#  Enhanced Secrecy in Optical Communication using Speckle from Multiple
Scattering Layers

Alfredo Rates  1,* Joris Vrehen  2 Bert Mulder  1 Wilbert L. IJzerman  2,3 and
Willem L. Vos1  1Complex Photonic Systems (COPS), MESA+ Institute for
Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The
Netherlands
2Signify NV, High Tech Campus 7, 5656 AE Eindhoven, The Netherlands
3Department of Mathematics and Computer Science, Eindhoven University of
Technology (TU/e), 5600 MB Eindhoven, The Netherlands
<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

We study the secrecy of an optical communication system with two scattering
layers, to hide both the sender and receiver, by measuring the correlation of
the intermediate speckle generated between the two layers. The binary message
is modulated as spatially shaped wavefronts, and the high number of
transmission modes of the scattering layers allows for many uncorrelated
incident wavefronts to send the same message, making it difficult for an
attacker to intercept or decode the message and thus increasing secrecy. We
collect 50,000 intermediate speckle patterns and analyze their correlation
distribution using Kolmogorov-Smirnov (K-S) test. We search for further
correlations using the K-Means and Hierarchical unsupervised classification
algorithms. We find no correlation between the intermediate speckle and the
message, suggesting a person-in-the-middle attack is not possible. This method
is compatible with any digital encryption method and is applicable for
codifications in optical wireless communication (OWC).

##  1 Introduction

Figure 1: Schematic of the light scattering systems studied here. (a) A single
incident light beam is sent through $N=1$ scattering slab, yielding a complex
speckle pattern as output. The speckle pattern transmitted through the slab is
described by a transmission matrix $Z$. (b) A single incident light beam is
sent through $N=2$ scattering slabs. The speckle pattern emanating from the
first slab is the intermediate pattern (described by transmission matrix $X$)
that is sent onto the second slab (with transmission matrix $Y$). The final
output is a new speckle pattern described by a transmission matrix
$Z^{\prime}$ with $Z^{\prime}=X\times Y$.

Scattering of light occurs in any 3D complex opaque material such as paint,
foam, or biological tissue, independent of its shape: whether slab, fiber, or
free-form [1, 2, 3, 4]. When light travels through a complex material, it
takes many different paths or channels inside the medium, whereby the light
performs a random walk with a typical step size called the mean free path [2].
Along these contorted paths, the light waves pick up a random distribution of
phase changes. Therefore, it is intuitively reasonable that a random
interference pattern appears in a target plane, called a speckle pattern, that
consists of a random arrangement of bright and dark areas. While a random
speckle interference also arises due to scattering from a rough surface [5],
the speckle due to light scattering in a 3D complex material has several
additional physical properties, notably, a number of intricate correlation
functions, see Refs. [2, 3]. Since the arrangement of speckle spots in an
observation plane is exceedingly difficult to predict due to the huge number
of degrees of freedom in a complex material, the complex light scattering in
such materials offers an attractive opportunity to encode information; indeed,
complex materials play a central role in optical physically unclonable
functions (PUF) and are employed in optical cryptography [6, 7, 8, 9, 10].

Although a speckle pattern is random, the intensities of many single speckle
spots have a well-defined exponential distribution, also known as Rayleigh
distribution, that is characterized by the average intensity. The transport of
scattered light through a complex material is successfully described by random
matrix theory from mesoscopic physics, that invokes a large transmission
matrix with many complex-valued elements [11, 12, 13, 14], illustrated in Fig.
1a). Knowledge of the transmission matrix has been successfully extracted and
even applied to imaging and transmitting encoded data [13, 15].

Statistics of the transmission matrix reveal the existence of “closed
channels”, with zero transmission, and “open channels”, with almost perfect
transmission [16, 17, 18, 19]. So even a thick complex scattering medium is
effectively transparent thanks to these states [20]. Modulation techniques
such as wavefront shaping (WFS) and mutual scattering profit from the high-
transmission states to control the transmission through the scattering medium,
thereby modulating the intensity distribution of speckle spots [21, 20, 22,
23]. This can be used, notably, to concentrate the intensity at a specific
focal point, transmit information, or change the overall transparency of the
scattering medium.

When the properties of the incident light are changed (e.g., positioned,
tilted, shaped, or frequency shifted), the light is scrambled in a different
way resulting in a new speckle pattern in the observation plane. When the
range of perturbations of the incident light is moderate, known as the memory
range, the new speckle pattern has remarkable non-zero correlations with the
original speckle pattern [24]. This effect is typically characterized by
intensity-intensity correlations between pairs of different positions in one
observation plane, known as C1 correlations [25, 2]. Such speckle and
intensity correlations find many applications, ranging from imaging through an
opaque screen [26, 27] to transmitting images and information through opaque
media [28].

In this work, we study speckle correlations in a different setting, where we
use two layers of scattering media and observe the intermediate speckle
pattern in between the scattering layers. Instead of measuring changes of a
speckle pattern in a target plane - as is described above - we collect two
speckle patterns in two different observation planes, namely one in between
the two scattering layers and the other after passing through both scattering
layers, illustrated in Fig. 1b). For applications to optical wireless
communication, we study if a binary message sent through these scattering
layers can be extracted or not by only observing the intermediate speckle
pattern. The presence of the first scattering layer is essential to spatially
scrambling the incident wavefront, effectively “hidding” the sending modulator
for direct inspection and subsequent decoding. The question we address in this
work is whether different incident wavefronts are correlated with each other,
knowing the message they result into. In other words, if the two transmitted
wavefronts correlate, does that mean the two intermediate wavefront are also
correlated?

We synthesize $N_{\rm W}=50,000$ different incident wavefronts and collect the
resulting intermediate and final speckle patterns. The message is encoded as
an intensity distribution in the final speckle pattern, where each speckle
pattern is assigned to either a binary 0 or a binary 1. Thus, we have
thousands of available wavefronts to send the same value. We measure the
correlation between intermediate speckle patterns, to study if the
intermediate speckle patterns resulting in the same message encoded in the
final speckle pattern are somehow correlated.

