Riemann’s zeta function
and Newton’s method:
Numerical experiments from acomplex-dynamical viewpoint ∗Tomoki Kawahira
Nagoya University, Nagoya, JAPANURL: http://math.nagoya-u.ac.jp/˜kawahira

Abstract. We apply some root ﬁnding algorithms to characterize the zerosof Riemann’s zeta. Finally we propose the Dynamical Riemann Hypothesis
which state the Riemann Hypothesis in terms of dynamical systems.

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Riemann’s zeta and primes

For s = σ + it ∈ C, one can easily see that the series
−1∞ 1111− sζ(s) = 1 + s + s · · · =23pnn=1
converges if σ > 1, where pn is the nth prime number. Indeed, ζ(s) isanalytic on {Re s = σ > 1} and by analytic continuation we consider it ameromorphic function ζ : C → C̄ with only one pole at s = 1, which issimple.
The Riemann Hypothesis. The most famous conjecture on Riemann’s
zeta function is: ζ has non-real(non-trivial) zeros only on the critical line
∗Poster presented at Kanazawa, June 20-24, 2005. Revised on Aug 29 2006. After I presented thisposter, I learned that several people were working on this subject before my experiment. Aimo Hinkkanen
published a paper on some properties on relaxed Newton’s method for Riemann’s zeta. Dierk Schleicher
and his student generated a picture of λ several years ago. Nick Sullivan generated some pictures of ν in2002. Here I do not intend to present new pictures, but I suggest a possible direction of investigations of
the zero-free region in the critical stripe.

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Re s = σ = 1/2 (the Riemann Hypothesis). If this conjecture is aﬃrmative,
we will have a nice result on the distribution of prime numbers;
pn+1 − pn = O(p1/2n log pn ).
This is better than any known results, for example;
pn+1 − pn = O(p0.525+)nfor any  > 0. To show the hypothesis, it is known that we only have to care
the zeros on the critical stripe S = {s ∈ C : 0 < Re s < 1}. In particular,
wider zero-free regions imply better estimates of distribution of primes.
For example, it is known that there exists a constant A > 0 such thatAs = σ + it ∈ S : σ ≥ 1 −(log(|t| + 1))2/3(log log(|t| + 1))1/3
is zero-free.

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Newton’s method

There are some root ﬁnding algorithms, but the most famous one would be
Newton’s method. From now on, we work with complex variable z = x + yi
instead of conventional s for ζ.
For a meromorphic function f : C → C̄, we deﬁne its Newton’s map Nf
byf (z)Nf (z) = z −  ,f (z)which is again meromorphic. One can easily check that f (α) = 0 iﬀ
Nf (α) = α. The idea of Newton’s method is: Start with an initial value z0
suﬃciently close to α. Then the sequence {zn } deﬁned by zn+1 = Nf (zn )converges (rapidly) to α.
More precisely, we have the following property:
If α is a simple zero of f , then Nf (α) = α and Nf (α) = 0. Thus
Nf (z) − α = O((z − α)2)(z → α).

If α is a multiple zero, then Nf (α) = α and |Nf (α)| < 1. Thus
|Nf (z) − α| ≤ C|z − α|2(z → α)for some 0 < C < 1.
Hence the precision of zn as an approximate value of α is exponentially or
linearly increasing according to the multiplicity of α.
Newton’s method as a dynamical systems. What makes this method
more intriguing is the theory of iteration of holomorphic function developed
by Fatou and Julia in early 1920s. For given z0 ∈ C, convergence of zn =Nfn (z0 ) (where Nfn is nth iteration of Nf ) is not guaranteed in general. To
investigate the behaver of such sequence, we consider the global dynamicalsystems
NfNfNfC̄ −→ C̄ −→ C̄ −→ · · ·given by iteration of Newton’s map. (As we will see, we need a spacial carefor poles of Nf .) For example, set f (z) := z 3 − 1. Then the iteration of its
Newton’s map gives the following picture (Figure 1):

Figure 1: Dynamics of Nf for f (z) = z 3 − 1.

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Blue, yellow, and green regions are the set of initial values z0 such√√that the orbit zn = Nfn (z0 ) converges to 1, −1+2 3i , and −1−2 3i respectively.
Shades distinguish the number of iteration to trap the orbit in small disks
around roots. The boundary of these regions has complicated structure
known as fractal. It is the Julia set of Nf , where the dynamics shows
chaotic behavior. In particular, orbits from the Julia set stay within theJulia set and never converge to the roots.
Newton’s method for meromorphic functions. If f is a rational function, then so is Nf thus it has no essential singularity. For a meromorphic
function f , its Newton’s map has an essential singularity at inﬁnity. Since
Nf (∞) is indeterminate, we must stop the iteration when the orbit landson a pole of Nf . In this particular setting, we deﬁne its Fatou set F (Nf )by:z0 ∈ F (Nf )⇐⇒ ∃U a nbd of z0 s.t.


