(2\mathrm{log}2)$ as $\sigma \to +\infty .$' This asymptote corresponds to ${\tau }_{k}^{\prime }.$

In addition, the lines of constant height ${{\mathcal{C}}}_{k}\equiv | \zeta | =1$ are crucial. Indeed, according to [7] they are defined for odd k as 'the connected component on which $| \zeta (s)| =1$ with asymptote $t=k\pi /(2\mathrm{log}2)$ as $\sigma \to +\infty$'. Hence, these lines run in between two consecutive separatrices that cross the complex plane. Since the Newton flow consists of lines of constant phase the flow lines are orthogonal to the lines of ${{\mathcal{C}}}_{k}.$

In order to connect our work with that of [7] we show in figure 3 the Newton flow in the neighborhood of the first three non-trivial zeros together with the curves ${{\mathcal{R}}}_{0}\;...\;{{\mathcal{R}}}_{12}$ and ${{\mathcal{C}}}_{1}\;...\;{{\mathcal{C}}}_{11}.$

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All flow lines end in a zero of the function. Hence, at the non-trivial zeros of ζ a trajectory with the phase given by an integer multiple of π terminates which is the continuation of the separatrix coming from the right. We now address the question if we can identify this line in the representation of ζ for $\sigma \to -\infty .$

From the analytic continuation, equation (7), of ζ we find

$$\begin{eqnarray}&&\zeta (-\sigma +{\rm{i}}\tau )=\chi (-\sigma +{\rm{i}}\tau )\zeta (1+\sigma -{\rm{i}}\tau ),\end{eqnarray} \tag{ 14 }$$

that is

$$\begin{eqnarray}&&\zeta (-\sigma +{\rm{i}}\tau )\cong \chi (-\sigma +{\rm{i}}\tau ),\end{eqnarray} \tag{ 15 }$$

where in the last step we have used the asymptotic value $\zeta (\sigma +{\rm{i}}\tau )\cong 1$ following from equation (9) for $\sigma \to \infty .$

With the help of the asymptotic expansion [2]

$$\begin{eqnarray}&&\chi (s)\cong {\left(\displaystyle \frac{2\pi }{\tau }\right)}^{\sigma +{\rm{i}}\tau -\frac{1}{2}}\;{{{\rm{e}}}^{}}^{{\rm{i}}(\tau +\frac{\pi }{4})}\end{eqnarray} \tag{ 16 }$$

valid in any fixed strip $\alpha \leqslant \sigma \leqslant \beta$ for $\tau \to \infty$ we find from equation (15) the expression

$$\begin{eqnarray}&&\zeta (-\sigma +{\rm{i}}\tau )\cong \mathrm{exp}\left[\left(\sigma +\displaystyle \frac{1}{2}\right)\mathrm{ln}\displaystyle \frac{\tau }{2\pi }\right]\\ &&\quad \times \mathrm{exp}\left\{-{\rm{i}}\left[\tau \left(\mathrm{ln}\displaystyle \frac{\tau }{2\pi }-1\right)-\displaystyle \frac{\pi }{4}\right]\right\}.\end{eqnarray} \tag{ 17 }$$

Hence, for a fixed but large negative value of σ the condition

$$\begin{eqnarray}&&{\tau }_{j}\left(\mathrm{ln}\displaystyle \frac{{\tau }_{j}}{2\pi }-1\right)-\displaystyle \frac{\pi }{4}=j\;\pi \end{eqnarray} \tag{ 18 }$$

for an integer j yields values of ${\tau }_{j}$ where ζ is real. However, at this point we have to recall that the expansion equation (16) is only valid for $1\ll \tau .$ Hence, the condition equation (18) only holds true starting from appropriately large values of j.

Some of the even values of j correspond to the separatrices running from the left to the right side of figure 1 and some of them end in a non-trivial zero.