##  2 Working principle and experimental methods

In light scattering theory, and in general in mesoscopic physics, a single
thick slab of scattering material with thickness $L$ is equivalent to $N$ thin
slabs with average thickness $(L/N)$ since the total thickness is the same:
$N(L/N)=L$. Here, we study $N=2$ as illustrated in Fig. 1. A scattering
material is represented as a random transmission matrix that couples the modes
of the incident light to the ones of the outgoing light, and the random
components of the transmission matrix represent the scattering events inside
the material [11, 2, 29]. Studying $N=2$ scattering slabs is represented as
the multiplication of two of these random matrices, which because it is a
linear system, results in yet another random transmission matrix. The
reasoning above implies that techniques such as wavefront shaping or mutual
scattering are also relevant and valid when multiple slabs are used,
independent of the distance between the slabs. Here, we study if there are
correlations between the intermediate speckle pattern that we observe between
the two scattering slabs on the one hand and the output speckle pattern on the
other hand.

Figure 2: Diagram of the experimental setup. The incident wavefront is
spatially phase-modulated using a Digital Mirror Device (DMD, not shown) [30].
The diffusers $d_{1}$ and $d_{2}$ are scattering materials forming speckle
patterns. Camera CCD1 records the final speckle pattern after both $d_{1}$ and
$d_{2}$ and CCD2 records the intermediate pattern after $d_{1}$ only. (L:
Lens, BS: Beamsplitter).

To study the speckle correlation, we used the experimental setup shown in Fig.
2. The initial light source is a frequency-doubled continuous wave green
($\lambda=532$ nm) Nd:YAG${}^{3}+$ laser (Coherent Compass 315M-100, 100mW).
The signal is encoded as a phase-modulated wavefront using a digital mirror
device (DMD, Vialux VX4100), in the same way as usually done in Wavefront
Shaping experiments [30]. The DMD applies a binary, ON-OFF modulation to the
wavefront, which we transform into a phase modulation using the Lee Holography
technique [31]. Using phase modulation rather than amplitude-only modulation
we achieve greater control over the scattering events. [31, 32] The modulated
wavefront is focused into a diffuser (Ground Glass Diffuser 220 Grit,
Thorlabs) using lens L1 ($f=75$ mm), and collimated using a microscope
objective (NA=0.3, Nedoptifia Zeist). This diffuser has a large diffusing
pattern and high transmission, and it is commonly used in scattering
experiments [26, 33]. To measure the light speckle between the diffusers, a
beamsplitter is used to pick up half of the signal, which is collected by a
charge-coupled device (CCD) camera (Guppy F-146B), to detect the intermediate
speckle pattern. The other half is focused on a second diffuser similar to the
first one using lens L3 ($f=50$ mm), and collected by a second CCD camera
(Stingray F-125), which we call the receiver. The CCD camera is placed in the
far field, at a distance of $30$ mm from the second diffuser.

Light coherence is a key factor in our experiment. We need spatial coherence
in order to obtain and measure the speckle pattern. In our experiment, we use
a well-defined laser beam as a source, meaning we have a high temporal and
spatial coherence. The laser Coherent Compass 315M-100 is estimated to have a
line width of around 10MHz, and a coherent length on the order of tens of
meters [34], by far sufficient for our purposes. Our experiment has a path
length on the order of one meter, thus ensuring the wavefront is spatially
coherent.

##  3 System characterization

The goal of our experiment is to send a signal through the two scattering
slabs, and study if there is any correlation between the intermediate speckle
pattern and the resulting pattern. As a starting point, we aim to obtain a
binary signal, i.e., only two levels: state 0 and state 1. Obviously, the
modulated incident wavefront has many more degrees of freedom than these two
states. For this reason, there exist multiple incident wavefronts, and thus
multiple intermediate speckle patterns, that can result in the same signal. We
use this property to see if different intermediate speckle patterns need to
have some degree of correlation in order to result in the same final state.

Figure 3: Examples of observed speckle patterns. (a-c) Speckle patterns
selected as state 0. (d-f) Speckle patterns selected as state 1. The colored
circles in each panel show the region of interest (ROI) of spot A (purple) and
spot B (green). (g) Average intensity normalized by the average intensity of
the background of spot A and spot B for each speckle pattern selected as state
0 or state 1.

To be robust against environmental noise and to have a large dynamic range, we
consider in the receiver plane only two regions of interest (ROI), called spot
A and spot B, as shown in Fig. 3. From the intensity distribution of the
speckle at the receiver, we estimate this area to be $\times 15$ larger than
the size of a single speckle. Only when the local average intensity of spot A
is high and the local intensity of spot B is low, a state 0 is received.
Conversely, if the local intensity of spot A is low and the local intensity of
spot B is high, state 1 is received, see Fig. 3g).

To select the speckle patterns corresponding to state 0 or state 1, we send a
total of $N_{\rm W}=50,000$ randomized wavefronts. For each wavefront, the
speckle pattern at CCD2 is collected along with the intensities at spot A and
spot B. From the definition of each state, we define arbitrary intensity
thresholds based on the joint intensity distribution of spot A and spot B,
shown in Fig. 4, where the thresholds are marked with dashed lines. For both
spot A and spot B, we set the thresholds at 20% and 80%. This means that a
speckle pattern is classified as state 0 only if the intensity at spot A is
higher than 80% of the distribution, and the intensity at spot B is lower than
20% of the distribution. The classification of wavefronts as state 1 follows
the same principle.

The red and blue regions in Fig. 4 highlight which speckle patterns are
accepted as state 0 or state 1, respectively. With the given thresholds, there
is a total of $N_{\rm W,0}=1839$ available wavefronts (3.7% of the total) to
get a state 0, and $N_{\rm W,1}=1774$ (3.5% of the total) to get a state 1. We
modulate the wavefront using a grid of $15\times 15$ segments, controlling the
phase of each segment. In our current realization, we modulate the phase from
$0$ to $2\pi$ in 16 steps. That means that the maximum number of different
wavefronts that we generate is equal to $N^{\prime}_{\rm W}=16^{225}$, which
is on the order of $\mathcal{O}(10^{307})$. The large complexity of the
scattering material is such, that small changes in the phase of a single
segment at the modulator produce large changes in the intensity distribution
at the receiver. This means that the number of available wavefronts can be
made arbitrarily large, at the expense of longer measurements and digital
memory.

Figure 4: Joint distribution of average intensities at spot A and at spot B at
the receiver for different incident wavefronts (green symbols). The horizontal
and vertical dashed lines are the intensity thresholds that define the states
0 or 1. Here, the red region indicates the accepted state 0 with high
intensity in spot A and low intensity in spot B. The blue region indicates the
accepted state 1 with low intensity in spot A and high intensity in spot B.

