Abstract
This note presents a geometric construction inspired by the maximum modulus principle for holomorphic
functions. Given a non-constant holomorphic function \(f\), we associate to each point \(z\) paths
obtained by following the points where \(|f|\) reaches its maximum on families of curves surrounding
\(z\).
Under a natural uniqueness assumption, these paths define a continuous structure and sweep regions of
the complex plane. The aim is to develop this construction, prove its basic continuity properties, and
derive a lower bound for \(|f|\) on certain regions swept by maximizing paths.
1. Geometric Framework
Let \(U\subset\mathbb C\) be a connected open set and let \(f:U\to\mathbb C\) be a non-constant
holomorphic function. We study the points where \(|f|\) reaches its maximum on a family of curves
surrounding a given center.
Definition — Admissible family of curves.
A continuous map \(\Theta:[0,1]\times[0,1]\to\mathbb C\) is admissible if, for every \(r\in[0,1]\),
\(J_r(\theta)=\Theta(r,\theta)\) satisfies: \(J_0\) is constant; for every \(r\in(0,1]\), \(J_r\)
is a Jordan loop; and if \(0\le r<s\le1\), then
\[
\operatorname{Im}(J_r)\subset \operatorname{Int}(J_s).
\]
Example — Circular family. For \(R>0\), the family
\[
\Theta(r,\theta)=Rr e^{2i\pi\theta}
\]
is admissible. The curves are circles centered at the origin with radius \(Rr\).
Definition — Admissible set of centers.
\[
U_\Theta=\bigl\{z\in\mathbb C:\ z+\Theta([0,1]^2)\subset U\bigr\}.
\]
Equivalently, \(z\in U_\Theta\) when all translated curves \(z+J_r\), \(r\in[0,1]\), are contained
in \(U\).
If \(f\) is entire, then \(U=\mathbb C\) and \(U_\Theta=\mathbb C\). If \(U=\mathbb C^*\) and
\(\Theta(r,\theta)=Rr e^{2i\pi\theta}\), then
\[
U_\Theta=\mathbb C\setminus\overline{B(0,R)}.
\]
2. Maximizing Selections
For \(z\in U_\Theta\) and \(r\in[0,1]\), consider the set of points on the curve \(z+J_r\)
where \(|f|\) is maximal:
\[
M_f(z,r)=
\operatorname*{argmax}_{\theta\in[0,1]}
|f(z+\Theta(r,\theta))|.
\]
This set is non-empty by compactness of \([0,1]\) and continuity of \(|f|\).
Definition — Maximizing map.
A map \(\gamma:[0,1]\to\mathbb C\) is called a maximizing map for \(|f|\) at \(z\), relative to
\(\Theta\), if for every \(r\in[0,1]\),
\[
\gamma(r)\in M_f(z,r).
\]
We denote by \(\Gamma_f(z)\), or simply \(\Gamma_z\), the set of maximizing maps at \(z\).
This definition does not assume that \(\gamma\) is continuous. Continuity will be obtained under a
uniqueness assumption. Thus the natural object is first the multivalued correspondence
\[
(z,r)\longmapsto M_f(z,r),
\]
and maximizing maps should be interpreted as selections of this correspondence.
Definition — Uniqueness.
The maximum is unique at \(z\in U_\Theta\) if, for every \(r\in[0,1]\), \(M_f(z,r)\) is reduced
to one point. In this case, we denote this point by \(\gamma_z(r)\).
For \(f(w)=w^n\), \(n\ge1\), and \(\Theta(r,\theta)=Rr e^{2i\pi\theta}\), uniqueness fails at
\(z=0\), since the maximum is attained at every point of each circle. If \(z\ne0\), the maximum is
attained in the direction of \(z\), and
\[
\gamma_z(r)=z+Rr\frac{z}{|z|}.
