# Structural mechanics of filamentous cyanobacteria

Mixon K. Faluweki<EMAIL_ADDRESS>School of Science and
Technology, Nottingham Trent University, Nottingham NG11 8NS, UK.  Malawi
Institute of Technology, Malawi University of Science and Technology, Limbe,
Malawi.  Lucas Goehring<EMAIL_ADDRESS>School of Science and
Technology, Nottingham Trent University, Nottingham NG11 8NS, UK.

###### Abstract

Filamentous cyanobacteria, forming long strands of connected cells, are one of
the earliest and most successful forms of life on Earth. They exhibit self-
organised behaviour, forming large-scale patterns in structures like biomats
and stromatolites. The mechanical properties of these rigid structures have
contributed to their biological success and are important to applications like
algae-based biofuel production. For active polymers like these cyanobacteria,
one of the most important mechanical properties is the bending modulus, or
flexural rigidity. Here, we quantify the bending stiffness of three species of
filamentous cyanobacteria using a microfluidic flow device, where single
filaments are deflected by fluid flow. This is complemented by measurements of
the Young’s modulus of the cell wall, via nanoindentation, and the cell wall
thickness. We find that the stiffness of the cyanobacteria is well-captured by
a simple model of a flexible rod, with most stress carried by a rigid outer
wall. Finally, we connect these results to the curved shapes that these
cyanobacteria naturally take while gliding, and quantify the forces generated
internally to maintain this shape. The measurements can be used to model
interactions between filamentous cyanobacteria, or with their environment, and
how their collective behaviour emerges from such interactions.

††preprint: APS/123-QED

##  I Introduction

Cyanobacteria, also known as blue-green algae, are one of the most successful
forms of life on Earth, with origins dating back over two billion years Schopf
and Packer (1987); Allwood  _et al._ (2006); Rasmussen  _et al._ (2008).
Indeed, it is believed that the ability of cyanobacteria to photosynthesise
and release oxygen caused significant changes to the Earth’s atmosphere during
the Precambrian, allowing for the eventual evolution of other life forms that
rely on that oxygen, including us Rasmussen  _et al._ (2008). Cyanobacteria
are ubiquitous, finding habitats in most water bodies and in extreme
environments such as the polar regions, deserts, brine lakes and hot springs
Walter  _et al._ (1976); Jones  _et al._ (2002); Wharton  _et al._ (1983).
They have also evolved surprisingly complex collective behaviours that lie at
the boundary between single-celled and multi-cellular life. For example,
filamentous cyanobacteria live in long chains of cells (see Fig. 1) that
bundle together into larger structures including biofilms, biomats and
stromatolites Whitton and Potts (2012); Stal (2012). These large colonies
provide a rigid, stable and long-term environment for their communities of
bacteria. In addition, cyanobacterial biofilms can be used as bioreactors to
produce a wide range of chemicals, including biofuels like biodiesel and
ethanol Farrokh  _et al._ (2019). However, despite their importance to the
history of life on Earth, and their commercial and environmental potentials,
there remain basic questions of how filamentous cyanobacteria move, respond to
their environment and self-organise into collective patterns and structures.
Here we will address some of these concerns, in particular measuring and
quantifying the key mechanical properties that underlie the behaviour of three
typical cyanobacteria species.

Figure 1:  Filamentous cyanobacteria under confocal fluorescence imaging. (a)
Under ideal conditions active gliding specimens of _O. lutea_ appear as long
thin curved filaments. (b) When rendered inactive, for example by being
briefly cooled, the same filaments adopt a more random shape. (c) Under higher
magnification _O. lutea_ is seen to be composed of one cell wide strands of
connected cells.

All known cyanobacteria lack flagella Rippka  _et al._ (1979), however many
filamentous species move on surfaces by gliding, a form of locomotion where no
physical appendages are seen to aid movement Hoiczyk (2000). The actual
mechanism behind gliding is not fully understood, although over a century has
elapsed since its discovery Hansgirg (1883); Drews (1959). One theory suggests
that gliding motion in cyanobacteria is mediated by the continuous secretion
of polysacharides through pores on individual cells Hosoi (1951); Walsby
(1968); Hoiczyk and Baumeister (1998). Another theory suggests that gliding
motion involves the use of type IV pili, polymeric assemblies of the protein
pilin Craig  _et al._ (2004), as driving engines of motion Duggan  _et al._
(2007); Risser  _et al._ (2014); Khayatan  _et al._ (2015). However, it is not
clear how the action of these pili would lead to motion, with some suggesting
they retract Schuergers  _et al._ (2015), while others suggest they push
Khayatan  _et al._ (2015), to generate forces. Other scholars have suggested
surface waves generated by the contraction of a fibril layer as the mechanism
behind gliding motion in _Oscillatoria_ Halfen and Castenholz (1970, 1971).
Recent work also suggests that shape fluctuations and capillary forces could
be involved in gliding motion Tchoufag  _et al._ (2019). As a challenge
underlying all these models, and any quantitative appreciation of their
motion, is the need to understand the range of forces that cyanobacteria can
generate as they move and change shape. Therefore, in this work we aim to
measure the bending stiffness, energy and axial stress associated with shape
fluctuations in filamentous cyanobacteria.

No matter what the origin of their motile forces, mechanically, filamentous
cyanobacteria can be considered to act as self-propelled semi-flexible rods.
This means that they belong to the class of active polymer or active nematic
systems, terms that describe collections of self-driven moving objects with a
highly elongated shape (see e.g. Marchetti  _et al._ (2013); Doostmohammadi
_et al._ (2018); Winkler and Gompper (2020)). For such systems the bending
stiffness–also sometimes called the flexural rigidity–characterises the
flexibility of the filaments, and influences their behaviour. As a recent
example, the motion of active polymers in porous media has been shown to
depend dramatically on their stiffness, ranging from filaments that smoothly
move through the pore spaces, to those that easily get trapped Mokhtari and
Zippelius (2019). This would be of relevance to the design of scaffolds for
microbial fuel cells Han  _et al._ (2016); Wang  _et al._ (2015), to determine
pore sizes that can effectively trap filaments but still allow for the flow of
dissolved gasses.

Through collective interaction, filamentous cyanobacteria self-organise into
colonies or biofilms, symbiotic communities found in a wide variety of
ecological niches. Their larger-scale collective structures are characterised
by diverse shapes including bundles, vortices and window-like patterns Shepard
and Sumner (2010), and similar patterns have been observed in fossil records
Sumner (1997); Allwood  _et al._ (2006). For filamentous cyanobacteria, the
mechanics of the filaments is known to contribute to self-organisation, for
example in determining how one filament will bend when in contact with other
filaments or obstacles Tamulonis and Kaandorp (2014). Further, biofilms and
biomats in general show some remarkably conserved macro-mechanical properties,
typically behaving as viscoelastic materials with a relaxation time of about
20 minutes Shaw  _et al._ (2004). However, there has been a lack of any
corresponding systematic measurement of the micro-mechanical properties of
filamentous cyanobacteria, to the point where such properties are typically
just assumed when needed in models (_e.g._ see Tamulonis and Kaandorp (2014)).
This paper seeks to fill that gap by providing carefully measured values of
the mechanical properties of individual filaments of cyanobacteria.

In general, we focus here on the equilibrium structural mechanics, or statics,
of filamentous cyanobacteria and we leave their dynamics for future work. In
particular, we look at the bending stiffness of the cyanobacteria, treating
them as slender bodies, or flexible rods. For this, custom microfluidic flow
cells were developed, in which we studied how the bacteria are reverisbly
deflected by the drag forces of fluid flow past them. We then connect their
bending stiffness to the mechanical response of their cell walls. For this, we
quantify the Young’s modulus of the filaments, through nanoindentation
techniques, and show that the stiffness of the cell walls provides most of the
filament’s resistance to bending. Finally, we look at the shapes of
cyanobacteria filaments in different scenarios and connect these shapes to
their internal distributions of forces. We study three species– _Kamptonema
animale_ , _Lyngbya lagerheimii_ , and _Oscillatoria lutea_ –all from the
order _Oscillatoriales_. These species are well-studied, easy to cultivate,
non-toxic and associated with mat formation and stromatolites (e.g. Hoiczyk
(2000); Boal and Ng (2010); Rippka  _et al._ (1979); Strunecky  _et al._
(2014); Shepard and Sumner (2010)). The similarity of the three species also
allows a check of the consistency of our measurements.

##  II Methods

Figure 2:  Modelling cyanobacteria bending and structure. (a) The filament of
bacteria is treated as a slender rod of length $L$, which bends under a drag
force arising from an external channel flow $\bm{u}$. (b) Close-up the
filament is approximated as a hollow cylinder of radius $r$ and wall thickness
$\Delta r$. The radius of curvature ($R=1/\kappa)$, tangent ($\hat{t}$) and
normal ($\hat{n}$) vectors are defined with reference to the neutral axis of
the cylinder, which follows path $s$.

###  II.1 Theory of a bending slender rod

Here, we outline the theoretical framework used to evaluate the mechanical
properties of filamentous cyanobacteria. A strand of bacteria is treated as a
slender elastic rod Landau and Lifshitz (1986), whose shape is defined along
an arc or contour length coordinate $s$ by its position $\bm{x}$, the
orientation $\theta(s)$ of the tangent vector $\bm{\hat{t}}=d\bm{x}/ds$, or
the curvature $\kappa(s)=d\theta/ds=|d^{2}\bm{x}/ds^{2}|$, as sketched in Fig.
2. The radius of curvature of the filament, $R(s)=1/\kappa$.

External forces and internal moments can both act to bend an elastic filament.
Specifically, if $w$ is an external force per unit length, applied normal to
the filament, this will generate a bending moment $M$, such that

$\frac{d^{2}M}{ds^{2}}=-w.$ (1)

This moment, along with any internally generated moments, will bend or flex
the filament, whose mechanical resistance is given by its bending stiffness or
flexural rigidity, $\beta$. In mechanical equilibrium, and assuming a linear
elastic response, this balance can be given as

$M=-\beta(\kappa-\kappa_{0})=-\beta\tilde{\kappa},$ (2)

where $\kappa_{0}$ is any intrinsic curvature (i.e. it describes the shape of
the filament in the absence of external forces), and where $\tilde{\kappa}$ is
the change in curvature away from this reference configuration. Equivalently,
the deformation energy, per unit length, is

$\mathcal{U}=\frac{1}{2}\beta\tilde{\kappa}^{2}.$ (3)

This bending means that one side of the filament, on the inside of the bend,
will be under compression, whereas the opposite side will be in tension or
extension.

In our experiments, external drag forces are generated by the flow of water
past a filament of cyanobacteria. The filament is treated as a flexible
cylinder of radius $r$ surrounded by water of density $\rho$ and dynamic
viscosity $\mu$, which is flowing towards the filament with a normal velocity
component $u_{\mathrm{n}}(s)$. As the water moves around the filament it
produces a drag force per unit length of

$w=\rho C_{\mathrm{D}}ru_{\mathrm{n}}^{2},$ (4)

where $C_{\mathrm{D}}$ is the drag coefficient. At low Reynolds number,
$\mathrm{Re}=2\rho ru_{\mathrm{n}}/\mu\ll~{}1$, the drag coefficient of a
cylinder is well-approximated Eames and Klettner (2017); Lamb (1911) by

$C_{D}\cong\frac{8\pi}{\mathrm{Re}\log(7.4/\mathrm{Re})}.$ (5)

In our system, using a filament radius of 2.2 $\mu$m and a maximum flow speed
of $\sim$3 mm s-1, we maintain Re$<$0.01. Thus, in this laminar limit the drag
force per unit length along any part of the filament is given by

$w(s)=\frac{4\pi\mu}{\log(7.4/\mathrm{Re})}u_{\mathrm{n}}(s).$ (6)

A solution for the flow field $\bm{u}$ expected in our channels, with
rectangular cross-section, is given in Appendix A.