##  4 Correlation distribution of intermediate speckles

Every wavefront used to send a message also generates an intermediate speckle
pattern between the slabs. When sending two messages separately, we generate
two intermediate speckles and we calculate the correlation between these two
intermediate speckles. We are interested in comparing three cases: the
correlation between two speckle patterns from state 0 (0-0 correlation), the
correlation between two speckle patterns from state 1 (1-1 correlation), and
the correlation between a speckle pattern from state 0 and another from state
1 (0-1 correlation).

Table 1: Example of pairs of intermediate speckle patterns used to calculate the Pearson correlation distribution for each relevant case. The left column indicates the case, while the right column shows each pair used for that case. The letters between parentheses refer to the sub-figure indexes of the speckle patterns shown in Fig. 3. Case | Pairs of speckle patterns
---|---
0-0 correlation | (a,b), (a,c), (b,c)
1-1 correlation | (d,e), (d,f), (e,f)
0-1 correlation |  | (a,d), (a,e), (a,f),
---
(b,d), (b,e), (b,f),
(c,d), (c,e), (c,f)

We illustrate the combinations of speckles in Table 1 using the panels indexes
of Fig. 3. To characterize the correlations between speckle patterns, we use
the Pearson correlation coefficient $C_{\rm P}$ from the Python library SciKit
[35], calculated as follows:

$C_{\rm P}\coloneqq\frac{\sum_{\rm i,j}(x_{\rm i,j}-\overline{x})(y_{\rm
i,j}-\overline{y})}{\sqrt{(\sum_{\rm i,j}(x_{\rm i,j}-\overline{x})^{2}(y_{\rm
i,j}-\overline{y})^{2}}},$ (1)

with $x_{\rm i,j}$ the value of pixel $(i,j)$ of the first speckle image,
$y_{\rm i,j}$ the value of the pixel $(i,j)$ of the second speckle, and
$\overline{x}$, $\overline{y}$ the average pixel values of the first and
second speckle, respectively.

We calculate the correlation $C_{\rm P}$ between two intermediate speckles
$s_{\rm int}$, e.g., $C_{\rm P}(s_{\rm int,1},s_{\rm int,2})$. Whether a
message is classified as a state 0 or a state 1 does not depend on the
intermediate speckle $s_{\rm int}$ but on the final speckle $s_{\rm f}$. The
final speckle is expressed as $s_{\rm f}=Ys_{\rm int}$, with the transmission
matrix of the second slab as $Y$ (see Fig. 1). We thus choose states 0 and 1
as the set of final speckles $s_{\rm f}$ that have a high correlation with the
target patterns $s^{\star}_{0}$ and $s^{\star}_{1}$, respectively. Following
what we showed in section 3, the target pattern $s^{\star}_{0}$ has a high
intensity in spot A, zero intensity in spot B, and average intensity elsewhere
. Conversely for $s^{\star}_{1}$. Thus, an intermediate speckle results in a
state 0 if $C_{\rm P}(Ys_{\rm int},s^{\star}_{0})\approx 1$, and in a state 1
if $C_{\rm P}(Ys_{\rm int},s^{\star}_{1})\approx 1$.

When measuring the correlations for the cases 0-0, 1-1, and 0-1, we
investigate if the intermediate speckle pattern correlate, given that their
resulting speckle correlate. If any correlation is needed to result in the
same state, we expect the correlation between two patterns from the same group
to be larger than between two patterns from different groups, e.g., we expect
the 0-0 and 1-1 correlations to be larger than the 0-1 correlation. For
perspective, previous studies of speckle correlation, such as the memory
effect, focus on the auto correlation of the resulting speckle $C_{\rm
P}(s_{\rm f,1},s_{\rm f,2})$ or in the correlation between the incoming and
outgoing pattern $C_{\rm P}(s_{\rm int,1},s_{\rm f,1})$.

When we calculate $C_{\rm P}$ between all $N_{\rm W}$ available intermediate
speckles, i.e., $C_{\rm P}$ between individual pairs of speckles with all the
combinations possible, we get a distribution of $C_{\rm P}$. In Fig. 5 we show
the correlation distribution for the three cases previously described, 0-0,
1-1, and 0-1. We see that, qualitatively, the distributions are very similar,
with the same peak position.

Table 2: K-S statistic between the different correlation distributions. Each K-S statistic is calculated between Dataset 1 and Dataset 2. The $p$-value for every case is on the order of $10^{-15}$. Dataset 1 | Dataset 2 | K-S statistic |  $p$-value
---|---|---|---
0-0 correlation | 1-1 correlation | 0.03 | 0.00
0-1 correlation | 0-0 correlation | 0.03 | 0.00
1-1 correlation | 0-1 correlation | 0.02 | 0.00

To compare these distributions quantitatively, we used the two-sample
Kolmogorov–Smirnov (K-S) test [36]. This test compares the empirical
distribution of two sets of observations. When the two observations are from
the same distribution, the K-S statistic tends to zero. The results of this
test are shown in Table 2. We see that the K-S statistic is close to zero and
that the $p$-value is lower than 5$\%$ ($\mathcal{O}(10^{-15})$), this tells
us that the correlation distributions of the three groups are
indistinguishable.

We see that the correlation distributions are not centered around zero. When
we claim there is no clear correlation between speckles from the same group,
it is because this correlation is the same as the correlation between speckles
from different groups. However, the fact that the correlation is not zero
leads to the question if there is any underlying relation between speckle
patterns that the Pearson correlation coefficient does not have access to. To
test this further, we used unsupervised classification algorithms.

Figure 5: Probability density histograms of speckle cross-correlation between
different signals: a) between all pairs of speckle patterns belonging to state
0, b) between all pairs of speckle patterns belonging to state 1, and c)
between speckle patterns from state 0 and speckle patterns from state 1. The
vertical dashed red lines indicate the peaks of the distributions.

##  5 Unsupervised classification of intermediate speckles

For our analysis of the intermediate speckle correlations, we use two
different classification algorithms: The K-means algorithm and the
Hierarchical clustering algorithm. We choose these two algorithms based on our
familiarity with them, their required input data, and the ability to visually
illustrate their classification process, so that we can have a better
understanding of the results. We use the open-access Scikit-learn Python
library to implement both algorithms. The procedure for classification is the
following: First, to reduce the computational time we reduce the resolution of
the picture by averaging the $5\times 5$ adjacent pixels. This is much smaller
than the speckle size, thus no information is lost. Then, we transform the
representation of the data: we consider each pixel as a different dimension
and the intensity of this pixel as the position of the realization in that
dimension. This forms a new high-dimensional space, called the feature space.
Due to the resolution of our camera ($1280\times 960$) and the reduced
resolution, the feature space has almost $50,000$ dimensions, where each point
is a realization (or speckle).