\]
For \(f(w)=e^w\), since \(|e^w|=e^{\operatorname{Re}(w)}\), the maximum is attained at the rightmost
point of the circle, so \(\gamma_z(r)=z+Rr\).
3. Monotonicity and Continuity
Proposition — Strict growth of the maximum.
Let \(z\in U_\Theta\) and let \(\gamma\in\Gamma_z\). Then
\[
r\in[0,1]\longmapsto |f(\gamma(r))|
\]
is strictly increasing.
Proof.
Let \(0\le r<s\le1\). By admissibility of \(\Theta\), the curve \(z+J_r\) is contained in the
interior of the Jordan curve \(z+J_s\). Since \(f\) is holomorphic and non-constant, the maximum
modulus principle implies that \(|f|\) cannot attain its maximum in the interior of the bounded
domain enclosed by \(z+J_s\). Consequently,
\[
\max_{\theta\in[0,1]} |f(z+\Theta(r,\theta))|
<
\max_{\theta\in[0,1]} |f(z+\Theta(s,\theta))|.
\]
By definition of \(\gamma\), this gives \(|f(\gamma(r))|<|f(\gamma(s))|\). Thus the map is
strictly increasing.
Lemma — Continuity of the maximal value.
The function
\[
m:U_\Theta\times[0,1]\to\mathbb R_+,\qquad
m(z,r)=\max_{\theta\in[0,1]} |f(z+\Theta(r,\theta))|
\]
is continuous.
Proof.
Let \((z_n,r_n)\to(z,r)\) in \(U_\Theta\times[0,1]\), and define
\(F(z,r,\theta)=|f(z+\Theta(r,\theta))|\). For \(n\) large enough, the points \((z_n,r_n)\) remain
in a compact subset \(K\subset U_\Theta\times[0,1]\). Since \(F\) is continuous, it is uniformly
continuous on \(K\times[0,1]\). Therefore,
\[
\sup_{\theta\in[0,1]} |F(z_n,r_n,\theta)-F(z,r,\theta)|\longrightarrow0.
\]
Moreover,
\[
|m(z_n,r_n)-m(z,r)|
\le
\sup_{\theta\in[0,1]} |F(z_n,r_n,\theta)-F(z,r,\theta)|.
\]
Hence \(m(z_n,r_n)\to m(z,r)\).
Theorem — Continuity under uniqueness.
Let \(E\subset U_\Theta\). Assume that, for every \(z\in E\) and every \(r\in[0,1]\), the maximum
defining \(M_f(z,r)\) is unique. Then
\[
E\times[0,1]\to\mathbb C,\qquad
(z,r)\longmapsto\gamma_z(r)
\]
is continuous.
Proof.
Let \((z_n,r_n)\to(z,r)\) in \(E\times[0,1]\). We prove that
\(\gamma_{z_n}(r_n)\to\gamma_z(r)\). Each term is of the form
\[
\gamma_{z_n}(r_n)=z_n+\Theta(r_n,\theta_n)
\]
for some \(\theta_n\in[0,1]\), and the set of all such points is contained in the image of a compact
set under the continuous map \((z,r,\theta)\mapsto z+\Theta(r,\theta)\). Thus the sequence admits
accumulation points.
Let \(w\) be an accumulation point and extract a subsequence such that
\(\gamma_{z_{n_k}}(r_{n_k})\to w\). Choose \(\theta_k\in[0,1]\) with
\[
\gamma_{z_{n_k}}(r_{n_k})=z_{n_k}+\Theta(r_{n_k},\theta_k).
\]
After extracting again, assume that \(\theta_k\to\theta\in[0,1]\). By continuity of \(\Theta\),
\[
w=z+\Theta(r,\theta).
\]
Furthermore, by continuity of \(m\),
\[
|f(w)|
=\lim_{k\to\infty}|f(\gamma_{z_{n_k}}(r_{n_k}))|
=\lim_{k\to\infty}m(z_{n_k},r_{n_k})
=m(z,r).