However, all odd values of j lead directly to a non-trivial zero. As a result, we can count the non-trivial zeros when we cast equation (18) for $j\equiv 2k+1$ into the form

$$\begin{eqnarray}&&k=\displaystyle \frac{{\tau }_{2k+1}}{2\pi }\left(\mathrm{ln}\displaystyle \frac{{\tau }_{2k+1}}{2\pi }-1\right)-\displaystyle \frac{5}{8}.\end{eqnarray} \tag{ 19 }$$

Hence, the number N of non-trivial zeros with imaginary parts smaller than T is given by ${\tau }_{2k+1}=T$ which yields

$$\begin{eqnarray}&&N\cong \displaystyle \frac{T}{2\pi }\left(\mathrm{ln}\displaystyle \frac{T}{2\pi }-1\right)-\displaystyle \frac{5}{8},\end{eqnarray} \tag{ 20 }$$

in complete agreement with the Riemann–von Mangoldt formula, equation (1).

We conclude this section by noting that the Newton flow of ζ provides us not only with some qualitative insight into the function, but also has a predictive power. Indeed, it allow us to rederive the asymptotic expression, equation (1), of the number of zeros. A similar calculation based on the lines where ζ is real can also be found in [9].

In figure 4 we show the Newton flow of ζ on the Riemann sphere in the domain governed by the pole and the first trivial zero of ζ. In order to focus on the essential parts, we have omitted close to the North pole most of the field lines creating a bold top.

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The point ${\vec{x}}_{p}\equiv (4/5,0,2/5)$ corresponding to the pole of ζ at s = 1, marked in the right picture by a black dot, is the source of all flow lines which terminate in the first trivial zero s = −2, represented by a red dot in the left picture. Here, we have also indicated by red dots the other trivial zeros.

The pole and the trivial zeros lie on a great circle, indicated in the pictures by the violet line. This circle is the result of the mapping equation (21) of the real axis of the complex plane onto the Riemann sphere. In analogy to the globe we call the image of the positive real axis with the pole of ζ on it the zero-degree, or the Greenwich meridian.

The opposite meridian, which is the image of the negative real axis, contains all the trivial zeros as also apparent from figure 5. Here, we present a view from the top almost along the vertical axis through the North pole. This figure also brings out the fact that the symmetry of the flow with respect to the real axis, mentioned in connection with figure 1, translates into a symmetry with respect to this great circle.

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The two separatrices circumventing in figure 1 the pole from the right and approaching the first two trivial zeros through the first trivial zero of ζ', depicted there by the green solid lines start on the sphere from the North pole, almost approach the equator in the first trivial zero of ζ' as shown on the left picture of figure 4 and then annihilate in the first two trivial zeros.

This line is the green cardioid-like curve of figure 5 confining the flow to the North pole. Indeed, this trajectory separates the flow lines emerging from the pole of ζ and dominating the behavior in the Southern hemisphere from those coming out of the North pole of the globe governing the Northern hemisphere.

The trefoil structure shown in figure 5 which is reminiscent of the club card, rotated here by 90°, is a consequence of the interplay between the trivial zeros along the 180° meridian and the non-trivial zeros located along a path (orange line) orthogonal to it. According to figure 1 they are separated from each other by a separatrix which runs from minus infinity to plus infinity. This separatrix translates into the next inner green curve located close to the North pole and defines the transverse structure of the club. Its top and the bottom parts are given by the approach of the trivial zeros and the excursion of the flow lines from the pole of ζ on the Southern hemisphere towards the North pole, respectively.

Since both types of zeros tend towards infinity the behavior on the North pole gets rather crowded. In figure 6 we magnify the Newton flow on the Riemann sphere along the positive critical line.

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From section 2 we recall that for a hyperbolic point corresponding to a simple zero of F' we have the crossing of two separatrices. To the right of the critical line there are infinitely many separatrices that enter the North pole reflecting the fact that we have infinitely many hyperbolic points at infinity characterized by the imaginary parts ${\tau }_{2j}^{\prime }$ defined by equation (11).

However, there are also infinitely many separatrices that leave the North pole. They result from the points in infinity with the imaginary parts ${\tau }_{2j+1}^{\prime }.$ The deeper reason for the crossing of infinitely many separatrices at the North pole is the fact that here we do not deal with a polynomial but an exponential function presented in equation (10).