To reduce even further the dimensionality of the problem and thus make the
problem addressable for a personal computer, we use the principal components
analysis (PCA) and only consider the first 100 principal components,
explaining $>90\%$ of the variance of the data. This new data representation
is finally used for the classification algorithm, where each principal
component is now a dimension of the feature space, and its value corresponds
to the position of that dimension. Now, the feature space has $N_{\rm d}=100$
dimensions.

For both algorithms, we use two different classes, intending to separate state
0 and state 1. For the K-Means algorithm, we use 100 initializations and a
maximum of $30,000$ iterations. For the Hierarchical clustering algorithm, the
agglomerate strategy is used. For the sake of generality, the classification
is also run without reducing dimensionality, i.e., without using PCA. For
these cases, the image was reduced by averaging the $10\times 10$ adjacent
pixels, considering the data memory compared with the previous cases.

We show the results of the classification in Table 3. In total, four
classification methods are used, two different algorithms and two different
data representations.

To get statistical information from the classification, each method is
repeated 5 times with sub-groups of half of the data. Furthermore, for each
sub-group, we repeat 5 more times to average over the initial random guess of
the algorithms. Note that we select an equal number of speckle patterns from
both groups, $N_{\rm W,0}=N_{\rm W,1}=1774$, so the data is evenly distributed
between the two groups. Table 3 presents the balanced accuracy of the
classification algorithms. The balanced accuracy is an average between the
true positive rate (TPR) and true negative rate (TNR), which is the rate of
predicted versus the total amount of positives and negatives, respectively
[37]. In our case, positive and negative represent a binary 0 and a binary 1.
We see that the balanced accuracy of all the methods we used is around 50%. As
this is a binary classification and only two options are possible, a random
classifier yields on average the same accuracy of 50%. This shows that the
classification is as effective as tossing a coin, thus we obtain no new
information from it. Furthermore, the right column of Table 3 shows the
percentage of 0’s predicted by the clustering. For the last method, all the
data points were classified as 0, which also yields 50% accuracy. All the
other methods separate the data exactly in half for every sub-group and every
repetition.

We thus find that there are no trivial correlations between intermediate
speckle patterns, regardless of their corresponding encoded message.
Furthermore, the different unsupervised classification algorithms are not able
to find any correlation or separation between the patterns. This means that
when measuring two intermediate speckle patterns, is not possible to know for
certain if they are encoding the same message or not in the final speckle
pattern. We believe that this knowledge is relevant for applications,
particularly in the field of optical wireless communication (OWC), to make
communication more secure. Therefore, in the following section, we described a
possible implementation of a communication link based on these findings.

Table 3: Balanced classification accuracy of the unsupervised methods under study.  Method | Average (%) | STD (%) | 0’s predicted (%)
---|---|---|---
K-Means | 51 | 4.07 | 50
Hierarchical | 49 | 7.58 | 50
K-means no PCA | 49 | 2.52 | 50
Hierarchical no PCA | 50 | 0.00 | 100

##  6 Proposed communication scheme

Based on the described scenario, we propose a new communication scheme based
on two layers of physical unclonable functions (PUFs). This scheme is depicted
in Fig. 6, which is inspired by the experimental setup shown in Fig. 2. Alice
sends a message to Bob through free space using visible or infrared coherent
light. The initial digital message is encoded as a phase-modulated wavefront
that is encrypted by the first PUF. When arriving at the destination, the
signal passes through a second PUF and the message is recovered as light
intensity by Bob. If an attacker, depicted in Figure 6 as Eve, intercepts the
signal, the message will not be recovered as the second PUF is not known.
Similarly, if Eve tries to send a false message to Bob, this will not be
considered as it does not pass through the first PUF.

The secrecy of the proposed system relies on the lack of correlation between
speckle patterns, as demonstrated before. If for every binary message, the
sender randomly selects one wavefront from a large library with many options,
the attacker is not able to separate which speckle patterns are related to a
state 0 and which ones are to a state 1. The proposed system is also resilient
for an attacker to send false messages or store the message for future
decryption. This is thanks to the first PUF, which scrambles the incoming
wavefront. If the first PUF is not present, the attacker has access to the
incident wavefront and they can measure, store, and replicate it. This is not
challenging because the spatial complexity of the incident wavefront is
limited by the spatial resolution of the modulator. Inversely, if the first
scattering layer is present, the attacker only has access to the intermediate
speckle, which has a much larger spatial complexity and it is highly
challenging to record with all its properties and fluctuations [6].
Furthermore, if the attacker wants to send a false message, they will need a
modulator device or optical system able to replicate the intermediate speckle
pattern with high resolution in order to result in the proper final speckle
pattern.

If the digital codification of the message forces Alice to repeat a specific
bit on a predictable basis, e.g., for identification, Eve can record the
speckle of these bits to identify each binary 0 and binary 1. In this case,
the number of messages Eve needs to record to obtain all speckles grows as
$\mathcal{O}(N_{\rm W}log(N_{\rm W}))$ [38]. An implementation without this
identification is possible by changing the digital codification to avoid any
predictable bit. Even more, based on the complexity of the scattering material
and the degrees of freedom at the wavefront modulation, the number of
available wavefronts can be made arbitrarily large, increasing the number of
messages needed. This also limits Eve to send a false message to Bob.

Figure 6: Scheme of the communication link. Alice sends a message to Bob
through the channel (free-space), which is encrypted using two Physical
Unclonable Functions (PUF).

Many alternatives have been studied to break the secrecy of PUFs employ
Machine Learning (ML) techniques, which have proved to be powerful tools for
these attacks [39, 40], at a large expense of time and digital memory from the
side of the attacker. In most of these studies, however, they needed to obtain
a Challenge-Response Pair (CRP) set for training a supervised algorithm. To do
so, it is assumed that Eve can send a challenge to the PUF and read the
response. In our case, the CRP corresponds to the system characterization
shown in Section 3. We obtain all the CRP sets offline and prior to the
communication link, as it is end-to-end encryption. Obtaining a CRP set means
that Eve has a subset of speckle patterns at CCD2 with their respective
classification at CCD1. In our system, an attacker does not have access to the
space between the second PUF and CCD1, and gaining access for an invasive
measurement poses a threat larger than eavesdropping, thus we disregard such
attacks in this study. If Eve does not have a CRP set, there are no training
data for a supervised algorithm. Therefore, we do not consider the use of
supervised classification methods to be a realistic scenario.