\]
Therefore \(w\in M_f(z,r)\). By uniqueness, \(w=\gamma_z(r)\). Every accumulation point is equal
to \(\gamma_z(r)\), so the whole sequence converges.
Corollary — Uniform stability.
Let \((z_n)_{n\in\mathbb N}\) be a sequence in \(U_\Theta\) such that \(z_n\to z\). If the maxima
are unique for every \(z_n\) and for \(z\), then
\[
\|\gamma_{z_n}-\gamma_z\|_\infty\longrightarrow0.
\]
Proof.
Apply the previous theorem to the compact set \(E=\{z\}\cup\{z_n:n\in\mathbb N\}\). The map
\((w,r)\mapsto\gamma_w(r)\) is continuous on \(E\times[0,1]\), hence uniformly continuous. Since
\(z_n\to z\),
\[
\sup_{r\in[0,1]}|\gamma_{z_n}(r)-\gamma_z(r)|\to0.
\]
4. Regions Swept by Maximizing Maps
Let \(\xi:[0,1]\to U_\Theta\) be a continuous path of centers such that the maxima associated with
\(\xi(t)\) are unique for every \(t\in[0,1]\). We can then consider the continuous family of paths
\[
(t,r)\longmapsto\gamma_{\xi(t)}(r).
\]
Definition — Loop associated with a path of centers.
The loop \(L_\xi:[0,1]\to\mathbb C\) is defined by
\[
L_\xi(s)=
\begin{cases}
\gamma_{\xi(4s)}(0), & 0\le s\le \frac14,\\[2mm]
\gamma_{\xi(1)}(4s-1), & \frac14\le s\le \frac12,\\[2mm]
\gamma_{\xi(3-4s)}(1), & \frac12\le s\le \frac34,\\[2mm]
\gamma_{\xi(0)}(4-4s), & \frac34\le s\le1.
\end{cases}
\]
We set
\[
S(L_\xi)=\operatorname{Im}(L_\xi)\cup
\{w\in\mathbb C\setminus\operatorname{Im}(L_\xi):\operatorname{Ind}(L_\xi,w)\ne0\}.
\]
If \(L_\xi\) is a Jordan loop, then \(S(L_\xi)\) is simply the image of the loop together with its
interior. The definition through the winding number also works when the loop has self-intersections.
Theorem — Sweeping theorem.
If \(\xi:[0,1]\to U_\Theta\) is a continuous path such that the maxima are unique at every point of
\(\xi([0,1])\), then
\[
S(L_\xi)
\subset
\bigl\{\gamma_{\xi(t)}(r):\ (t,r)\in[0,1]^2\bigr\}.
\]
Proof.
Define \(H:[0,1]^2\to\mathbb C\) by \(H(t,r)=\gamma_{\xi(t)}(r)\). By the continuity theorem under
uniqueness, \(H\) is continuous. The boundary of the square \([0,1]^2\), positively oriented, is
mapped by \(H\) onto \(L_\xi\), up to reparametrization.
Let \(w\notin H([0,1]^2)\). Then
\[
(t,r)\longmapsto \frac{H(t,r)-w}{|H(t,r)-w|}
\]
is well-defined and continuous on the square. Its restriction to the boundary is null-homotopic,
since the square is contractible. Hence the winding number of \(L_\xi\) around \(w\) is zero:
\[
\operatorname{Ind}(L_\xi,w)=0.
\]
By contraposition, if \(\operatorname{Ind}(L_\xi,w)\ne0\), then \(w\in H([0,1]^2)\). The points of
\(\operatorname{Im}(L_\xi)\) also belong to \(H([0,1]^2)\), so \(S(L_\xi)\subset H([0,1]^2)\).
Corollary — Lower bound on the swept region.
Under the assumptions of the sweeping theorem, for every \(w\in S(L_\xi)\),
\[
|f(w)|\ge \min_{t\in[0,1]}|f(\xi(t))|.