Finally, one of our aims is to compare the bending stiffness of a strand of
cyanobacteria to the contributions expected from its cell wall. For this, if
we treat the cell wall as a thin, hollow cylinder of radius $r$, thickness
$\Delta r$, and Young’s modulus $E$, it will have a bending stiffness

$\beta=\pi Er^{3}\Delta r$ (7)

assuming that $\Delta r\ll r$.

###  II.2 Culture and sample preparation

Three species of filamentous cyanobacteria were used: _Kamptonema animale_
(SAG 1459-6), _Lyngbya lagerheimii_ (SAG 24.99), and _Oscillatoria lutea_ (SAG
1459-3). The were maintained in a medium of BG11 broth (Sigma-Aldrich) diluted
to a ratio of 1:100 with deionised water. Following Lorentz et al. Lorenz  _et
al._ (2005) they were incubated at $20\pm 1^{\circ}$C, with warm-white LEDs
(colour temperature of 2800 K) providing a light intensity of $10\pm 2$
$\mu$mol m-2 s-1 on a 16h day + 8h night cycle. For sampling, material was
transferred into a 100 ml glass bottle half-filled with medium and shaken
mildly so that the filaments were separated, but not otherwise damaged.
Samples were then drawn out, typically in a 0.5 ml syringe, for experimental
use.

###  II.3 Flow cell for bending measurements

Figure 3:  Microfluidic flow cell. The sketch to the left gives a schematic of
the flow cell, showing how the components were connected. The close-up view to
the right highlights the interactions between a cyanobacteria filament and the
flow in the channel. The filament is introduced through a narrow inlet, with
tight corners to pin it in place and anchor it at the point along the wall
where it enters into the main channel. The view here of the chip in use is a
superposition of two white-light microscopy images of the rest and deflected
configurations of a single filament, as it is pushed by a laminar flow
$\bm{u}$ travelling along the $y$-axis of the channel.

The microfluidic devices (Fig. 3) used here to measure the bending stiffness
of cyanobacteria are inspired by similar devices used for the characterisation
of the elastic properties of pollen tubes Nezhad  _et al._ (2013) and _E.
coli_ Amir  _et al._ (2014); Caspi (2014). Microfluidic chips were produced by
soft lithography techniques (e.g. see Madou (2002)). Briefly, designs were
printed on a chrome-quartz photomask, from which a template was made in a UV-
cured photoresist (Microresist SU-8 3025 and 3050). Individual chips were made
by pouring poly(dimethylsiloxane) (PDMS, Dow Sylgard 184) over the template
and degassing the result under vacuum before curing at 75∘C for 1 hour. The
chip was then cut out and holes punched at the inlet and outlet points,
through which tubing was connected. Finally, the chip was primed in an oxygen
plasma, together with a PDMS-coated glass slide, and these two components were
bonded together to form a complete device.

The chips are designed around a simple rectangular flow channel, much wider
(300 $\mu$m) than it is tall (two designs with channel heights 72 $\pm$ 2 and
134 $\pm$ 3 $\mu$m), with well-separated inlets and outlets allowing for an
even flow to develop (channel length of 30 mm). The inlet was connected to a
syringe pump while tubing from the outlet led to a beaker to collect waste.
Part-way along the channel is an injection point where a filament of
cyanobacteria can be threaded into the channel, from a syringe containing a
dilute suspension of the cyanobacteria; the syringe was operated by hand until
only one filament was allowed to protrude into the flow channel. In order to
enter the channel, the cyanobacteria filament had to bend around a narrow
U-bend, designed to anchor it at the channel wall. A sketch of the design,
along with a snapshot of a chip in use, are given in Fig. 3.

The setup used meant that for any experiment a single isolated strand of
cyanobacteria was under observation in the flow cell, well-anchored to one
wall and crossing close to perpendicular across most of the channel. This
filament was then pushed on and deflected by a flow of water in the channel,
using average flow speeds between 0.48 and 2.59 mm s-1. For analysis, images
of the filament at equilibrium and deflected configurations were collected
using a confocal laser scanning microscope (Leica TCS SP5). By scanning the
focal plane, this method also allowed us to measure the height at which the
filament entered the channel. Taking advantage of the auto-fluorescence of the
chlorophyll-a in the cyanobacteria Millie  _et al._ (2002), we used 514 nm
laser light for excitation and observed the resulting emission through a
bandpass filter from 620 to 780 nm. The images were thresholded and
skeletonised in Matlab, in order to have the filament shapes expressed as a
set of pixel coordinates along a path $s$ of length $L$, under different flow
conditions. Finally, deflection profiles were calculated from the change in
the positions of points at the same relative distance along a filament, $s/L$,
before and after a flow was turned on.

Table 1:  Summary of results, showing the bending stiffness $\beta$, the reduced modulus $E^{*}$ and Young’s modulus $E$ of the cell walls, and the average active curvature $\kappa$ and cross-sectional radius $r$ of the cyanobacteria filaments. In all cases the uncertainty range gives the standard deviation of the measurements and the number of independent measurements are shown in parenthesis. Species |  $\beta$ (N m2) |  $E^{*}$ (MPa) |  $E$ (MPa) |  $\kappa$ (m-1) |  $r$ ($\mu$m)
---|---|---|---|---|---
_K. animale_ |  $4.8\pm 2.9\times 10^{-17}$ (9) |  53 $\pm$ 8 (13) |  40 $\pm$ 6 (13) |  470 $\pm$ 304 (119) |  2.2 $\pm$ 0.1 (29)
_L. lagerheimii_ |  $6.0\pm 5.0\times 10^{-17}$ (9) |  27 $\pm$ 6 (8) |  20 $\pm$ 4 (8) |  452 $\pm$ 322 (98) |  2.2 $\pm$ 0.1 (23)
_O. lutea_ |  $2.6\pm 1.6\times 10^{-17}$ (7) |  35 $\pm$ 7 (10) |  26 $\pm$ 6 (10) |  537 $\pm$ 228 (154) |  2.1 $\pm$ 0.1 (21)

###  II.4 Nanoindentation and cell wall properties

To measure the mechanical response of the cell wall, we performed
nanoindentation experiments with an atomic force microscope (Brucker Dimension
Icon), operating in the quantitative nanomechanical mapping (QNM) mode (e.g.
see Young  _et al._ (2011); Wright and Armstrong (2006) for method details).
We used an RTESPA-150 cantilever with a nominal spring constant of $5$ N m-1,
which is suitable for $5\leq E\leq 500$ MPa. Each cantilever was calibrated
for sensitivity against a a sapphire test surface and we then employed a
relative method of measurement by comparison to a PDMS reference sample. For
this the reference sample was probed to a selected indentation depth between
25-50 nm (typically 50 nm), and the effective tip radius and cantilever spring
constant adjusted to match its known Young’s modulus of 3.5 MPa. Cyanobacteria
filaments were then deposited on a wet glass slide and probed to the same
indentation depth, while the cantilever scanned across an area. To ensure that
the filaments reamained well-hydrated during scanning (and to prevent an
artificial increase in $E$ with time, due to drying out Yao  _et al._ (1999)),
scanning times were kept to less than 5 minutes, and the scanner was covered
to minimise air flow. Elastic modulus measurements were extracted by fitting
the Derjaguin-Muller-Toporov (DMT) model Derjaguin  _et al._ (1975) to the
resulting force-distance curves, using Bruker’s QNM software. This model takes
into consideration the tip radius and, for a hard tip probing a soft surface,
gives the reduced modulus, $E^{*}=E/(1-\nu^{2})$, where $\nu$ is the Poisson
ratio.

Transmission electron microscopy (TEM) was used to obtain images of the
internal structure of the cyanobacteria filaments. Samples were prepared
following methods in Strunecky  _et al._ (2014), procedure (b). In this
instance the resin was TAAB 813 (TAAB, Aldermaston) and uranyl acetate was
substituted for EM Stain 336 (Agar Scientific Ltd., Stansted) which in both
cases was used according to the manufacturer’s instructions. Once fixed,
embedded, sectioned and stained, sections were examined with a JEM2100Plus
(JEOL UK, Welwyn Garden City) operating at 120 kV. Electron micrographs were
digitised using a Rio16 (Gatan UK, Abingdon) camera operated using Digital
Micrograph (3.32.2403.0) and exported to .tiff for analysis.

###  II.5 Filament shape

Freely gliding cyanobacteria filaments naturally adopt a curved shape, as
shown in Fig. 1a, whereas inactive filaments have a more irregular, meandering
shape. We quantified these shapes by measuring the curvature of over one
hundred free-gliding (active) and inactive filaments in each of the three
species studied. In the case of active filaments, dilute suspensions were
transferred to a well plate (well diameter of 1.5 cm, with a surface coverage
of $\sim$ 1 filament mm-2) and then left undisturbed in the incubator for 24 h
before measurement. For inactive filaments, the procedure was similar except
that dilute suspensions were transferred to a medium with temperature of 2∘C
and stored at this temperature for 4 hours; during imaging, ice was also added
to the suspension to keep the temperature low. After the experiments, the
inactivated filaments recovered their motility after warming back up. Images
were taken using a confocal microscope, as in Section 2II.3, and were
processed in Matlab. Global curvatures were determined by fitting circles to
skeletonised filaments using Pratt’s method Nikolai Chernov (2021) (2021);
Pratt (1987). For each filament we also independently evaluated the local
orientation $\theta$ along the path length $s$ by fitting tangent vectors to
each point along the thresholded image skeleton, using a sampling window of 30
pixels (90 $\mu$m). Finally, for each species we measured the cross-sectional
radius of at least 150 filaments, by manual measurements of high-magnification
optical microscope images, using ImageJ Schindelin  _et al._ (2012).

##  III Results

Our aims are to quantify the mechanical properties of typical species of
filamentous cyanobacteria, and to connect these properties to the shapes they
naturally take and the forces that they can generate internally. Here, we
report our observations of their bending stiffness $\beta$, Young’s modulus
$E$, curvature $\kappa$ and cross-sectional radius $r$. A summary of the key
results is given in Table 1.

###  III.1 Bending stiffness

The bending responses of the three species of filamentous cyanobacteria were
measured using the flow cells described in Section 2II.3 and sketched in Fig.
3. A flow test consisted of a series of alternating flowing and stopped
conditions, such as those given in Fig. 4(a). In order to minimise any
systematic effects of plastic deformation (as observed in e.g. Amir  _et al._
(2014) for _E. coli_), the sequence of flow speeds used during any experiment
was randomised, and between each condition the flow was turned off to allow
the filament to relax back to an equilibrium or rest position. For each test
the measured displacement profile ${\delta}(s)$ gives the difference in
position between the bent and rest configurations of the filament, from where
it enters the flow cell, $s=0$, to its tip at $s=L$.

Figure 4:  Measurements of the bending stiffness of filamentous cyanobacteria.
(a) Each flow test consisted of a sequence of different flow speeds, separated
by stopped flow conditions, while we tracked the deflection of a single
filament of cyanobacteria in the channel. As shown here, the total deflection
was proportional to the applied flow. (b) Each deflection profile was fit to
the expected elastic deformation, with the magnitude of the elastic response
(effectively, $\beta$) and a solid-body rotation, $\alpha$, as fitting
parameters; shown here are the fits for one example filament. (c) After
accounting for any rotation through a global correction, a bending stiffness,
$\beta(s)$, was then extracted at every point along the filament. (d) We show
the final distribution of the values of $\beta$ observed for each of the three
species of filamentous cyanobacteria studied. On each box, the whiskers
indicate extreme points while the red line is the median and the bottom and
top edges of the box indicate the 25th and 75th percentile, respectively.
Figure 5:  Cyanobacteria elasticity was measured by nanoindentation. (a) A
summary here shows the Young’s modulus measurements of the three species of
filamentous cyanobacteria, probed to 50 nm depth. On each box, the whiskers
indicate extreme points while the red line is the median and the bottom and
top edges of the box indicate the 25th and 75th percentile, respectively. The
inset gives an example map of the reduced modulus $E^{*}$ for _O. lutea_. (b)
Repeating the tests for different indentation depths showed no significant
effect on the measured $E$, over the range of depths used. Error bars
represent standard deviations of measurements.