Nevertheless, an attacker may try to classify the speckle patterns using
unsupervised algorithms. The results from Section 5 show that none of the
tested unsupervised methods are significantly better than a random
classification, which suggests that the data are not classifiable given the
intensity speckle pattern obtained from the channel. Hence, the communication
is secure against an attacker in free-space.

##  7 Applications and limitations

We initially envision our scheme in an indoor communication scenario as a one-
way communication. A central device would include the modulator device with
the first PUF layer, and the receiving device would have a pair of photodiodes
representing the regions of interest ROI A and ROI B, with its own integrated
PUF. In this system, the central device can send many messages to different
devices using different wavefronts, while the receiver is an inexpensive and
fast device. The calibration can be done for specific positions of the
receiver, e. g., cubicles in an open office. If the layout is not changed, the
calibration only needs to occur once per device. We believe our method opens
many possibilities to both test applicability of the scheme, and study the
scattering properties of complex media.

Several questions and limitations arise from our system, which need to be
addressed to scale up this scheme. One limitation of our scheme is the bit
rate. If only two ROIs are considered, the receiver can consist of two fast
photodiodes, thus not limiting the speed. The bottleneck arises from the
refresh rate of the DMD, which for our DMD model is around 20 kHz. This
limitation can be remedied by sending multiple bits in one wavefront (e.g.,
sending 50,000 bits per image results in 1Gbit/s). In this scenario, the
camera at the receiver is needed, which will limit the speed as well. Another
solution for higher speed is using another faster modulation technique, like
Free-Space Electro-Optic Modulators, which can get up to 100 MHz [41].

Another limitation of our system is the source. The need to have a coherent
light source hinders us to extend our technique to applications such as light
fidelity (Li-Fi). This restriction is present in any technology that wants to
take advantage of luminaries already installed. Nevertheless, it has been
demonstrated that modulation techniques such as WFS are applicable to LED
sources [42]. Extending our method to LED sources would simplify the
architecture. Furthermore, it was recently demonstrated that is possible to
use a random material as a PUF having a screen projector as a source [43],
which is a commercial combination between an ultra-high-performance (UHP)
lamp, which is incoherent, and a DMD.

Since our system is based on light scattering, it may be highly sensitive to
noise, misalignment, or additional scattering events. This sensitivity depends
on the power of the signal, the distance, and the dynamics of the medium.
Importantly, we use a diffuser with around 80% of transmission and a
collimating lens, so the use of PUFs does not drastically increase the losses
of the system. This suggests that the restrictions on noise and power are
similar to any OWC scheme. Furthermore, if the PUF is static and reliable, the
dynamics of the medium may fall under the memory effect of the system. It
remains to be tested to what extent these factors limit the applicability of
the scheme.

In case the medium changes significantly or we want to extend this to moving
users, the calibration needs to be done online. One way of implementing this
is adding a secondary classical channel where the users can share the CRP.
More studies are needed to test the secrecy of such implementation.

Finally, the main advantage of our system is that the secrecy is imposed by
the number of measurements needed from an attacker to learn the message. This
is more restrictive than only computational power, as traditional digital
encryption. While the results presented in this manuscript indicate no
correlation between the signal retrieved by the attacker and the final
message, it is important to acknowledge that it is impossible to account for
all possible scenarios that could potentially benefit the attacker.

For future tests, we conceive that an attacker may benefit from other
detection configurations that may give more information. We think of at least
three possible extensions to the proposed detection configuration: (a) measure
the complex information of the speckle (both amplitude and phase), (b) measure
in the conjugated plane of the sample, and (c) measure the reflective speckle
from the receiver PUF. The reflective speckle pattern is particularly
relevant, as it has been successfully used in imaging through scattering media
[44, 45, 46]. It has been proven that the reflective speckle pattern has a
certain correlation with the transmitted speckle pattern [47], similar to the
correlations between the incident and transmitted wavefront mentioned before
and may prove useful to decipher the message in our communication scheme.
Because we calibrate the system beforehand, if the commented configurations
give useful information to the attacker (e.g., the reflection-transmission
correlations), we believe it is possible to account for it beforehand by
filtering further which available wavefronts are available to send the
message. Furthermore, from the practical point of view, one attractive part of
the present scheme is that the detection of the message by the receiver can be
done easily, fast, and cheaply. This is because the receiver only needs two
photodiodes to measure the average intensity. In turn, the commented possible
configurations impose high technical difficulties. Further studies are needed
to evaluate how obtaining this new information may affect the secrecy of the
scheme.

##  8 Conclusion

In this paper, we have studied the correlation between speckle patterns when
passing through multiple slabs of scattering media. We spatially modulate the
phase of the incoming light and we send a signal through two diffusers,
measuring the resulting speckle pattern both between and after the two
diffusers. The signal is encoded as changes in light intensity at two regions
of interest (ROI) at the receiver, where multiple modulated incoming
wavefronts may result in the same message. We have studied the correlation
between speckle patterns when sending different messages. Therefore, we use
the Pearson correlation coefficient and two unsupervised classification
algorithms. In all cases, we observe that there is no correlation between the
intermediate speckle pattern and the resulting pattern (or message). This
method is attractive for optical wireless communication (OWC) schemes,
particularly in line-of-sight communication and wireless indoor communication.

Funding Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO-TTW
P15-36).

Acknowledgments We thank Cornelis Harteveld for expert technical support and
sample preparation, and Ad Lagendijk, Allard Mosk, Chris Toebes, and Pepijn
Pinkse for stimulating discussions. This work was supported by the MESA+
Institute section Applied Nanophotonics (ANP).

Disclosures The authors declare no conflicts of interest.

Data availability The data used for this publication are publicly available in
the Zenodo database [48].

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# Complex valued multiplicative functions with bounded partial sums

Marco Aymone

###### Abstract.

We present a class of multiplicative functions $f:\mathbb{N}\to\mathbb{C}$
with bounded partial sums. The novelty here is that our functions do not need
to have modulus bounded by $1$. The key feature is that they pretend to be the
constant function $1$ and that for some prime $q$,
$\sum_{k=0}^{\infty}\frac{f(q^{k})}{q^{k}}=0$. These combined with other
conditions guarantee that these functions are periodic and have sum equal to
zero inside each period. Further, we study the class of multiplicative
functions $f=f_{1}\ast f_{2}$, where each $f_{j}$ is multiplicative and
periodic with bounded partial sums. We show an omega bound for the partial
sums $\sum_{n\leq x}f(n)$ and an upper bound that is related with the error
term in the classical Dirichlet divisor problem.