\]
If \(w\in S(L_\xi)\setminus\xi([0,1])\) and \(w\) is reached with a radial parameter \(r>0\),
then the inequality is strict.
Proof.
By the sweeping theorem, there exists \((t_0,r_0)\in[0,1]^2\) such that
\(w=\gamma_{\xi(t_0)}(r_0)\). By strict growth,
\[
|f(w)|=|f(\gamma_{\xi(t_0)}(r_0))|
\ge |f(\gamma_{\xi(t_0)}(0))|.
\]
Since \(J_0\) is constant and equals \(0\) in the normalized setting,
\(\gamma_{\xi(t_0)}(0)=\xi(t_0)\). Therefore
\[
|f(w)|\ge |f(\xi(t_0))|
\ge \min_{t\in[0,1]}|f(\xi(t))|.
\]
If \(r_0>0\), strict growth gives the strict inequality.
5. Conditional Application to the Riemann Zeta Function
We now apply the preceding construction conditionally to the Riemann zeta function. Let
\[
\Theta(r,\theta)=5r e^{2i\pi\theta}
\]
and let \(f=\zeta\), viewed as a holomorphic function on \(\mathbb C\setminus\{1\}\). Then
\[
U_\Theta=\mathbb C\setminus\overline{B(1,5)}.
\]
Theorem — Conditional implication.
Assume that there exists \(T_0\ge0\) such that, for every \(t\ge T_0\), the maxima of \(|\zeta|\)
on the circles
\[
\left\{\frac12+it+5e^{2i\pi\theta}:\theta\in[0,1]\right\}
\]
are unique. Assume also that the corresponding maximizing selections satisfy, for all sufficiently
large \(t\),
\[
\operatorname{Re}\bigl(\gamma_{\frac12+it}(1)\bigr)<0.
\]
Then \(\zeta\) has only finitely many zeros in the critical strip outside the critical line.
Proof.
Choose \(T\) large enough so that the uniqueness assumption and endpoint condition hold for all
\(t\ge T\). For \(n\ge1\), consider the vertical path
\[
\xi_n(u)=\frac12+i(T+nu),\qquad u\in[0,1].
\]
By uniqueness, the associated maximizing maps are continuous. By the endpoint condition, the outer
endpoints \(\gamma_{\xi_n(u)}(1)\) lie strictly to the left of \(\operatorname{Re}(s)=0\) for all
\(u\in[0,1]\), provided \(T\) is sufficiently large.
The loop \(L_{\xi_n}\) therefore surrounds a vertical region containing a large rectangle in the
critical strip. The height of this rectangle tends to infinity with \(n\). By the lower bound on
swept regions, for every point \(w\) in this swept region,
\[
|\zeta(w)|\ge \min_{u\in[0,1]}|\zeta(\xi_n(u))|.
\]
Away from the central path, the inequality is strict as soon as the corresponding radial parameter
is positive.
Using the symmetry relation \(\zeta(\overline{s})=\overline{\zeta(s)}\), together with the functional
symmetry of the critical strip, the same argument applies symmetrically to the opposite side.
Consequently, all zeros outside the critical line must be contained in a bounded subset of the
critical strip. Since \(\zeta\) is holomorphic and not identically zero on the critical strip, its
zeros are isolated. A bounded region contains only finitely many isolated zeros unless they accumulate
in the domain, which is impossible for a non-zero holomorphic function.
References
Complex analysis and maximum principle
-
Lars V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, 1979.
-
John B. Conway, Functions of One Complex Variable I, 2nd ed., Springer, 1978.
-
Walter Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, 1987.
-
Elias M. Stein and Rami Shakarchi, Complex Analysis, Princeton University Press, 2003.
Winding number and planar topology
-
Sébastien Boisgérault,
The Winding Number.
-
Theodore W. Gamelin, Complex Analysis, Springer, 2001.
Riemann zeta function