To find the bending stiffness, $\beta$, the displacement profiles were
compared to the bending expected from the drag of fluid across the filament.
For this, we assumed a Poiseuille flow profile in a closed channel with a
rectangular cross-section. The normal component of the impinging flow was
calculated for points along the bent filament position (see Appendix A). The
drag force $w(s)$ was then determined according to Eq. 6, and used to find the
applied moment $M(s)$ by integrating Eq. 1, assuming free boundary conditions
for the end of the filament in the channel ($M(L)=M^{\prime}(L)=0$). Further
integrating the moment leads to a prediction for $\beta\delta$, assuming
clamped boundary conditions where the filament enters the flow cell
($\delta(0)=\delta^{\prime}(0)=0)$. However, we also considered the
possibility of a small solid-body rotation of the filament by an angle
$\alpha$ around its anchor point. To account for this we performed a least-
squares fit of the predicted deformation to the observed deflection profile,
with $\alpha$ and $\beta$ as fitting parameters. Effectively, this decomposes
the observed deflection profile into a bending and rotation mode, and fits the
magnitudes of each. The results of this fit are demonstrated in Fig. 4(b).
Finally, by fixing $\alpha$ and then taking the ratio of the predicted and
measured bending profiles we found the value of $\beta$ that would correspond
to the observed bending of every point along the filament, as shown in Fig.
4(c). From these we calculated an average value and standard deviation for
$\beta$ for each test.

For each species we tested seven to nine independent strands of cyanobacteria,
and each strand was subjected to up to six different flow conditions. Results
were averaged (error-weighted mean) to give a representative value of $\beta$
for each individual strand, and the subsequent population averages for each
species are reported in Table 1 and Fig. 4(d). There was considerable
variation between individuals, up to about one order of magnitude, but the
average bending stiffnesses of all three species were very similar to each
other.

These results were further analysed to look for evidence of plastic or
nonlinear responses. For example, as the cyanobacteria filaments can change
their own shape, their response to an external flow might relax over time due
to a redistribution of internally generated forces. We tested for this effect
in our data by looking for correlations between the measured bending stiffness
and the order in which flows were applied. As shown in Fig. 4(a), each
filament was typically subject to six different flow rates applied over a few
minutes. Spearman’s rank-order correlation test showed no significant
correlation between the order in which the flows were applied and the measured
$\beta$ (K. animale, $r_{s}(51)=-0.70,p=0.23$; L. lagerheimii,
$r_{s}(41)=0.40,p=0.75$; O. lutea, $r_{s}(36)=0.26,p=0.66$). We conclude that,
at least over the experimental timescales, there is no evidence for any
plastic response to the shear flows.

We also performed statistical tests to check for any systematic effects of the
filament length, or flow speed, on the measured bending stiffness. The various
cyanobacteria filaments extended lengths $L$ between 190 and 295 $\mu$m into
the flow cell, but there was no significant correlation between $L$ and
$\beta$ (Pearson correlation coefficient: K. animale, $r(51)=0.07,p=0.86$; L.
lagerheimii, $r(41)=0.17,p=0.66$; O. lutea, $r(36)=0.65,p=0.11$). There is,
however, a moderate positive correlation between flow speed and $\beta$ (K.
animale, $r(51)=0.47,p<0.01$; L. lagerheimii, $r(41)=0.35,p=0.02$; O. lutea,
$r(36)=0.63,p<0.01$). Although this correlation is not particularly strong, it
may indicate a degree of strain stiffening in the filaments, a type of
response known from a variety of biopolymers Storm  _et al._ (2005), for
example.

Finally, we note that complementary three-point bending measurements of O.
lutea and K. animale were performed in parallel to this study by a
collaborative group, and are reported in Kurjahn  _et al._ (2022). While
broadly consistent with the results here, and overlapping much of the same
range as we report, they observed slightly higher average stiffnesses of 1.3
and 1.1 $\times$ 10-16 N m2, respectively.

###  III.2 Nanoindentation and cell wall properties

The bending stiffness of a slender rod, such as a filament of cyanobacteria,
is directly related to the elastic properties of its constituents. Assuming
that the cytoplasm does not support a significant load, most of the bending
moment can be expected to be carried by rigid structures like the cell wall.
Here we test this assumption, and evaluate the mechanical properties of the
cell walls of our species of filamentous cyanobacteria.

The main structural component of the bacterial cell wall is peptidoglycan, a
stiff cross-linked polymer; this means that the cell wall behaves elastically,
with measurements of its Young’s modulus $E$ typically ranging between about
5-50 MPa Yao  _et al._ (1999); Longo  _et al._ (2012); Deng  _et al._ (2011);
Thwaites and Mendelson (1989); Mendelson and Thwaites (1989); Auer and Weibel
(2017). Although classed as gram-negative, cyanobacteria have particularly
thick cell walls, extending to tens of nanometres or more Hoiczyk and
Baumeister (1995); Hoiczyk and Hansel (2000).

Maps of the reduced modulus $E^{*}=E/(1-\nu^{2})$ were collected by atomic
force microscopy using quantitative nanomechanical mapping techniques (see
Section 2II.4), as demonstrated in the inset to Fig. 5(a). For each map
average values and standard deviations were calculated over an area of a cell
that avoided any imaging artifacts like scarring and focused on the centre of
a filament, to minimise the effects of surface curvature on measurements
Wright and Armstrong (2006). The results were converted into measurements of
the Young’s modulus, $E$, by assuming $\nu=0.5$, as appropriate for soft
biological materials Wright and Armstrong (2006); Touhami  _et al._ (2003).
Measurements from at least $8$ different filaments were analysed for each
species, at a fixed indentation depth of 50 nm. To capture the full
distribution of measured values, a box plot of $E$ is given in Fig. 5(a). All
three species have similar Young’s modulii, with most observations in the
range of 25–50 MPa, although _K. animale_ potentially has a slightly higher
modulus than the other two species. These results are similar to those found
for _E. coli_ (35–60 MPa) Yao  _et al._ (1999) but noticeably higher than
_Bacillus subtilis_ (3 MPa) Thwaites and Surana (1991). To check for any
depth-dependence of $E$, measurements were repeated with indentation depths of
25, 40 and 50 nm. As shown in Fig. 5(b), there is no clear trend of the
measured elastic modulus with indentation depth, supporting the interpretation
that we are accurately probing the elastic properties of the cell wall.

From the measured Young’s modulii, we can also estimate an effective cell wall
thicknesses, $\Delta r$, under the assumption that the majority of the
filament stiffness comes from the cell wall. Using Eq. 7, the estimated wall
thicknesses are given in Table 2, with a value of 35$\pm$8 nm for _K. animale_
for example. We can compare these estimates to direct observations of the cell
wall structures using TEM, with example micrographs shown in Figure 6. We
first note that the thickness of the full cell envelope is 47 $\pm$ 4 nm, 45
$\pm$ 3 nm, and 69 $\pm$ 6 nm for _K. animale_ , _L. lagerheimii_ and _O.
lutea_ , respectively. Similar thicknesses, of $35\pm 5$ nm, can be seen for
TEM images of _K. animale_ in Strunecky et al. Strunecky  _et al._ (2014).
However, the cell envelope in cyanobacteria is a layered structure, consisting
of a thick peptidoglycan layer separated by inner and outer membranes Hoiczyk
and Hansel (2000). Of these, the peptidoglycan is expected to be the stiffest
layer, and if measured alone has shown thicknesses of 18 $\pm$ 2 nm, 14 $\pm$
2 nm, and 18 $\pm$ 3 nm for _K. animale_ , _L. lagerheimii_ and _O. lutea_ ,
respectively. Thus, while the bending stiffness measurements are largely
consistent with the interpretation of the mechanical response of the
cyanobacteria as a hollow cylinder, the peptidoglycan layer will need some
additional support to fully supply this role. This might come, for example,
from the walls between cells, along the length of the filament, which have not
been included in this simple model. Practically, however, these observations
show that assuming that the entire cell wall is a uniform load-bearing layer
provides a good estimate of the bacteria’s mechanical properties.

Figure 6:  Cross-sectional images of filamentous cyanobacteria, taken using
transmission electron microscopy. (a) _K. animale_. (b) _L. lagerheimii_. (c)
_O. lutea_. Arrows indicate: PG, peptidoglycan layer; CM, cytoplasmic
membrane; OM, outer membrane; and EL, external layer.

###  III.3 Filament curvature and shape

Finally, in order to relate the mechanical properties of filamentous
cyanobacteria to their shape and ability to generate forces, we observed
filaments under optimal conditions ($\sim$20∘C) and when chilled to reduce
metabolism and mobility ($\sim$2∘C). At room temperature the filaments were
active and glided steadily along the bottom of their containers, adopting a
curved shape, as shown in Fig. 1(a). When cooled, the bacteria were inactive
and displayed a more irregular, meandering shape, as in Fig. 1(b).

Filament shapes were characterised by looking at how their relative
orientation $\theta$, taken from tangent vectors, varies along their length
$s$. Four typical cases of active filaments of O. lutea are explored in Fig.
7, and the other two species behaved similarly. In Fig. 7(a) we see that the
filaments each have a relatively constant curvature, $\kappa=d\theta/ds$, so
that their shapes are well-approximated by circular arcs. We therefore fit
circles directly to the filament shapes, as demonstrated in Fig. 7(b), to
measure their average curvature. Curvature distributions were collected from
about a hundred individual filaments for the active and inactive cases of each
species. These are summarised in Fig. 8. The actively gliding members of all
three species have a preferred curvature of around 0.5 mm-1. For the inactive
filaments there was no preferred overall curvature, and the measured curvature
distributions instead showed a peak, or median value, around zero. It is clear
that for the active case, internal forces within the cyanobacteria filaments
break the symmetry, generating the pattern of compression and tension required
to maintain a curved shape. Indeed, this bias seems related to their motion,
as we also observed a strong preference for clockwise, over counter-clockwise,
motion in all three species.

Figure 7:  Freely gliding filamentous cyanobacteria adopt a curved shape. (a)
A plot of the orientation angle vs. position along a filament shows that the
filament shape has a constant rate of change of angle, i.e. a constant
curvature. Shown are four typical cases of O. lutea, corresponding to the
profiles shown in the upper left. There is a variation of curvature between
filaments. (b) Examples of active, gliding filamentous cyanobacteria (black)
showing curved configurations along with best-fit circles. Figure 8:
Curvature distributions for filamentous cyanobacteria, as measured by fitting
the filament shape to a circular arc, for (a) active gliding filaments at 20∘C
and (b) inactive filaments at 2∘C.

##  IV Discussion

Table 2:  Summary of derived quantities based on results in Table 1, showing the estimated cell wall thickness $\Delta{r}$, maximum cell wall stress $\sigma_{s}$, and bending energy per unit length $\mathcal{U}$. In all cases the uncertainty range gives the standard error of estimates propagated from uncertainties in the quantities in Table 1. Species |  $\Delta{r}$ (nm) |  $\sigma_{s}$ (kPa) |  $\mathcal{U}$ (J m-1)
---|---|---|---
_K. animale_ |  $35\pm 8$ |  $41\pm 13$ |  5.2 $\pm$ 1.2 $\times 10^{-12}$
_L. lagerheimii_ |  $90\pm 28$ |  $20\pm 9$ |  6.1 $\pm$ 2.0 $\times 10^{-12}$
_O. lutea_ |  $34\pm 8$ |  $29\pm 10$ |  3.7 $\pm$ 0.9 $\times 10^{-12}$

As outlined in the introduction, the mechanical origins of the gliding motion
of filamentous cyanobacteria has remained obscure, despite several alternative
models being proposed Hosoi (1951); Walsby (1968); Duggan  _et al._ (2007);
Risser  _et al._ (2014); Halfen and Castenholz (1970, 1971). One factor behind
this has been a lack of data about the mechanical response of the bacteria,
which can be used to constrain such models. Here, we will discuss how
measurements of the stiffness and other mechanical properties of the
cyanobacteria can help understand the forces at play inside a moving chain of
cyanobacteria cells. We have already seen in Section 3III.2 that the bending
stiffness of cyanobacteria can be largely attributed to a rigid cell envelope,
and we will expand this interpretation here to consider the magnitude of the
internal forces needed to sustain the curved shape of active filaments, along
with an estimate of the internal bending energy stored in this curvature.
Using a similar energy scale, we will also discuss the disordered shape of
inactive bacteria, in the context of a persistence length of a randomly driven
flexible rod.