A separatrix approaching the North pole is always followed by an emerging one and we can interpret the two as one and the same trajectory especially since the value of ζ at the North pole along all of these separatrices is unity. However, this trajectory has a kink at the North pole corresponding to the different imaginary parts ${\tau }_{2j+1}^{\prime }$ and ${\tau }_{2j}^{\prime }.$

In contrast, the separatrices on the left of the critical line all start from the North pole. On the ones where ζ is positive ζ tends towards plus infinity as we reverse the path and approach the North pole. These separatrices are the ones which enter the North pole from the right. They then leave the North pole to the right and are met in the non-trivial zero by flow lines leaving the North pole to the left. On one of them (blue line) ζ is negative and tends to minus infinity as we backtrack the flow line towards the North pole.

This behavior is interrupted at the second non-trivial zero of ζ. Here a hyperbolic point appears slightly to the right of the critical line as shown in figure 6 and two separatrices that both emerge to the left of the North pole cross each other. Along these lines ζ is not real but given by the phase of ζ at the hyperbolic point. This phenomenon repeats itself infinitely many times as we consider non-trivial zeros with ever increasing imaginary parts.

So far we have concentrated on the very special class of flow lines emerging from the left of the North pole where ζ is real and the phase is either zero or π. However, there are infinitely many flow lines covering continuously all phases from zero to $2\pi .$ Hence, along these lines ζ assumes all values as we backtrack them towards the North pole. Since there are infinitely many zeros this continuous change of the phase repeats itself infinitely many times. The resulting rapid oscillations close to the North pole are in complete accordance with the theorem of Picard [39] and reflect the fact that the North pole is an essential singularity.

As the critical line approaches the North pole it becomes a tangent to the 90° meridian. Moreover, the non-trivial zeros and the associated separatrices and lines where ζ is negative accumulate at the North pole. As the imaginary parts of the non-trivial zeros increase these lines now cover an ever smaller domain of the polar cap. This localization is a consequence of the stereographic projection defined by equation (21).

There is one more feature that adds to the complexity of the flow close to the North pole. As the imaginary part τ grows, that is as we approach the polar cap, the number of zeros caught between two separatrices running from the left to the right increases. This property is not apparent from figures 1 and 4 but is clearly visible in figure 10 of [4]. As a result, more and more zeros get cramped in less and less space and it seems that the complexity of the Newton flow gets intractable.

We are therefore wondering if the use of the Riemann sphere rather than the familiar complex plane can offer an advantage in the discussion of the Riemann hypothesis.

Indeed, in the Newton flow of ζ in the complex plane displayed in figure 1 these lines range in their real parts from minus infinity to plus infinity and the asymptotic formulae equations (9) and (17) are only valid at infinity and not in the critical strip. However, after the stereographic projection of ζ we might be able to find elementary expressions for the Newton flow lines close to the North pole which are valid along the complete line. Such an approach might even help to unravel the accumulation of the zeros predicted by the Riemann–von Mangoldt formula, equation (1). Unfortunately this problem goes beyond the scope of the present paper.

We start our discussion by first analyzing the asymptotics of the Newton flow of the sine function. Here we concentrate on two aspects: (i) the starting point at large positive and large negative imaginary parts of the argument $s\equiv \sigma +{\rm{i}}\tau$ with vertical tangent, and (ii) the distribution of phases.

In the center of figure A1 we display the numerically obtained Newton flow

$$\begin{eqnarray}&&\dot{s}(t)=-\displaystyle \frac{{\mathcal{S}}(s(t))}{{\mathcal{S}}^{\prime} (s(t))}\end{eqnarray} \tag{ A.1 }$$

created by the sine function

$$\begin{eqnarray}&&{\mathcal{S}}(s)\equiv \mathrm{sin}s\equiv \displaystyle \frac{1}{2{\rm{i}}}\left({{{\rm{e}}}^{}}^{{\rm{i}}s}-{{{\rm{e}}}^{}}^{-{\rm{i}}s}\right)\end{eqnarray} \tag{ A.2 }$$

and its first derivative

$$\begin{eqnarray}&&{\mathcal{S}}^{\prime} (s)=\displaystyle \frac{1}{2}\left({{{\rm{e}}}^{}}^{{\rm{i}}s}+{{{\rm{e}}}^{}}^{-{\rm{i}}s}\right)\end{eqnarray} \tag{ A.3 }$$

leading to

$$\begin{eqnarray}&&\dot{s}(t)={\rm{i}}\;\displaystyle \frac{{{{\rm{e}}}^{}}^{{\rm{i}}s}-{{{\rm{e}}}^{}}^{-{\rm{i}}s}}{{{{\rm{e}}}^{}}^{{\rm{i}}s}+{{{\rm{e}}}^{}}^{-{\rm{i}}s}}.\end{eqnarray} \tag{ A.4 }$$