##  1\. Introduction.

We say that $f:\mathds{N}\to\mathds{C}$ is multiplicative if $f(nm)=f(n)f(m)$
whenever $n$ and $m$ are relatively prime, and we say that such $f$ is
completely multiplicative if this relation holds for all $n$ and $m$.
Therefore, a multiplicative function $f$ is determined by its values at prime
powers.

We say that $f:\mathds{N}\to\mathds{C}$ has bounded partial sums if there
exists a constant $C>0$ such that for all $x\geq 1$, $|\sum_{n\leq x}f(n)|\leq
C$; otherwise we say that $f$ has unbounded partial sums.

Resolving the Erdős discrepancy problem, Tao [9] showed that a complex valued
completely multiplicative function $f$ with $|f|=1$ has unbounded partial
sums. Further, Tao gave a partial classification of all multiplicative
functions $f$ taking only values $\pm 1$ with bounded partial sums. To state
this partial classification, we need to introduce the language of pretentious
number theory [2]: Given two complex valued multiplicative functions $f$ and
$g$ taking values in the unit disk, we say that $f$ pretends to be $g$ or that
$f$ is $g$-pretentious if the “distance” between $f$ and $g$ given by

$\mathbb{D}(f,g;x):=\left(\sum_{p\leq
x}\frac{1-Re(f(p)\overline{g(p)})}{p}\right)^{1/2}$

is $O(1)$ as $x\to\infty$, where in the sum above $p$ stands for a generic
prime number.

The multiplicative function $f:\mathds{N}\to\\{-1,1\\}$ such that
$f(2^{k})=-1$ for all $k\geq 1$ and $f(p^{k})=1$ for all primes $p\geq 3$ and
all powers $k\geq 1$, then $f$ is the periodic function $f(n)=(-1)^{n+1}$
which clearly has bounded partial sums. In [9], Tao showed that if
$f:\mathds{N}\to\\{-1,1\\}$ is multiplicative and has bounded partial sums,
then $f$ is $1$-pretentious and at powers of $2$, $f(2^{k})=-1$ for all $k\geq
1$. Later, Klurman [5] completely classified such multiplicative functions
with bounded partial sums by proving that they must be periodic of some period
$m$ and $\sum_{n=1}^{m}f(n)=0$. This last result had been known as the Erdős-
Coons-Tao conjecture.

When we allow that a multiplicative function $f$ takes complex values, then
there is no known criterion to determine when $f$ has bounded partial sums,
therefore we must analyze case by case. For instance, in [1] and [6] it has
been proved that a multiplicative function $f$ supported on the squarefree
integers such that at primes $f(p)=\pm 1$, has unbounded partial sums. On the
other hand, without any restriction we can easily construct examples of
multiplicative functions $f:\mathds{N}\to\mathds{C}$ with bounded partial
sums. A non-trivial way to construct such examples exists if we impose
conditions on the values $f(p)$ at primes $p$ such that $\sum_{n\leq x}|f(n)|$
is bounded below by $cx$ for all sufficiently large $x$, for some positive
constant $c$. Here we aim to do this.

######  Theorem 1.1.

Assume that $f:\mathds{N}\to\mathds{C}$ is multiplicative, has bounded partial
sums and $\sum_{p}\frac{|1-f(p)|}{p}<\infty$. Then there exists a prime $q$
such that

(1) $\sum_{k=0}^{\infty}\frac{f(q^{k})}{q^{k}}=0.$

######  Remark 1.1.

We impose the condition $\sum_{p}\frac{|1-f(p)|}{p}<\infty$ to keep the
intuiton behind pretentiouness in the case that $f$ takes values outside the
unit disk. For example, if $p_{n}$ is the $n$-th prime and
$f(p_{n})=1+(-1)^{n}\in\\{0,2\\}$, the partial sums $\sum_{p\leq
x}\frac{1-f(p)}{p}$ are $O(1)$ as $x\to\infty$ while the values $f(p)$ are
always distant from $1$.

It is interesting to observe that if $f$ is real-valued and $f^{2}\leq 1$,
then, since $f(1)=1$, (1) can only be satisfied when $q=2$ and $f(2^{k})=-1$
for all $k\geq 1$. But we have many options to satisfy (1) when we allow that
$f$ takes complex values.

######  Theorem 1.2.

If a multiplicative function $f:\mathds{N}\to\mathds{C}$ has period $m$,
$f(m)\neq 0$ and has bounded partial sums, then the following three conditions
are satisfied.
i. For some prime $q|m$, $\sum_{k=0}^{\infty}\frac{f(q^{k})}{q^{k}}=0.$
ii. For each $p^{a}\|m$, $f(p^{k})=f(p^{a})$ for all $k\geq a$.
iii. For each $\gcd(p,m)=1$, $f(p^{k})=1$, for all $k\geq 1$.
Conversely, if $f:\mathds{N}\to\mathds{C}$ is multiplicative and the three
conditions above are satisfied, then $f$ has period $m$ and has bounded
partial sums.

An intermediate step in the proof of the Erdős-Coons-Tao conjecture [5] is a
result similar to Theorem 1.2 – Proposition 4.4 of [5], where it is assumed
that $f^{2}\leq 1$. Our contribution here is the observation that the proof of
Proposition 4.4 of [5] allow us to deal with the case where $|f|$ is not
necessarily bounded by $1$.

We stress that the condition that $f$ does not vanish at its period, $f(m)\neq
0$, is pivotal to deduce the three conditions above. Indeed, a non-principal
Dirichlet character is a classical example of a periodic (completely)
multiplicative function with bounded partial sums that vanishes at its period,
and does not satisfy either i. and iii. However, the three conditions above
allow us to produce examples of periodic multiplicative functions with bounded
partial sums, despite the fact that $f$ vanishes or not at its period.

######  Example 1.1.

Let $f$ be multiplicative and define for all primes $p\neq 3$, $f(p^{k})=1$
for all powers $k\geq 1$, and at powers of $3$: $f(3)=2$, $f(9)=-15$ and
$f(3^{k})=0$ for all $k\geq 3$. Then $f$ has period $27$, $f(27)=0$, and has
bounded partial sums.