The cyanobacteria studied here move with a slow gliding motion, at speeds of
about 3 $\mu$m s-1 and slowly rotate as they glide, in a corkscrew-like motion
Hoiczyk (2000); Halfen and Castenholz (1970); Hoiczyk and Baumeister (1995);
Burchard (1981). Their curved shape does not change significantly as they
advance, suggesting that the shape is dynamically maintained by something like
a compressional wave that accompanies the rotational motion. However, their
well-defined curvature largely disappears, on average, when the cells are
inactivated by lowering their temperature, further evidencing that their shape
results from an internally generated distribution of forces.

In any event, assuming that the filament behaves as a hollow cylinder of
radius $r$ and wall thickness $\Delta r$, we can estimate the magnitude of the
active forces needed to maintain a curvature $\kappa$ as

$\sigma_{s}=\frac{\kappa\beta}{\pi r^{2}\Delta r}=\kappa Er.$ (8)

This gives the maximum stress that would be felt in a hollow cylindrical beam
under uniform bending, or alternatively an estimate of the maximum internal
stress in the cell wall that would be needed to give rise to the observed
curvature. Based on the empirically determined values given in Table 1, we can
use Eq. 8 to estimate $\sigma_{s}$ in the three species of cyanobacteria
studied here. As shown in Table 2, these internal stresses are typically of
the order of tens of kPa. The energy stored in the bent shape can also be
estimated, by Eq. 3. As shown in Table 2, these internal bending energies
should reach magnitudes of a few pJ m-1, per unit length along the filament.

Although our focus here has been on the static mechanics of cyanobacteria,
these axial stresses can be linked to the various models of gliding motion. As
one example, Halfen and Castenholz Halfen and Castenholz (1970, 1971)
suggested that the fibril layer, which consists of helical structures lying
between the peptidoglycan layer and the outer membrane Read  _et al._ (2007),
contracts and in the process sends waves in the direction opposite to that of
motion. In the framework of this model, our measurements here give the
magnitude of the stresses and bending energies that would need to be generated
by these contractile waves, allowing more quantitative predictions to be
developed for the origins of the gliding motion.

Finally, an alternative characterisation of the shape of filaments is through
their persistence length, $P$, which describes the distance over which
correlations in the local filament orientation or direction are lost Doi and
Edwards (1986). This metric has been used to demonstrate that modern
cyanobacteria have a similar persistence length to fossil specimens from the
Precambrian Boal and Ng (2010), for example, or to estimate the bending
stiffness of microtubules Gittes  _et al._ (1993). Mathematically, $P$ can be
defined through the relation

$\langle\cos(\phi)\rangle=e^{-\Delta s/2P}$ (9)

where $\phi=\theta(s+\Delta s)-\theta(s)$ is the change in filament
orientation over a distance $\Delta s$ along its contour length, the factor of
$2$ in the exponent is appropriate for filaments confined to a surface Gittes
_et al._ (1993) and the angled brackets represent an average over the contour
$s$ and an ensemble of filaments.

Persistence lengths and angular correlations for all three cyanobacteria
species studied here were extracted from the skeletonised filament images
obtained in Section 3III.3. For the active case of gliding cyanobacteria, as
we showed in Fig. 7, the filaments adopt the shape of a circular arc rather
than a disordered shape. In this case we would expect that $\phi(\Delta
s)=\kappa\Delta s$, instead of following Eq. 9. In inactive filaments,
however, the persistence-length analysis is well-defined, as the filaments are
more disordered in their shape. Figure 9 shows the persistence length
measurements for inactive (non-motile, chilled) filaments, which are all in
the range of 5–10 mm. This is of a similar magnitude, although slightly larger
than, the values of 1.3–3.9 mm previously measured for $P$ in other species of
Oscillatoria Boal and Ng (2010).

Interestingly, these shape fluctuations can also be linked to an energy
density along the length of the filaments. For thermally driven systems of
slender filaments, such as microtubules, $P=\beta/k_{B}T$, where $k_{B}$ is
Boltzmann’s constant, and $T$ the system temperature Doi and Edwards (1986);
Gittes  _et al._ (1993); Boal and Ng (2010). Cyanobacteria are too large to be
in this limit–the persistence lengths based on the thermal energy $k_{B}T$ and
our measurements of $\beta$ would be several kilometers. However, an analogous
relationship may still hold, if the shape is determined by some actively
generated but incoherent distribution of forces. In this case we would
anticipate a stored strain energy along a segment of length $P$ to be of order
$\beta/P$, or that the total strain energy density in the filament is
$U\simeq\beta/P^{2}$. Given our measurements of $\beta$ and $P$ this is about
1 pJ m-1 for all three species studied here, which is surprisingly similar to
the bending energies of the gliding, uniformly curved specimens.
Alternatively, this means that on average the magnitude of the locally-defined
curvature $|d\theta/ds|$ is similar in both cases. A plausible interpretation
of this is that, when the filaments are not moving, the cells maintain some
degree of the internal stresses that would otherwise be coordinated into e.g.
a contractile wave, but that these forces are instead uncoordinated.

Figure 9:  Measured persistence length in inactive filaments of _K. animale_ ,
_L. lagerheimii_ , and _O. lutea_ , evaluated by analysing 164, 162 and 79
filaments for each species, respectively. Fits for the persistence length uses
equation 9. The shaded region represents extreme points of the distributions.

##  V Conclusions

In this work we have quantified the mechanical properties of three related
species of filamentous cyanobacteria, namely _K. animale_ , _L. lagerheimii_ ,
and _O. lutea_. These species all belong to the order _Oscillatoriales_ and
are of similar size and motile behaviour. This allows for robust validation of
the internal consistency of our methods. As a further consistency check, we
note some contemporaneous observations using three-point bending tests from a
collaborating lab Kurjahn  _et al._ (2022). Our observations focused on
quantifying and understanding the source of the bending stiffness or flexural
rigidity in filamentous cyanobacteria, as well as looking at their
implications on the shape and motion of these organisms.

We found that the mechanical responses of all three species were similar to
each other. Indeed, for many measurements there was just as much variation
between filaments of the same species as there was between species. The
average bending stiffness ranged between 2.6-6.0$\times 10^{-17}$ N m2, as
measured by monitoring the bending of filaments under flow in a custom-built
microfluidic flow cell. These observations were complemented by measurements
of the elastic modulus of the cell wall, through AFM techniques, and
measurements of the thickness of the cell wall and its component layers,
through TEM. By comparing the bending stiffness to that expected from a hollow
rigid cylinder, we were able to show that the resistance of filamentous
cyanobacteria to bending is largely due to the mechanical stiffness of their
characteristically thick cell walls.

To link the bending stiffness to the shape and motion of the cyanobacteria, we
characterised the shapes of filaments when they were actively gliding at room
temperature, and when their motion was halted by reducing the temperature of
their environment. While active, isolated filaments adopted a uniformly curved
shape, like circular arcs. The curvature distributions were shifted noticeably
away from zero, with mean curvatures of about 0.5 mm-1 in all cases. This bias
seems to be connected to their motion, which shows a strong preference for
clockwise over counter-clockwise rotation, as the filaments glide.

Finally, exploring these results in line with one of the theories of gliding
motion, which suggests that shape fluctuations or internally generated waves
drive filaments in gliding motion, we estimated the bending energy and maximum
axial stress in the cell walls of the filaments. Our results shows that
activity in filamentous cyanobacteria can induce axial stresses of up to tens
of kPa in the cell wall. These results are applicable to wide range of areas
where mechanical properties of filamentous cyanobacteria are needed, such as
modelling biofilm self-organisation and in the control of bio-fouling or
design of bio-reactors.

##  VI ACKNOWLEDGEMENTS

The authors would like to thank Dr. Graham J. Hickman for TEM sample
preparation and imaging, and Dr. Hickman and Kathryn Kroon for technical
support of confocal microscopy. The microscopy facilities were provided by the
Imaging Suite at the School of Science and Technology at Nottingham Trent
University. The authors would also like to thank Marco G. Mazza, Jan Cammann
and Stefan Karpitschka for valuable discussions. Finally, the authors thank
the SAG culture collection in Göttingen, and specifically Maike Lorenz, for
support and advice including in the supply and maintenance of the
cyanobacteria culture.

##  Appendix A Appendix

### Flow profile inside the channel

To calculate the flow profile around the filaments we make use of Boussinesq’s
series solution for Poiseuille flow in a rectangular channel Boussinesq
(1868); White (2006). The flow along a channel of width $w$ and height $h$,
with a coordinate system centred at its midpoint, such that $-w/2\leq x\leq
w/2$ and $-h/2\leq z\leq h/2$, can be given by

$u_{y}=\frac{4w^{2}U_{0}}{\pi^{3}k}\sum^{\infty}_{n=1,3,5,...}\bigg{(}\frac{(-1)^{(n-1)/2}}{n^{3}}\\\
\times\bigg{[}1-\frac{\cosh(n\pi z/w)}{\cosh(n\pi h/2w)}\bigg{]}\cos(n\pi
x/w)\bigg{)}.$ (10)

Here, $U_{0}$ is the average flow speed in the channel, set by the rate of the
external syringe pumps, and $k$ is an effective permeability given by

$k=\frac{w^{2}}{12}\bigg{(}1-\frac{192w}{\pi^{5}h}\sum^{\infty}_{n=1,3,5,...}\frac{\tanh(n\pi
h/2w)}{n^{5}}\bigg{)}.$ (11)

This solution is appropriate for laminar flows. For the fastest flows in our
devices $U_{0}$ = 3.5 mm s-1. Using the hydraulic diameter $D_{H}=2wh/(w+h)$
to calculate the Reynolds number $\mathrm{Re_{c}}=\rho U_{0}D_{H}/\mu$ of the
channel flow gives a maximum $\mathrm{Re_{c}}\approx 0.5$, well within the
laminar limit.

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# Distribution of the zeros of polynomials near the unit circle

Mithun Kumar Das  Mathematics Section, The Abdus Salam International Centre
for Theoretical Physics, Str. Costiera, 11, 34151 Trieste, Italy
and
School of Mathematical Sciences, National Institute of Science Education and
Research, A CI of Homi Bhabha National Institute, Jatni, Khurda, 752050,
India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We estimate the number of zeros of a polynomial in $\mathbb{C}[z]$ within any
small circular disc centered on the unit circle, which improves and
comprehensively extends a result established by Borwein, Erdélyi, and Littmann
[4] in 2008. Furthermore, by combining this result with Euclidean geometry, we
derive an upper bound on the number of zeros of such a polynomial within a
region resembling a gear wheel. Additionally, we obtain a sharp upper bound on
the annular discrepancy of such zeros near the unit circle. Our approach
builds upon a modified version of the method described in [4], combined with
the refined version of the best-known upper bound for angular discrepancy of
zeros of polynomials.