Nfn |U n≥0 is deﬁned and a normal family

The Julia set J(Nf ) is the complement C − F (Nf ).

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Applying the method to zeta.

Now let us apply Newton’s method to Riemann’s zeta. For the meromorphic function ζ : C → C̄, we set
ν(z) := Nζ (z) = z −ζ(z).
ζ  (z)We also apply the method to the functions
η(z) := (z − 1)ζ(z)and

1z(1 − z)π z/2 Γ(z/2)ζ(z),2where ξ(z) a classical zeta-related function with symmetry ξ(z) = ξ(1 − z).
ξ(z) =Since η(z) and ξ(z) are entire functions, we may expect better dynamics
for
ξ(z)
η(z)
and
λ(z) := z −  .
µ(z) := z − η (z)ξ (z)4Now let us go to the gallery!
Pictures for ν. The ﬁrst picture is on the dynamics of ν. The coloring
indicates the number of iteration to trap the orbits in attracting ﬁxedpoints:
0 = orange < yellow < green < blue < purple < red = maximum.
Probably points colored in red are close to the Julia set.

Figure 2: The orange dots are arrayed on −2N and the critical line. The picture in thebottom is a magniﬁcation near the origin. Probably the sequence of orange dots near
{Im z = 4.5} are preimages of −2N.

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Pictures for µ Next we show the pictures of the dynamics of µ. The Julia
set of µ seems much simpler.

Figure 3: The Julia set of µ(z). The pictures in the second row are colored to distinguishthe ﬁxed points to converge. The pictures on the right shows the details of a prospectivepole of µ(z) (“A head of chicken”).

Figure 4: Head of another chicken in diﬀerent colorings.

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Figure 5: Chickens for µ(z). Heads appear constantly in this range, though the zeros get
denser as their imaginary parts increase.

Pictures for λ. Finally we go to λ. One can easily check that the Newton’smap λ has a symmetry with respect to the point z = 1/2. The dynamics
seems the simplest, but the calculation for λ is the heaviest.

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Figure 6: Julia sets for λ(z). The dynamics seems very simple: Probably each layer has
conformally the same dynamics as z → z 2 on the unit disk.

Figure 7: Julia set for λ(z) (large scaled).

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4Further applications of the theory of dynamicalsystems

For these Newton’s maps, we have:ﬁxed-point-free region of ν, µ, and λ =⇒ zero-free region of ζ.
For example, if ν(D) ∩ D = ∅ for some region D then ζ is zero-free on D.
Hence the problem of zero-free region is translated to be rather topological.
More generally, to Riemann’s zeta function ζ(z), one may apply otherroot ﬁnding algorithms, or functions of the formνh (z) = z + h(z)ζ(z)with holomorphic h(z) which does not vanish on the critical stripe S minus
the critical line. We temporarily say such functions are generalized νfunctions. For these functions, zeros of ζ(z) do not necessarily attract
nearby points. However, ζ(α) = 0 still implies that α is a ﬁxed point of νh .
Now our translation of the Riemann Hypothesis is:
Dynamical Riemann Hypothesis. Any fixed points of generalized ν-functions in the critical stripe S are arrayed on thecritical line.
This is slightly stronger than the original.
For example, even the dynamics of
ν1 (z) = z + ζ(z)is intriguing to work with. Of course, the entire function
νz−1 (z) = z + (z − 1)ζ(z)may be easier to work with. I hope the Dynamical Riemann Hypothesis
brings new perspective for the theory of zeta functions.

Acknowledgment. I would like to thank K. Matsumoto for giving me
this opportunity to present this work, and some valuable comments. His
articles on Riemann’s zeta were helpful also.
95References

Books about Riemann’s zeta function:[1] H.M. Edwards. Riemann’s Zeta Function. Academic Press, 1974.
[2] A. Ivić. The Riemann Zeta-Function. Wiley. 1985.
Books about complex dynamics:[3] A.F. Beardon. Iteration of Rational Functions. Springer-Verlag, 1991.
[4] L. Carleson and T. Gamelin. Complex Dynamics. Springer-Verlag,
1993.
[5] X-H. Hua and C-C. Yang. Dynamics of Transcendental Functions.
Gordon and Breach Science Publishers, 1998.
[6] J. Milnor. Dynamics in one complex variable: Introductory lectures.
vieweg, 1999.
A survey on dynamics of meromorphic functions:[7] W. Bergweiler. Iteration of meromorphic functions. Bull. Amer. Math.
Soc., 26(1999), 151–188.
For the theory of root finding algorithms;[8] P. Henrici. Elements of Numerical Analysis. Wiley, 1964.

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