Due to the periodicity of ${\mathcal{S}}$ with period $2\pi$ we confine ourselves in figure A1 to the interval $-\pi /2\leqslant \sigma \leqslant 3\pi /2.$

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The flow originates exclusively from the top and the bottom of the figure and approaches the zeros which are indeed on the real axis. We emphasize that this figure is of course no proof but only serves as a guide for developing a proof. For this purpose we now verify analytically several features of this flow.

We start by noting that for $1\ll \tau$ the flow, equation (A.4), reduces to the relation

$$\begin{eqnarray*}&&\dot{s}(t)\cong -{\rm{i}}={{{\rm{e}}}^{}}^{-{\rm{i}}\pi /2},\end{eqnarray*}$$

which shows that for large positive imaginary parts τ of s all flow lines, independent of the real part σ, have a vertical tangent and start towards the real axis.

Likewise, for large negative imaginary parts, that is for $1\ll -\tau ,$ we obtain

$$\begin{eqnarray*}&&\dot{s}(t)\cong {\rm{i}}={{{\rm{e}}}^{}}^{{\rm{i}}\pi /2},\end{eqnarray*}$$

which indicates that independent of the real part σ the field lines have again a vertical tangent pointing towards the real axis.

We conclude this section by considering the phases of these flow lines. From the definition, equation (A.2), of ${\mathcal{S}}$ we obtain immediately the asymptotic representation

$$\begin{eqnarray}{\mathcal{S}}(s)\cong \left\{\begin{array}{lll}\displaystyle \frac{1}{2}\mathrm{exp}\left[{\rm{i}}\left(\displaystyle \frac{\pi }{2}-\sigma \right)\right]{{{\rm{e}}}^{}}^{\tau } & \mathrm{for} & 1\ll \tau \\ \displaystyle \frac{1}{2}\mathrm{exp}\left[{\rm{i}}\left(\sigma -\displaystyle \frac{\pi }{2}\right)\right]{{{\rm{e}}}^{}}^{| \tau | } & \mathrm{for} & 1\ll -\tau .\end{array}\right.\end{eqnarray} \tag{ A.5 }$$

Hence, for the flow lines initiating from the top of the complex plane the phase decreases linearly as we increase τ, whereas for the ones originating from the bottom the phase increases linearly. This behavior of $\mathrm{arg}(\mathrm{sin}s)$ is apparent in the top and the bottom parts of figure A1, respectively.

Next we turn to the phases of the separatrices which are determined by the phases of ${\mathcal{S}}$ at the points ${s}_{j}^{\prime }$ in the complex plane where ${\mathcal{S}}^{\prime}$ given by (A.3) vanishes. From the well-known trigonometric relation

$$\begin{eqnarray}&&{{\mathcal{S}}}^{2}+{\mathcal{S}}{^{\prime} }^{2}=1\end{eqnarray} \tag{ A.6 }$$

following from equations (A.2) and (A.3) we find with ${\mathcal{S}}^{\prime} ({s}_{j}^{\prime })=0$ the condition

$$\begin{eqnarray*}&&{\mathcal{S}}{({s}_{j}^{\prime })}^{2}=1,\end{eqnarray*}$$

or

$$\begin{eqnarray*}&&{\mathcal{S}}({s}_{j}^{\prime })={{{\rm{e}}}^{}}^{{\rm{i}}j\pi }.\end{eqnarray*}$$

Hence, the phases of the separatrices are integer multiples j of π.

We emphasize that at this point we do not know the locations ${s}_{j}^{\prime }$ where ${\mathcal{S}}^{\prime}$ vanishes, but due to the trigonometric relation, equation (A.6), we could obtain the phases. Since they are constant along the whole flow line we can distinguish separatrices from regular flow lines by their phases at their starting points for large imaginary parts, that is at infinity.