###### Key words and phrases:

Zeros of polynomials, uniform distribution, location of zeros

###### 2010 Mathematics Subject Classification:

30C15, 11K38

##  1\. Introduction

Let $P_{n}$ be a polynomial in $\mathbb{C}[z]$ of degree $n$. Our primary
focus in this study is to understand the distribution of the zeros of $P_{n}$
on the complex plane, emphasizing their distribution within and around the
unit disk. One of the most extensively studied classes of polynomials is the
family of unimodular polynomials denoted as $\mathcal{K}_{n}$. These
polynomials are defined as follows:

$\displaystyle\mathcal{K}_{n}:=\Big{\\{}P_{n}:P_{n}(z)=\sum_{k=0}^{n}a_{k}z^{k},\>a_{k}\in\mathbb{C},\,|a_{k}|=1\Big{\\}}.$
(1.1)

If we further restrict the coefficients to only $-1$ and $1$, such polynomials
are called Littlewood polynomials, named in honor of Littlewood, who
extensively studied various analytical properties (the mean value, the number
of zeros, etc.) related to polynomials and power series with constrained
coefficients lying on the complex unit circle $|z|=1$ (see [10, 11]). The
study of complex zeros of Littlewood polynomials and polynomials with
coefficients $\\{-1,\,0,\,1\\}$ is an old subject, dating back to the
pioneering work of Bloch and Pólya [2]. Their research established that, on
average, such polynomials of degree $n$ have at most
$\operatorname{O}(\sqrt{n})$ real roots. Modern investigations into the
distribution of zeros often originate from the seminal result of Erdős and
Turán[7], which addresses the angular equidistribution of zeros and is
applicable to polynomials within the $\mathbb{C}[z]$. In a later section, we
will delve into a more detailed discussion of this matter. In their work [3],
Borwein and Erdélyi explore a broader perspective by considering a
generalization of the class $\mathcal{K}_{n}$ as follows:

$\displaystyle\mathcal{G}_{n}:=\Big{\\{}P_{n}:P_{n}(z)=\sum_{k=0}^{n}a_{k}z^{k},\>a_{k}\in\mathbb{C},\,|a_{0}|=|a_{n}|=1,\,|a_{k}|\leq
1\Big{\\}}.$ (1.2)

They demonstrate that any polygon with vertices on the unit circle $|z|=1$
contains at most $\operatorname{O}(\sqrt{n})$ zeros of such polynomials. Such
a result has been established in [14, 15] for random polynomials.

In 2008, Borwein, Erdélyi, Littmann [4] studied the distribution of zeros for
any polynomial in $\mathcal{G}_{n}$ that lies within small circles centered on
the unit circle. Specifically, they established the following result:

######  Theorem 1 (Theorem 1, [4]).

Let $P_{n}\in\mathcal{G}_{n}$ be a polynomial of degree $n$. Then any disk of
radius $33\pi\log{n}/\sqrt{n},(\leq 1)$ with the center on the unit circle
$|z|=1$ contains at least $8\sqrt{n}\log{n}$ zeros of $P_{n}$.

For random polynomials, Pritsker and Sola [14], as well as Pritsker and Yeager
[15], have derived the expectation of the above result asymptotically. In
their work [14], they considered the polynomial
$P_{n}(z)=\sum_{k=0}^{n}C_{k}z^{k}$ with coefficients that are iid complex
random variables with absolutely continuous distribution and
$\mathbb{E}(|C_{0}|^{t})<\infty$ for any $t>0$. Then, they showed that the
expected number of zeros inside any disc of radius $r<2$ with center on the
unit circle $|z|=1$ is given by

$\displaystyle\frac{2}{\pi}\arcsin(r/2)n+\operatorname{O}(\sqrt{n\log{n}})\quad\mbox{
as }n\rightarrow\infty.$

##  2\. New results

After the pioneering work of Borwein, Erdélyi and Littmann [4] on the
polynomial class $\mathcal{G}_{n}$, no further extensions of general
polynomials in $\mathbb{C}[z]$ have been explored, except for the results
mentioned above about to random polynomials. In this study, we establish a
lower bound for the number of zeros of polynomials in $\mathbb{C}[z]$ situated
within a circular disk, with its center located on the unit circle
$\mathbb{T}$. Furthermore, we introduce a parametrization to the radius of
this circular disk. Defining some notations regarding the $p$-norm of
polynomials on the unit circle $\mathbb{T}$ is necessary to present our
results. Let

$\displaystyle
P_{n}(z)=c_{n}\prod_{j=1}^{n}\big{(}z-\alpha_{j}\big{)}=\sum_{j=0}^{n}c_{j}z^{j}$
(2.1)

be a polynomial of degree $n$, with $c_{0}\neq 0$ and roots
$\alpha_{j}=\rho_{j}\,e(\theta_{j})$, $j=1,2,\ldots,n$, where
$e(\theta):=e^{2\pi i\theta}$ and throughout the paper we use the same
notation. For $0<p<\infty$, a $p$-norm of $P_{n}$ is defined on unit circle by

$\|P_{n}\|_{p}=\left(\frac{1}{2\pi}\int_{\mathbb{T}}|P_{n}\big{(}e^{i\theta}\big{)}\big{|}^{p}\,\text{\rm
d}\theta\right)^{\frac{1}{p}}=\left(\int_{\mathbb{R}/\mathbb{Z}}|P_{n}\big{(}e(\theta)\big{)}\big{|}^{p}\,\text{\rm
d}\theta\right)^{\frac{1}{p}}.$

Thus, for $p=2$, we get $\|P_{n}\|_{2}^{2}=|c_{n}|^{2}+\ldots+|c_{0}|^{2}$.
The $\sup$-norm of $P_{n}$on $\mathbb{T}$ is defined by
$\|P_{n}\|_{\infty}=\sup_{\mathbb{T}}|P_{n}\big{(}e^{i\theta}\big{)}\big{|}$.
Furthermore, let $E$ be the subset of $[0,1]$ which is given by
$E=\\{\theta\in[0,1]:|P_{n}(e(\theta))|<1\\}$ and for $0<p\leq\infty$, defined
the logarithmic $p$-norm by

$\displaystyle
B_{p}(P_{n})=\begin{cases}\log{\left(\frac{\|P_{n}\|_{\infty}}{\sqrt{|c_{0}c_{n}|}}\right)}&\mbox{
if }p=\infty,\\\
(1-|E|)\log{\left(\frac{\|P_{n}\|_{p}}{\sqrt{|c_{0}c_{n}|}}\right)}+\frac{1}{ep}&\mbox{
if }0<p<\infty.\end{cases}$ (2.2)

Now we state our new results as follows:

######  Theorem 2.

Let $P_{n}(z)=\sum_{k=0}^{n}c_{k}z^{k}$ be a polynomial of degree $n$ such
that $P_{n}(0)\neq 0$ and $\theta$ be a fixed real number in
$[\frac{1}{2},\,1]$. Then any disk of radius
$\gamma_{n}=7{\frac{(2B_{\infty}(P_{n}))^{\theta}}{\sqrt{n}}}\leq 1$ with
center on the unit circle $|z|=1$, contains at least
$\sqrt{n}(2B_{\infty}(P_{n}))^{\theta}$ zeros of $P_{n}$.

Furthermore, if $p\in(0,\,\infty)$ and $P_{n}$ satisfy the conditions
$|c_{n}c_{0}|\geq 1$ and $\|P_{n}\|_{p}\geq 1$, then any disk of radius
$\gamma_{n}=9{\frac{(2B_{p}(P_{n}))^{\theta}}{\sqrt{n}}}\leq 1$ with center on
the unit circle $|z|=1$, contains at least $\sqrt{n}(2B_{p}(P_{n}))^{\theta}$
zeros of $P_{n}$.

The subsequent result presents an analog of Theorem 2 for the class of
polynomials $P_{n}\in\mathcal{G}_{n}$.

######  Theorem 3.

Let $P_{n}\in\mathcal{G}_{n}$ a polynomial and $\theta\in[\frac{1}{2},\,1]$ be
any fixed real number. Then any open disk of radius
$\gamma_{n}=9{\frac{(\log{n})^{\theta}}{\sqrt{n}}}\leq 1$ with centre on
$|z|=1$ contains at least $\sqrt{n}(\log{n})^{\theta}$ zeros of $P_{n}$.

######  Remark 4.

In Theorem 3, if we set $\theta=1$, we can draw a meaningful comparison with
Theorem 1. Notably, when the condition $\gamma_{n}\leq 1$ holds, we are
permitted to work with polynomials of degree as small as $6170$, whereas
Theorem 1 imposes a strict lower limit of $2307256$ for the polynomial degree.
Furthermore, the disk’s radius described in Theorem 3 is approximately $11.5$
times smaller than the radius mentioned in Theorem 1. However, in our case,
the number of zeros within the disk is only $1/8$-th of the number of zeros
within the disk discussed in [4].

Further, if we decide to maintain the same radius, denoted by
$\gamma_{n}=33\pi{\frac{\log{n}}{\sqrt{n}}}$, then the disk of radius
$\gamma_{n}$ contains at least $31\sqrt{n}\log{n}$ zeros of $P_{n}$.

Now, we look at an application of Theorem 2 in order to estimate the number of
zeros inside a certain gear wheel type region. In mathematics, a gear wheel
(also known as a gear or cogwheel) is typically defined in the context of
geometry and trigonometry. Gears are used to transmit motion and force from
one rotating object to another, and they have a wide range of applications in
mechanical engineering, physics, and other fields. A geometric construction
for the gear wheel region we consider here is the following: Create a sizeable
circular disk, denoted as $C$, and symmetrically position $T$ smaller circles
along the boundary of $C$, maintaining a uniform width gap $d$ between any two
adjacent smaller circles. After resolving any intersections that may occur
during this process, the result is a gear with $T$ teeth, each tooth
possessing a width of $d$. Please refer to Figure 1, generated using
Mathematica software for visual reference. We prove the following result on
zero distribution result of $P_{n}$ on the gear wheel.

######  Theorem 5.

Let $P_{n}(z)=\sum_{k=0}^{n}c_{k}z^{k}$ be a polynomial of degree $n$ such
that $P_{n}(0)\neq 0$. Set
$\gamma_{n}=7{\frac{(2B_{\infty}(P_{n}))^{\theta}}{\sqrt{n}}}\leq\frac{1}{2}$
with $\theta\in[\frac{1}{2},\,1]$ and $0\leq\delta<1$. Then the gear wheel
region on the unit disc with approximately
$\frac{\pi}{1+\delta}(\arcsin(\frac{\gamma_{n}}{2}\sqrt{4-\gamma_{n}}))^{-1}$
teeth and each of width
$2\delta\arcsin\left(\frac{\gamma_{n}}{2}\sqrt{4-\gamma_{n}^{2}}\right)$,
contain at most $n\Big{(}1-\frac{0.97\pi}{7(1+\delta)}\Big{)}$ zeros of
$P_{n}$.

Further, if $P_{n}$ satisfy the conditions $|c_{n}c_{0}|\geq 1$ and
$\|P_{n}\|_{p}\geq 1$ and if we assume
$\gamma_{n}=9{\frac{(2B_{p}(P_{n}))^{\theta}}{\sqrt{n}}}\leq\frac{1}{2}$ then
the same gear wheel contains at most
$n\Big{(}1-\frac{0.97\pi}{9(1+\delta)}\Big{)}$ zeros of $P_{n}$.

######  Remark 6.

If $P_{n}\in\mathcal{G}_{n}$ we can choose
$\gamma_{n}=9{\frac{(\log{n})^{\theta}}{\sqrt{n}}}\leq\frac{1}{2}$. If
$\theta=1$, the inequality $\gamma_{n}\leq\frac{1}{2}$ is valid for $n\geq
35582$. Thus, the associated gear wheel contains at most $66\%$ zeros of the
polynomial $P_{n}.$

Figure 1. Left: Gear wheel of 78 teeth with width 0
($\gamma_{n}\approx.04,\delta=0$); Right: Gear wheel of 48 teeth with width
$\pi/120$ ($\gamma_{n}\approx\pi/60,\delta\approx.2577$).