######  Example 1.2.

Let $f$ be multiplicative and define for all primes $p\neq 5$, $f(p^{k})=1$
for all powers $k\geq 1$, and at powers of $5$: $f(5)=\pi$,
$f(5^{k})=-20-4\pi$ for all $k\geq 2$. Then $f$ has period $25$, $f(25)\neq
0$, and has bounded partial sums.

We point out that our class of examples in Theorem 1.2 is not the only one
with bounded partial sums. Indeed we can construct very easily examples of
non-periodic multiplicative functions with bounded partial sums by a standard
convolution argument: If $g:\mathds{N}\to\mathds{C}$ is multiplicative and
$\sum_{n=1}^{\infty}|g(n)|<\infty$, and if $h:\mathds{N}\to\mathds{C}$ has
bounded partial sums, then $f=g\ast h$ also has bounded partial sums, where
$\ast$ stands for Dirichlet convolution. In particular, $h$ can be as in
Theorem 1.2 or a non-principal Dirichlet character $\chi$.

Now we turn our attention to multiplicative functions
$f:\mathds{N}\to\mathds{C}$ of the form $f=f_{1}\ast f_{2}$, where each
$f_{j}$ is multiplicative and periodic with bounded partial sums. We begin by
observing that if each $f_{j}$ satisfies the conditions i-iii of Theorem 1.2,
then $f$ has unbounded partial sums.

Before we state our next result, we recall the notation $f(x)=\Omega(g(x))$,
where $g(x)>0$ for all $x>0$. This means that
$\limsup_{x\to\infty}\frac{|f(x)|}{g(x)}>0$.

######  Theorem 1.3.

Let $f_{1}$ and $f_{2}$ be two multiplicative functions satisfying conditions
i-iii of Theorem 1.2. Let $f=f_{1}\ast f_{2}$. Then there exists a constant
$d>0$ such that

$\sum_{n\leq x}f(n)=\Omega\left(\exp\left(d\frac{\log x}{\log\log
x}\right)\right).$

A key argument in the proof of the result above is that $f(n)=\tau(n)$
whenever $\gcd(n,m)=1$ for some $m$, where $\tau(n)$ is the divisor function:
$\tau(n)=\sum_{d|n}1$. The omega result is then obtained by using classical
estimates for the maximal value of $\tau$.

Our next question concerns upper bounds for the partial sums of $f=f_{1}\ast
f_{2}$ as in Theorem 1.3. We begin by recalling the classical estimate for the
partial sums of the divisor function:

(2) $\sum_{n\leq x}\tau(n)=x\log x+(2\gamma-1)x+\Delta(x),$

where $\Delta(x)$ is an error term and $\gamma$ is the Euler-Mascheroni
constant.

In the past 200 years there were a lot of attempts to obtain sharp estimates
for the error term $\Delta(x)$. This is classically known as the Dirichlet
divisor problem, where one seeks to obtain estimates for the exponent

(3) $\alpha:=\inf\\{a>0:\Delta(x)=O_{a}(x^{a})\\},$

where the notation $O_{a}$ means that the implied constant may depend on the
parameter $a$.

It is common knowledge that $\alpha\geq 1/4$ (Hardy [3] and Landau,
independently), but its exactly value is unknown. It is conjectured that
$\alpha=1/4$. The best upper bound up to date is due to Huxley [4] (2003):
$\alpha\leq 131/416\approx 0.314$. For a nice historical account on this
problem we refer to the book of Tenenbaum [10].

Before we state our next result we recall some classical notation. Here $\mu$
is the Möbius function.

######  Theorem 1.4.

Let $f_{1}$ and $f_{2}$ be two multiplicative functions satisfying conditions
i-iii of Theorem 1.2, and let $m_{1}$ and $m_{2}$ be the periods of $f_{1}$
and $f_{2}$, respectively. Let $f=f_{1}\ast f_{2}$. Then, for all
$x>m_{1}m_{2}$

$\sum_{n\leq
x}f(n)=\sum_{n|m_{1}m_{2}}f\ast\mu\ast\mu(n)\Delta\left(\frac{x}{n}\right).$

######  Corollary 1.1.

Let $f$ be as in Theorem 1.4 and $\alpha$ defined by (3). Then, for all
$\epsilon>0$

$\sum_{n\leq x}f(n)=O_{\epsilon}(x^{\alpha+\epsilon}).$

In particular, by the result of Huxley:

$\sum_{n\leq x}f(n)=O_{\epsilon}(x^{131/416+\epsilon}).$

Thus, we have a considerable gap between our omega result (Theorem 1.3) and
our upper bound above. Even for the simplest case
$f_{1}(n)=f_{2}(n)=(-1)^{n+1}$ it seems to be hard to obtain sharp estimates
for the partial sums of $f=f_{1}\ast f_{2}$. We speculate that $\sum_{n\leq
x}f(n)=\Omega(x^{1/4})$, and in the final section of this paper we prove this
omega bound for some particular cases of non-vanishing periodic multiplicative
functions. We also discuss a possible approach to the general case.

##  2\. Proofs of the main results

###  2.1. Notation

We use both $f(x)\ll g(x)$ and $f(x)=O(g(x))$ whenever there exists a constant
$C>0$ such that $|f(x)|\leq C|g(x)|$ for all $x$ in a set of parameters. When
not specified, this set of parameters will be the range in which $x$ is
sufficiently large. Further, $\ll_{\delta}$ means that the implicit constant
may depend on $\delta$. The standard $f(x)=o(g(x))$ means that $\lim_{x\to
a}\frac{f(x)}{g(x)}=0$. Sometimes $a$ can be $\infty$. We write $\mathcal{P}$
for the set of primes and $p$ for a generic element of $\mathcal{P}$. The
notation $p^{k}\|n$ means that $k$ is the largest power of $p$ for which
$p^{k}$ divides $n$. Dirichlet convolution is denoted by $\ast$.

###  2.2. Proof of Theorems 1.1 and 1.2

We begin with the following.

######  Lemma 2.1.

If $f:\mathds{N}\to\mathds{C}$ is multiplicative and has bounded partial sums,
then $\sup_{n}|f(n)|<\infty$ and for each $\epsilon>0$, there exists a $M>0$
such that if $p\geq M$, then $|f(p^{k})|\leq 1+\epsilon$, for all $k\geq 1$.