Indeed, from the asymptotic representation, equation (A.5), of ${\mathcal{S}}$ we find that the phases $j\pi$ arise for

$$\begin{eqnarray*}&&{\sigma }_{j}\equiv (2j+1)\displaystyle \frac{\pi }{2},\end{eqnarray*}$$

as indicated in the top and the bottom of figure A1 by green triangles.

Hence, in the interval $-\pi /2\leqslant \sigma \leqslant \pi /2$ defined by two consecutive separatrices the phases of the flow lines originating from the top decrease linearly from π to zero. In contrast, the flow lines starting from the bottom increase linearly from $-\pi$ to zero. Hence, in this interval all flow lines cover a complete range of $2\pi$ which indicates that this domain must include a simple zero.

We are now in the position to complete our proof based on the Newton flow that all zeros s_j of ${\mathcal{S}}$ lie on the real axis. For this purpose we assume that there is one s₀ that does not satisfy this property. In this case s₀ has a non-vanishing imaginary part. However, the condition

$$\begin{eqnarray}&&\mathrm{sin}{s}_{0}=0,\end{eqnarray} \tag{ A.7 }$$

implies

$$\begin{eqnarray*}&&\mathrm{sin}{s}_{0}^{*}=0,\end{eqnarray*}$$

and thus the complex conjugate ${s}_{0}^{*}$ of s₀ is a zero as well.

Hence, we obtain two zeros, which is in contradiction with the prediction of a single zero due to the behavior of the Newton flow lines. As a consequence, the zero s₀ has to lie on the real axis.

We conclude by briefly comparing and contrasting the Newton flows of ζ and ${\mathcal{S}}$. Whereas in the introduction of this appendix we have emphasized the similarities, we now focus on the differences. Moreover, we identify the ingredient crucial for our proof of the distribution of zeros of ${\mathcal{S}}$.

Obviously, the zeros of ${\mathcal{S}}$ along the real axis play the role of the non-trivial zeros along the critical axis of ζ. However, due to the symmetry of ${\mathcal{S}}$ with respect to the real axis and the periodicity with a single period $2\pi$ there are no trivial zeros of ${\mathcal{S}}$.

Moreover, we emphasize that it is this symmetry that enforces the Newton flow of ${\mathcal{S}}$ to approach the zeros from the top and the bottom of the complex plane. In contrast, the flow of ζ exclusively comes from the left and either goes straight to the non-trivial zeros, or passes them and eventually gets turned around by the separatrices at infinity.

However, the most important difference between the Newton flows of ${\mathcal{S}}$ and ζ stands out in the asymptotic behavior of the phases and their grouping by the separatrices. Whereas in ${\mathcal{S}}$ these phases decrease or increase linearly, in ζ the growth is faster due to the presence of the logarithm in equation (17).

Moreover, the separation of the individual zeros by the separatrices is much more intricate in the case of ζ. Indeed, for ${\mathcal{S}}$ the phases of two consecutive separatrices always differ by π which is not the case for ζ as apparent from the second and third non-trivial zero of ζ depicted in figure 1 and amplified in figure A2 . Here the top green line consists of two separatrices that are rather close together: the one where ζ is real runs through the picture from the left to the right. The other one is slightly below and comes from the left but eventually turns around and approaches a zero of the derivative of ζ marked by a green triangle. The narrow path between these two separatrices leaves enough space for the flow in the light blue domain that needs to approach the second non-trivial zero from the right.

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Due to this flow the phases of the flow lines caught between the lower separatrix going through the picture and the one approaching the hyperbolic point indicated in figure A2 by a green triangle cannot cover a complete interval of $2\pi .$ Hence, in contrast to ${\mathcal{S}}$ where the flow domains in the complex plane are separated from each other, in ζ we find nested regions. For example, the domain (yellow region) for the third non-trivial zero is embedded in the region (light blue) of the second non-trivial zero as indicated in figure A2.

We conclude by emphasizing that the crucial element of our proof for ${\mathcal{S}}$ was the trigonometric relation, equation (A.6), which allowed us to determine the phases of the separatrices without knowing the locations of the points where the first derivative vanishes. Unfortunately, at the moment we are not aware of an analogous relation for ζ.