###  2.1. Height or mean of polynomials

We consider here three notions of size, or height, of a polynomial $P_{n}(z)$
of degree $n$. The simple one is given by

$\|P_{n}\|_{\infty}=\max_{|z|=1}\,|P_{n}(z)|.$

The Mahler measure (geometric mean) $M(P_{n})$ of $P_{n}$ is defined as the
following equivalent forms by using Jensen’s formula,

$M(P_{n})=\exp{\left(\frac{1}{2\pi}\int_{\mathbb{T}}\log\\!\left(\big{|}P_{n}\big{(}e^{i\theta}\big{)}\big{|}\right)\,\text{\rm
d}\theta\right)}=\exp{\left(\int_{\mathbb{R}/\mathbb{Z}}\log\\!\left(\big{|}P_{n}\big{(}e(\theta)\big{)}\big{|}\right)\,\text{\rm
d}\theta\right)}=|c_{n}|\prod_{j=1}^{n}\max\\{1,|\alpha_{j}|\\}.$

The logarithmic Mahler measure, denoted by $m(P_{n})$, is the natural
logarithm of $M(P_{n})$. Another one is defined as follows:

$M^{+}(P_{n})=\exp{\left(\frac{1}{2\pi}\int_{\mathbb{T}}\log^{+}\\!\left(\big{|}P_{n}\big{(}e^{i\theta}\big{)}\big{|}\right)\,\text{\rm
d}\theta\right)}=\exp{\left(\int_{\mathbb{R}/\mathbb{Z}}\log^{+}\\!\left(\big{|}P_{n}\big{(}e(\theta)\big{)}\big{|}\right)\,\text{\rm
d}\theta\right)},$

where $\log^{+}{x}=\max\\{0,\,\log{x}\\}$ and then set
$m^{+}(P_{n})=\log{M^{+}(P_{n})}$. From the definition, one observed that

$M(P_{n})\leq M^{+}(P_{n})\leq\|P_{n}\|_{\infty}.$ (2.3)

For a monic polynomial $P_{n}$, the Mahler measure $M(P_{n})\geq 1$ implies
that the logarithmic Mahler measure $m(P_{n})\geq 0$. A classical theorem of
Kronecker implies that $M(P_{n})=1$ if and only if $P_{n}\in\mathbb{Z}[x]$ is
a product of cyclotomic polynomials and the monomial $x$. An old question asks
by Lehmer [9] if there exist polynomials with Mahler measure between $1$ and
$1+\epsilon$ for arbitrary $\epsilon>0$, and he notes that the polynomial

$L(x)=x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1$

has $M(L)=1.1762808\ldots$. Several extensions have been done. But following
[1, 6, 20] we have

$\displaystyle
m(P_{n})>\frac{1}{4}\Big{(}\frac{\log\,\log{n}}{\log{n}}\Big{)}^{3}$ (2.4)

for other general polynomials.

###  2.2.  Discrepancy of zeros of polynomials

This section will discuss a modified analysis of the angular distribution of
the zeros of the polynomial $P_{n}(z)$ given in (2.1). This refinement serves
as a fundamental tool for deriving the new results mentioned earlier. A
standard way to define the normalized zero-counting measure for such a
polynomial is

$\tau_{n}=\frac{1}{n}\sum_{k=1}^{n}\delta_{\alpha_{k}}.$

A sector $S(\alpha,\,\beta)$ on $\mathbb{C}$ is defined as

$S(\alpha,\,\beta)=\\{z\in\mathbb{C}:0<\alpha\leq\arg{z}<\beta\leq 2\pi\\}.$
(2.5)

Then, the discrepancy between the normalized zero counting measure in
$S(\alpha,\,\beta)$ and the Lebesgue measure of the arc on the unit circle in
this sector is known as the angular discrepancy. It is denoted by
$\mathcal{D}(P_{n})$ and given by

$\displaystyle\mathcal{D}(P_{n})=\sup_{[\alpha,\beta]\subset\mathbb{T}}\left|\tau_{n}{(S(\alpha,\,\beta))}-\frac{\beta-\alpha}{2\pi}\right|.$

In a pioneering work in 1950 concerning the angular discrepancy of zeros,
Erdős, and Turán [7] proved that

$\mathcal{D}(P_{n})\leq C\sqrt{\frac{B_{\infty}(P_{n})}{n}}$ (2.6)

with $C=16$. In 1954, Ganelius [8] established (2.6) with the constant
$C=\sqrt{2\pi/k}=2.5619\ldots$, where
$k=1/1^{2}-1/3^{2}+1/5^{2}-\ldots=0.9159\ldots$ denotes the Catalan’s
constant. In 1992, Mignotte [13] refined Ganelius’s result by establishing the
stronger inequality

$\mathcal{D}(P_{n})\leq
C\sqrt{\frac{1}{n}\,m^{+}\left(\frac{P_{n}}{\sqrt{|c_{0}c_{n}|}}\right)}.$
(2.7)

with the same constant $C=\sqrt{2\pi/k}=2.5619\ldots$. In 2019, Soundararajan
[19] improved this result by establishing (2.7) with the constant
$C=\frac{8}{\pi}=2.5464\ldots$. Only recently, Carneiro et al. [5], where the
author of this article is one of the co-authors, improved the upper bound in
(2.7) by establishing the following result.

######  Theorem 7 (Carneiro et al. [5], (2021)8).

Let $P_{n}(z)=\sum_{k=0}^{n}c_{k}z^{k}$ be a polynomial of degree $n$ such
that $P_{n}(0)\neq 0$. Then we have

$\mathcal{D}(P_{n})\leq\frac{4}{\sqrt{\pi}}\sqrt{\frac{1}{n}\,m^{+}\left(\frac{P_{n}}{\sqrt{|c_{0}c_{n}|}}\right)}.$

However, more recently, in a preprint, Shu and Wang [18] obtained a sharp
discrepancy upper bound obtained in terms of the weaker norm. Precisely, they
proved

######  Theorem 8 (Shu-Wang(2021) [18]).

Let $P_{n}(z)=\sum_{k=0}^{n}c_{k}z^{k}$ be a polynomial of degree $n$ such
that $P_{n}(0)\neq 0$. Then

$\mathcal{D}(P_{n})\leq\sqrt{\frac{2}{n}B_{\infty}(P_{n})}.$ (2.8)

Note that in Theorem 7 and Theorem 8, the computation of the asymptotic
formulas for the associated norms is more complicated than the $p$-norms,
$0<p<\infty$, and thus we need to use a trivial upper bound only. In the
following proposition, we refine the Theorem 7 in terms of $p$-norm.

######  Proposition 9.

Let $p\in(0,\infty)$ and $P_{n}$ be a polynomial of degree $n$ such that
$P_{n}(0)\neq 0$ and $\|P_{n}\|_{p}\geq 1$. Then

$\mathcal{D}(P_{n})\leq\frac{4}{\sqrt{\pi}}\sqrt{\frac{B_{p}(P_{n})}{n}}.$
(2.9)

Although, the leading constant in the upper bound in Proposition 9 may be
larger in compare to (2.8), but we have $\|P_{n}\|_{p}\leq\|P_{n}\|_{\infty}$.

Conditionally, we obtain the following explicit result with the distribution
of $P_{n}$ on the unit circle.

######  Corollary 10.

Let $P_{n}\in\mathcal{K}_{n}$ such that $\|P_{n}\|_{p}\geq 1$. Then

$\left|\tau_{n}(S(\alpha,\,\beta))-\frac{\beta-\alpha}{2\pi}\right|\leq\frac{2\sqrt{2}}{\sqrt{\pi}}\sqrt{\frac{(1-|E|)\log{(n+1)}+{e}^{-1}}{n}}.$

Further, if $\theta$ is drawn randomly from $(0,1)$, and
$P_{n}(e(\theta))/\sqrt{n+1}$ converges in distribution (and in moments) to a
standard complex Gaussian. Then, for sufficiently large $n$, we have

$\left|\tau_{n}(S(\alpha,\,\beta))-\frac{\beta-\alpha}{2\pi}\right|<\frac{2\sqrt{2}}{\sqrt{\pi
n}}\sqrt{\Big{(}1-\frac{1}{4}\operatorname{erf}^{2}{\Big{(}\sqrt{\frac{2}{n+1}}\Big{)}}\Big{)}\log{(n+1)}+e^{-1}},$

where $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt$.

Now, we look at a particular class of polynomials and find the effective upper
bound of the discrepancy.

######  Example 11 (Rudin–Shapiro polynomials).

The Rudin–Shapiro polynomials are defined recursively, as follows:
$P_{0}(z)=1=Q_{0}(z)$ and for $k=0,\,1,\,2,\,\ldots$,

$P_{k+1}(z)=P_{k}(z)+z^{2^{k}}Q_{k}(z)\quad\mbox{and}\quad
Q_{k+1}(z)=P_{k}(z)-z^{2^{k}}Q_{k}(z).$

We see that $P_{k}$ and $Q_{k}$ are two Littlewood polynomial of degree $n-1$,
where $n:=2^{k}$. By the Parseval formula, $\|P_{n}\|_{2}=2^{k/2}$ . By using
this we obtain a discrepancy upper bound in Proposition 9

$\frac{2\sqrt{2}}{\sqrt{\pi}}\sqrt{\frac{(1-|E|)k\log{2}+e^{-1}}{n-1}}.$

Recently, Rodgers [17] proved Saffari’s conjecture and this gives that
$|E|\sim(2)^{-(k+1)}$ as $k\rightarrow\infty$. Thus, for sufficiently large
$k$, the discrepancy upper bound in Proposition 9 is

$\left|\tau_{n}(S(\alpha,\,\beta))-\frac{\beta-\alpha}{2\pi}\right|\leq\frac{2}{\sqrt{\pi}}\sqrt{\Big{(}2-\frac{1}{n-1}\Big{)}\frac{k\log{2}+e^{-1}}{n-1}}.$
(2.10)

###  2.3. Annular discrepancy of zeros of a polynomial

Our next result directly applies angular discrepancy, drawing inspiration from
recent works [14, 16, 15]. We establish an upper bound for the discrepancy
between the counting measure of zeros of the polynomial $P_{n}$ within an
annular sector and the uniform measure on the unit circle $\mathbb{T}$. To
provide context, let’s define the annulus $\mathcal{A}_{\rho}$ and the annular
sector $\mathcal{A}_{\rho}(\alpha,\beta)$ on the complex plane $\mathbb{C}$,
where $0<\rho<1$:

$\mathcal{A}_{\rho}=\\{z\in\mathbb{C}:\rho<|z|<1/\rho\\}\,\mbox{ and
}\,\mathcal{A}_{\rho}(\alpha,\,\beta)=\\{z\in\mathbb{C}:\rho<|z|<1/\rho,\,0<\alpha\leq\arg{z}<\beta\leq
2\pi\\}.$ (2.11)

Then, we obtain the following result.

######  Theorem 12.

Let $P_{n}(z)=\sum_{k=0}^{n}c_{k}z^{k}$ be a polynomial of degree $n$ such
that $|c_{n}c_{0}|\geq 1$ and
$\frac{\|P_{n}\|_{\infty}}{\sqrt{|c_{0}c_{n}|}}\leq e^{\epsilon n}$ for
sufficiently small $\epsilon>0$. Then for sufficiently large $n$ (depending on
$\epsilon$ and $\rho$), there exists a suitably small $\epsilon^{\prime}>0$
such that

$\left|\tau_{n}(\mathcal{A}_{\rho}(\alpha,\,\beta))-\frac{\beta-\alpha}{2\pi}\right|\leq\left(\frac{4}{\sqrt{\pi}}+\epsilon^{\prime}\right)\sqrt{\frac{B_{p}(P_{n})}{n}}.$

Further, for $P_{n}\in\mathcal{G}_{n}$ and sufficiently large $n$ (depending
on $\epsilon$ and $\rho$), there exists a suitably small
$\epsilon^{\prime\prime}>0$ such that

$\left|\tau_{n}(\mathcal{A}_{\rho}(\alpha,\,\beta))-\frac{\beta-\alpha}{2\pi}\right|\leq\sqrt{\frac{\log{(n+1)}}{n}}(C+\epsilon^{\prime\prime}),$

where $C$ is given by

$C=\min\left\\{\sqrt{\frac{8}{\pi}(1-|E|+\frac{1}{e\log{(n+1)}}},\,\sqrt{2}\right\\}\leq
1.974\ldots.$

Note that if $|E|<1-\pi/4$ then $C=\sqrt{2}$. The inequality
$\frac{\|P_{n}\|_{\infty}}{\sqrt{|c_{0}c_{n}|}}\leq e^{\epsilon n}$ with
suitably small $\epsilon$ is essentially the condition for angular
equidistribution zeros of $P_{n}$ (see Section 1 in [19]).

##  3\. Proof of results

In order to prove the main results in Section 2, i.e., Theorem 2, 3, and 5, we
need to establish Proposition 9, Theorem 12 and Proposition 13 at the
beginning.

###### Proof of Proposition 9.

Let us set $E^{c}=[0,1]\setminus E$. Then, by the hypothesis
$\|P_{n}\|_{p}\geq 1$ implies that $|E^{c}|\neq\emptyset$. Thus, we write

$m^{+}(P_{n})=\int_{E^{c}}\log{|P_{n}(e(\theta))|}d\theta=\frac{|E^{c}|}{p}\left(\frac{1}{|E^{c}|}\int_{E^{c}}\log{|P_{n}(e(\theta))|^{p}}d\theta\right).$

By using the concavity of the logarithmic function along with the Jensen’s
inequality on the integral above, we obtain

$m^{+}(P_{n})\leq\frac{|E^{c}|}{p}\log\left(\frac{1}{|E^{c}|}\int_{E^{c}}{|P_{n}(e(\theta))|^{p}}d\theta\right)=\frac{|E^{c}|}{p}\log{\frac{1}{|E^{c}|}}+(1-|E|)\log{\|P_{n}\|_{p}}.$

But $\max_{|E^{c}|\in(0,1]}|E^{c}|\log{\frac{1}{|E^{c}|}}=\frac{1}{e}$, and
hence, we have

$m^{+}(P_{n})\leq(1-|E|)\log{\|P_{n}\|_{p}}+\frac{1}{ep}.$ (3.1)

The above argument is essentially a part of the proof [14, Proposition 2.2].
We conclude the result by combining (3.1) and Theorem 7. ∎

###### Proof of Corollary 10.

For $P_{n}\in\mathcal{K}_{n}$, we have $\log{|c_{0}c_{n}|}=0$ and
$\|P_{n}\|_{2}=\sqrt{n+1}$ and hence we obtain the first part of the corollary
form Proposition 9.

Since $P_{n}(e(\theta))/\sqrt{n+1}$ converges to a standard complex Gaussian,
for sufficiently large $n$ we get

$\displaystyle\operatorname{meas}\left\\{\theta\in[0,\,1]\,:\,\frac{1}{\sqrt{n+1}}P_{n}(e(\theta))\in
R\right\\}=\frac{1}{\pi}\int_{R}e^{-x^{2}-y^{2}}dx\,dy,$

where $R$ is any rectangle in $\mathbb{C}$. Let $R$ be a square of sides
$\sqrt{\frac{2}{n+1}}$ and centered at the origin, then by simple Euclidean
geometry, we have

$|E|>\frac{1}{\pi}\int_{R}e^{-x^{2}-y^{2}}dx\,dy=\frac{1}{4}\operatorname{erf}^{2}{\Big{(}\sqrt{\frac{2}{n+1}}\Big{)}}.$

Hence, the result follows. ∎

###### Proof of Theorem 12.

We recall the annulus $\mathcal{A}_{\rho}$ annular sector
$\mathcal{A}_{\rho}(\alpha,\,\beta)$ from (2.11). In the following
proposition, we obtain an annular discrepancy upper bound of zeros of a
polynomial, which is a better upper bound in comparison to [14, Proposition
2.1].

######  Proposition 13.

Let $P_{n}(z)=\sum_{k=0}^{n}c_{k}z^{k}$ be a polynomial of degree $n$ such
that $|c_{n}c_{0}|\geq 1$ and for $p\in(0,\,\infty)$, recall $B_{p}(P_{n})$
from Proposition 9. For any $\rho\in(0,\,1)$ and $0\leq\alpha<\beta<2\pi$, we
have

$\tau_{n}(\mathbb{C}\setminus\mathcal{A}_{\rho})\leq\frac{2}{n(1-\rho)}m\left(\frac{P_{n}}{\sqrt{|c_{0}c_{n}|}}\right)\leq\frac{2}{n(1-\rho)}B_{p}(P_{n})$
(3.2)

and

$\left|\tau_{n}(\mathcal{A}_{\rho}(\alpha,\,\beta))-\frac{\beta-\alpha}{2\pi}\right|\leq\sqrt{\frac{2}{n}\min\left\\{\frac{8}{\pi}B_{p}(P_{n}),\,B_{\infty}(P_{n})\right\\}}+\frac{2}{n(1-\rho)}B_{p}(P_{n})$
(3.3)

Assume the mild condition $\frac{\|P_{n}\|_{\infty}}{\sqrt{|c_{0}c_{n}|}}\leq
e^{\epsilon n}$ for suitably small $\epsilon>0$. Then we must find a suitably
small $\epsilon^{\prime}>0$ such that
$\sqrt{B_{p}(P_{n})/n}\leq\frac{1-\rho}{2}\epsilon^{\prime}$. Specifically, we
can choose
$\epsilon^{\prime}=\frac{2}{1-\rho}\sqrt{(1-|E|)\epsilon+\frac{1}{epn}}$, and
this would be sufficiently small whenever $n\geq n_{\epsilon,\rho}$. Thus, for
$p\in(0,\,\infty)$, (3.3) of Proposition 13 gives

$\displaystyle\left|\tau_{n}(\mathcal{A}_{\rho}(\alpha,\,\beta))-\frac{\beta-\alpha}{2\pi}\right|$
$\displaystyle\leq\sqrt{\frac{2}{n}\min\left\\{\frac{8}{\pi}B_{p}(P_{n}),\,B_{\infty}(P_{n})\right\\}}+\epsilon^{\prime}\sqrt{\frac{B_{p}(P_{n})}{n}}\leq\left(\frac{4}{\sqrt{\pi}}+\epsilon^{\prime}\right)\sqrt{\frac{B_{p}(P_{n})}{n}}.$

For $p=2$ we have $\|P_{n}\|_{2}\leq\sqrt{n+1}$ for $P_{n}\in\mathcal{G}_{n}$.
This gives $B_{2}(P_{n})=\frac{1}{2}((1-|E|)\log{(n+1)}+e^{-1})$. Also, we get
the upper bound $B_{\infty}(P_{n})\leq\log{(n+1)}$. Thus, (3.3) of Proposition
13 gives

$\displaystyle\left|\tau_{n}(\mathcal{A}_{\rho}(\alpha,\,\beta))-\frac{\beta-\alpha}{2\pi}\right|$
$\displaystyle\leq\sqrt{\frac{\log{(n+1)}}{n}}$
$\displaystyle\times\left(\min\left\\{\sqrt{\frac{8}{\pi}(1-|E|+(e\log{(n+1)})^{-1})},\,\sqrt{2}\right\\}+\bigg{(}1-|E|+\frac{1}{e\log{(n+1)}}\bigg{)}\epsilon^{\prime}\right).$

Now, we choose
$\epsilon^{\prime\prime}=\bigg{(}1-|E|+\frac{1}{e\log{(n+1)}}\bigg{)}\epsilon^{\prime}$
in above to complete the proof. ∎

###### Proof of Proposition 13.

Let $D_{\rho}$ be a disk which is defined by
$D_{\rho}=\\{z\in\mathbb{C}:|z|<\rho\\}$ where $0<\rho<1$. Then we have

$\displaystyle\tau_{n}(\mathbb{C}\setminus\mathcal{A}_{\rho})=\tau_{n}(\mathbb{C}\setminus
D_{\frac{1}{\rho}})+\tau_{n}(D_{\rho}).$

By appealing [14, Lemma 5.1] in above equality we obtain the first inequality
of (3.2), i.e.,

$\tau_{n}(\mathbb{C}\setminus\mathcal{A}_{\rho})\leq\frac{2}{n(1-\rho)}m\left(\frac{P_{n}}{\sqrt{|c_{0}c_{n}|}}\right).$
(3.4)

To prove the second inequality of (3.2), at first, we recall $E$ from
Proposition 9 and we have

$m(P_{n}/\sqrt{|c_{0}c_{n}|})\geq 0.$

Then we obtain

$\displaystyle
m\bigg{(}P_{n}/\sqrt{|c_{0}c_{n}|}\bigg{)}=\int_{E^{c}}\log{|P_{n}(e(\theta))|}d\theta+\int_{E}\log{|P_{n}(e(\theta))|}d\theta-\frac{1}{2}\log|c_{0}c_{n}|.$

But from the definition of $E$ we have
$\int_{E}\log{|P_{n}(e(\theta))|}d\theta\leq 0$. Combining this with (3.1) and
using the condition $|c_{n}c_{0}|\geq 1$, we get

$m(P_{n}/\sqrt{|c_{0}c_{n}|})\leq\int_{E^{c}}\log{|P_{n}(e(\theta))|}d\theta-\frac{1-|E|}{2}\log|c_{0}c_{n}|\leq
B_{p}(P_{n}),$ (3.5)

for any $p\in(0,\,\infty)$. This completes the proof of (3.2).

To prove the inequality (3.3), we are following the same strategy as in the
proof of Proposition 2.1 in [14]. That is, we write

$\tau_{n}(\mathcal{A}_{\rho}(\alpha,\beta))=\tau_{n}(S(\alpha,\beta))-\tau_{n}(S(\alpha,\,\beta)\setminus\mathcal{A}_{\rho}(\alpha,\beta)).$

By using triangle inequality and the inequality
$\tau_{n}(S(\alpha,\,\beta)\setminus\mathcal{A}_{\rho}(\alpha,\beta))\leq\tau_{n}(\mathbb{C}\setminus\mathcal{A}_{\rho})$,
and then combining (3.4) with Proposition 9 and (2.8), we obtain the result. ∎

###### Proof of Theorem 2.

For the polynomial $P_{n}$, following (2.8) we get

$\displaystyle\Big{|}n\tau_{n}(S(\alpha,\,\beta))-\frac{\beta-\alpha}{2\pi}n\Big{|}\leq\sqrt{2nB_{\infty}(P_{n})},$
(3.6)

where
$B_{\infty}(P_{n})=\log{\left(\frac{\|P_{n}\|_{\infty}}{\sqrt{|c_{0}c_{n}|}}\right)}$.
For $0<\alpha<1$, Proposition 13 gives

$\displaystyle
n\tau_{n}(\mathbb{C}\setminus\mathcal{A}_{1-\alpha})\leq\frac{2}{\alpha}m\left(\frac{P_{n}}{\sqrt{|c_{0}c_{n}|}}\right)\leq\frac{2}{\alpha}B_{\infty}(P_{n}).$
(3.7)

Now, set $\delta_{n}=\frac{\pi
a}{\sqrt{n}}\left(2B_{\infty}(P_{n})\right)^{\theta}$, where
$\theta\in\big{[}\frac{1}{2},\,1\big{]}$, $a>1$ is a positive real number and
$z_{0}=e^{i\varphi}$ for some $\varphi\in[0,\,2\pi)$. From (3.6) we have

$\displaystyle n\tau_{n}(\mathbb{C}\setminus
S(\varphi-\delta_{n},\,\varphi+\delta_{n}))\leq
n-\frac{\delta_{n}n}{\pi}+\sqrt{2nB_{\infty}(P_{n})}\leq
n-(a-1)\sqrt{n}\left(2B_{\infty}(P_{n})\right)^{\theta}.$

The sector $S(\varphi-\delta_{n},\,\varphi+\delta_{n})$ contains atleast
$(a-1)\sqrt{n}(2B_{\infty}(P_{n}))^{\theta}$ zeros of $P_{n}$. Thus, the
number of zeros in $\mathcal{A}_{1-\alpha}\cap
S(\varphi-\delta_{n},\,\varphi+\delta_{n})$ is atleast

$\displaystyle(a-1)\sqrt{n}(2B_{\infty}(P_{n}))^{\theta}-\frac{2}{\alpha}B_{\infty}(P_{n}).$

Taking $\alpha=\frac{b}{\sqrt{n}}(2B_{\infty}(P_{n}))^{1-\theta}$ on the above
we get

$n\tau_{n}(\mathcal{A}_{1-\alpha}\cap
S(\varphi-\delta_{n},\,\varphi+\delta_{n}))\geq(a-b-1)\sqrt{n}(2B_{\infty}(P_{n}))^{\theta}.$

Without lose of generality, if we take $z_{0}=1$, we find a circle of radius

$r=\max\\{\operatorname{dist}\big{(}1,(1-\alpha)e^{i\delta_{n}}\big{)},\,\operatorname{dist}\big{(}1,(1-\alpha)^{-1}e^{i\delta_{n}}\big{)}\\},$

which contains the intersection $\mathcal{A}_{1-\alpha}\cap
S(-\delta_{n},\,\delta_{n})$ and simple calculation gives

$\displaystyle r$
$\displaystyle=\max\Big{\\{}\Big{(}\frac{4\sin^{2}(\delta_{n}/2)}{1-\alpha}+\frac{\alpha^{2}}{(1-\alpha)^{2}}\Big{)}^{\frac{1}{2}},\Big{(}\alpha^{2}+4(1-\alpha)\sin^{2}(\delta_{n}/2)\Big{)}^{\frac{1}{2}}\Big{\\}}$
$\displaystyle=\Big{(}\frac{4\sin^{2}(\delta_{n}/2)}{1-\alpha}+\frac{\alpha^{2}}{(1-\alpha)^{2}}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq\Big{(}\frac{\delta_{n}^{2}}{1-\alpha}+\frac{\alpha^{2}}{(1-\alpha)^{2}}\Big{)}^{\frac{1}{2}}.$

The last inequality uses the inequality $\sin{x}\leq x$ for any non-negative
$x$ with $0\leq x\leq 1/2$. Since $\delta_{n}\leq 1$ and $1-\theta\leq\theta$,
the definition of $\delta_{n}$ gives

$\alpha\leq\frac{b}{\sqrt{n}}(2B_{\infty}(P_{n}))^{\theta}\leq\frac{b}{\pi a}$

and hence we get

$\displaystyle r\leq\frac{1}{\sqrt{n}}(2B_{\infty}(P_{n}))^{\theta}\pi
a\sqrt{\frac{\pi a}{\pi a-b}+\frac{b^{2}}{(\pi a-b)^{2}}}.$ (3.8)

Note that if we choose $a=2.195$ and $b=0.195$ then we have $a-b-1=1$ and the
inequality (3.8) reduce to

$r<\frac{7}{\sqrt{n}}(2B_{\infty}(P_{n}))^{\theta}=\gamma_{n}.$

Second Part. For $0<\alpha<1$ and $p\in(0,\infty)$, Proposition 13 gives

$n\tau_{n}(\mathbb{C}\setminus\mathcal{A}_{1-\alpha})\leq\frac{2}{\alpha}B_{p}(P_{n}).$
(3.9)

Along with (3.9), the remaining proof follows from the first part with a
certain change in the choice of parameters. In the first part, if we work with
Proposition 9 in the place of (3.6) and (3.9) in the place of (3.7) and then
proceeding further with the choice $\delta_{n}=\frac{\pi
a}{\sqrt{n}}\left(2B_{p}(P_{n})\right)^{\theta}$ and
$\alpha=\frac{b}{\sqrt{n}}(2B_{p}(P_{n}))^{1-\theta}$ we get a circular disk
with radius $r$ contains atleast
$(a-b-2\sqrt{2/\pi})\sqrt{n}(2B_{p}(P_{n}))^{\theta}$. Here $r$ is bounded by

$r\leq\frac{1}{\sqrt{n}}(2B_{p}(P_{n}))^{\theta}\pi a\sqrt{\frac{\pi a}{\pi
a-b}+\frac{b^{2}}{(\pi a-b)^{2}}}.$

Now, we choose $a=2.8$, $b=1.8-2\sqrt{2/\pi}=0.20423\ldots$ such that
$a-b-2\sqrt{2/\pi}=1$ and we have

$r<\frac{9}{\sqrt{n}}(2B_{p}(P_{n}))^{\theta}=\gamma_{n}.$

This completes the proof. ∎

###### Proof of Theorem 3.

We follow the proof of Theorem 2 with some adjustments to derive this result.
Since $P_{n}\in\mathcal{G}_{n}$ we have $P_{n}(z)=\sum_{k=0}^{n}a_{k}z^{k}$
with $|a_{0}|=|a_{n}|=1$ and $|a_{k}|\leq 1$. Thus, $\log{|a_{0}a_{n}|}=0$ and
hence

$\log{\|P_{n}\|_{2}}=\frac{1}{2}\log{\left(\int_{0}^{1}|P_{n}(e(k\theta))|^{2}d\theta\right)}=\frac{1}{2}\log{\left(\sum_{k=0}^{n}|a_{k}|^{2}\right)}\leq\frac{1}{2}\log{(n+1)}.$
(3.10)

For any polynomial $P_{n}\in\mathcal{G}_{n}$, by (3.10) we obtain a trivial
bound

$B_{2}(P_{n})\leq\frac{1}{2}\log{(n+1)}+\frac{1}{2e}.$

The proof follows from the proof of the second part in Theorem 2 with a few
modifications. We replace the upper bounds of $B_{2}(P_{n})$ in (3.9) and
Proposition 9 by $\frac{1}{2}\log{(n+1)}+\frac{1}{2e}$ and will use for our
purpose. Furthermore, we set the parameters $\delta_{n}$ and $\alpha$ as
follows:

$\delta_{n}=\frac{2.8\pi}{\sqrt{n}}(\log(n+1)+1/e)^{\theta}\mbox{ and
}\alpha=\frac{0.2}{\sqrt{n}}(\log{(n+1)}+1/e)^{1-\theta}.$

With such changes, following the proof of Theorem 2 and using the inequality
$\log{(n+1)}>\log{n}$, we obtain the required result. ∎

###### Proof of Theorem 5.

At first, we want to compute the length of an arc of the unit circle that lies
inside the circle of radius $\gamma_{n}$, whose center is at some point of the
unit circle. Let $O_{1}$ be such point and $O$ be the center of the unit
circle. Also, denote the intersecting points of two circles by $A$ and $B$.
Let $\theta$ denote the angle $\angle{AOB}$ and $\Gamma$ be the arc length of
$AB$ of the unit circle. Since $O_{1}$ is the middle point of the arc $AB$ on
the unit circle, we have $\angle{O_{1}OA}={\theta}/{2}$. Thus, the arc length
of $O_{1}A=\theta/2$. Now, the area of the triangle, $\triangle
O_{1}OA=\frac{1}{2}\sin{(\theta/2)}$, since the side $O_{1}A=\gamma_{n}$. On
the other hand, by the Heron’s formula, we get

$\displaystyle\triangle O_{1}OA=[s(s-1)^{2}(s-\gamma_{n})]^{\frac{1}{2}}$

where $s$ is the semi-perimeter of the triangle, i.e., $s=1+\gamma_{n}/2$. As
a result, we get

$\displaystyle\triangle
O_{1}OA=\frac{\gamma_{n}}{2}\left(1-\frac{\gamma_{n}^{2}}{4}\right)^{\frac{1}{2}}=\frac{1}{2}\sin{(\theta/2)}.$

Since the $\Gamma=1\cdot\theta=\theta$, we get

$\displaystyle\Gamma=2\arcsin\left(\frac{\gamma_{n}}{2}\sqrt{4-\gamma_{n}^{2}}\right).$

Now, in order to make the unit disk a gear wheel, we have to draw circles
consecutively of radius $\gamma_{n}$ with the center at the boundary of the
unit disk such that the gap between any two consecutive small circles is
$\delta\Gamma$, for some $\delta\in[0,1]$. Then, we have to remove the region
of the disk that is inside of each of the small circles. Thus, the total
number of the teeth of the gear wheel equals the total number of gaps between
consecutive circles drawn with the center at the boundary of the unit disk.
Let $G$ be the such number. Then

$\displaystyle
G=\frac{2\pi}{\Gamma(1+\delta)}=\frac{\pi}{1+\delta}\left(\arcsin\left(\frac{\gamma_{n}}{2}\sqrt{4-\gamma_{n}^{2}}\right)\right)^{-1}.$

By Theorem 2, the number of zeros of $P_{n}(z)$ inside the gear wheel is at
most

$\displaystyle
n-G\sqrt{n}(2B(P_{n}))^{\theta}=n-\frac{\pi}{1+\delta}\sqrt{n}(2B(P_{n}))^{\theta}\left(\arcsin\left(\frac{\gamma_{n}}{2}\sqrt{4-\gamma_{n}^{2}}\right)\right)^{-1},$
(3.11)

where $\gamma_{n}=\frac{9}{\sqrt{n}}(2B(P_{n}))^{\theta}\leq\frac{1}{2}$ and
width of each tooth is

$\displaystyle
2\delta\arcsin\left(\frac{\gamma_{n}}{2}\sqrt{4-\gamma_{n}^{2}}\right).$

A sharp version of Shafer-Fink’s inequality to bounding $\arcsin$-function by
Malešević in [12, Theorem 1.3] gives

$\displaystyle\arcsin{x}\leq\frac{\pi x}{2+(\pi-2)\sqrt{1-x^{2}}},\quad 0\leq
x\leq 1.$

Using this inequality in (3.11), we get the number of zeros of $P_{n}(z)$
inside the gear wheel at most

$\displaystyle
n-\frac{\pi}{1+\delta}\frac{\sqrt{n}(2B(P_{n}))^{\theta}}{\gamma_{n}}\left(1-\frac{\pi-2}{2\pi}\gamma_{n}^{2}\right)\left(\sqrt{1-\frac{\gamma_{n}^{2}}{4}}\right)^{-\frac{1}{2}}$
$\displaystyle=n-\frac{\pi
n}{7(1+\delta)}\left(1-\frac{\pi-2}{2\pi}\gamma_{n}^{2}\right)\left(\sqrt{1-\frac{\gamma_{n}^{2}}{4}}\right)^{-\frac{1}{2}},$

where the last passage uses the definition of $\gamma_{n}$. Since
$\gamma_{n}\leq\frac{1}{2}$ we get that

$\displaystyle\left(1-\frac{\pi-2}{2\pi}\gamma_{n}^{2}\right)\left(\sqrt{1-\frac{\gamma_{n}^{2}}{4}}\right)^{-\frac{1}{2}}>0.97.$

Thus, the result follows. ∎

##  4\. Conclusion

In our results, we establish bounds for quantities in terms of $B_{p}(P_{n})$,
as defined in equation (2.2), for all values of $p$ within the open interval
$(0,\infty)$. These quantities contain the measure of the set $E$, wherein the
absolute value of the polynomial on the unit circle is smaller than one.
Corollary 10 and Example 11 shed light on the size of the set $|E|$, which is
remarkably small, particularly for polynomial degrees that are sufficiently
large. Consequently, a pressing question arises: Can we identify a class of
polynomials for which the size of $|E|$ remains constant?

Furthermore, our results are on the deterministic polynomials; however, it is
worth noting that these results can be readily extended to random polynomials
by using ideas from [14, 15, 16].

##  5\. Acknowledgement

The author would like to thank the referees for reviewing the manuscript in
detail and giving valuable comments. The author also would like to express
gratitude to Prof. R. Balasubramanian for providing valuable suggestions. The
author is supported by DST, Government of India under the DST-INSPIRE Faculty
Scheme with Faculty Reg. No. IFA21-MA 168.

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