###### Proof.

Let $C>0$ be such that $\left|\sum_{n\leq x}f(n)\right|\leq C$ for all $x\geq
1$. Assume by contradiction that $f$ is not $O(1)$. Thus there exists a
sequence of integers $x_{k}\to\infty$ such that $|f(x_{k})|\to\infty$. Since

$|f(x_{k})|-\left|\sum_{n\leq x_{k}-1}f(n)\right|\leq\left|\sum_{n\leq
x_{k}}f(n)\right|\leq C,$

we obtain a contradiction for large $k$. Thus $f$ must be $O(1)$. Now if there
are an infinite number of distinct primes $p_{1},p_{2},...$ such that for some
powers $k_{1},k_{2},...$, $|f(p_{j}^{k_{j}})|>1+\epsilon$, then $|f(n_{l})|$
become arbitrarily large for $n_{l}=p_{1}^{k_{1}}\cdot...\cdot p_{l}^{k_{l}}$,
and thus $f$ is not $O(1)$. ∎

###### Proof of Theorem 1.1.

Assume that $f$ has bounded partial sums. Therefore, the Dirichlet series
$F(s):=\sum_{n=1}^{\infty}\frac{f(n)}{n^{s}}$ is analytic in the half plane
$Re(s)>0$. By Lemma 2.1 above there exists a constant $C>0$ such that
$|f(n)|\leq C$, and hence, for $Re(s)>1$, $F(s)$ is given by the Euler product

$F(s)=\prod_{p\in\mathcal{P}}\sum_{k=0}^{\infty}\frac{f(p^{k})}{p^{ks}}.$

Now we split the Euler product in primes below and above $M$, where $M$ is
such that for all primes $p\geq M$, $|f(p^{k})|\leq 1+\epsilon$ for all $k\geq
1$. For the tail product we have that for $\sigma>1$

$\frac{1}{\zeta(\sigma)}\prod_{p>M}\sum_{k=0}^{\infty}\frac{f(p^{k})}{p^{k\sigma}}=\prod_{p\leq
M}\left(1-\frac{1}{p^{\sigma}}\right)\prod_{p>M}\left(1+\frac{f(p)-1}{p^{\sigma}}+\frac{O(1)}{p^{2\sigma}}\right).$

Therefore, since we assume that $\sum_{p}\frac{|1-f(p)|}{p}<\infty$, by making
$\sigma\to 1^{+}$ above we conclude that the limit exists, and since
$\zeta(\sigma)=\frac{1}{\sigma-1}+O(1)$, there exists a constant
$c\in\mathds{C}\setminus\\{0\\}$ such that

$\prod_{p>M}\sum_{k=0}^{\infty}\frac{f(p^{k})}{p^{k\sigma}}=\frac{c+o(1)}{\sigma-1},$

as $\sigma\to 1^{+}$. Thus, as $F$ is analytic at $s=1$, we conclude that as
$\sigma\to 1^{+}$, the finite product

$\prod_{p\leq M}\sum_{k=0}^{\infty}\frac{f(p^{k})}{p^{k\sigma}}=O(\sigma-1),$

and hence

$\prod_{p\leq M}\sum_{k=0}^{\infty}\frac{f(p^{k})}{p^{k}}=0,$

and this can happen only if some Euler factor equals to $0$. ∎

The proof of the next result follows the lines of Proposition 4.4 of [5].

###### Proof of Theorem 1.2.

Assume that $f:\mathds{N}\to\mathds{C}$ is multiplicative, has period $m$,
$f(m)\neq 0$ and has bounded partial sums. Then for all $k\geq 1$,
$f(km)=f(m)$. In particular, since $f(m)\neq 0$, for each $k$ coprime with
$m$, $f(k)=1$. Now write $m$ as a power of distinct primes, say
$p_{1}^{a_{1}},...,p_{l}^{a_{l}}$, where each $a_{j}\geq 1$. Since $f(m)\neq
0$, we obtain that each $f(p_{j}^{a_{j}})\neq 0$. Thus, by setting
$k=p_{j}^{t}$, the equation $f(km)=f(m)$ implies that
$f(p_{j}^{a_{j}+t})=f(p_{j}^{a_{j}})$. Thus we have shown that conditions ii-
iii are satisfied.

Observe that, since $f$ has period $m$ and bounded partial sums, we have that
$\sum_{n\leq m}f(n)=0$.

Now notice that if $\gcd(n,m)=d$, then $f(n)=f(d)$. This is because for each
$p^{a}\|n$ such that $\gcd(p,m)=1$, we have that $f(p^{a})=1$, and if
$p^{b}\|m$ with $b\geq 1$, we have that $f(p^{a})=f(p^{c})$ where
$c=\min(a,b)$. Thus we can write

$\displaystyle\sum_{n\leq m}f(n)=\sum_{d|m}\sum_{\begin{subarray}{c}n\leq m\\\
\gcd(n,m)=d\end{subarray}}f(n)=\sum_{d|m}f(d)\varphi(m/d)=f\ast\varphi(m),$

where $\varphi$ is the Euler’s totient function. Since $f$ and $\varphi$ are
multiplicative, we have that $f\ast\varphi$ is multiplicative. Recall that
$\varphi(p^{a})=p^{a}(1-1/p)$. Thus for each $p^{a}\|m$ with $a\geq 1$, we
have that

$\displaystyle f\ast\varphi(p^{a})$
$\displaystyle=f(p^{a})+f(p^{a-1})p\left(1-\frac{1}{p}\right)+f(p^{a-2})p^{2}\left(1-\frac{1}{p}\right)+...+p^{a}\left(1-\frac{1}{p}\right)$
$\displaystyle=p^{a}\left(1-\frac{1}{p}\right)\left(\sum_{k=0}^{a-1}\frac{f(p^{k})}{p^{k}}+\frac{f(p^{a})}{p^{a}(1-1/p)}\right).$

But since $f(p^{a})=f(p^{k})$ for all $k\geq a$, we have that

$\frac{f(p^{a})}{p^{a}(1-1/p)}=\sum_{k=a}^{\infty}\frac{f(p^{k})}{p^{k}}.$

Thus,

(4) $\sum_{n\leq
m}f(n)=\varphi(m)\prod_{p|m}\sum_{k=0}^{\infty}\frac{f(p^{k})}{p^{k}},$

and hence condition i. must be satisfied.

Now assume conditions i-iii. Then as above, if $\gcd(a,m)=d$, then
$f(a)=f(d)$, and if $n\equiv a\mod m$, then $\gcd(n,m)=\gcd(a,m)$, and hence
$f$ has period $m$. Now with conditions ii-iii we can arrive at (4), and with
condition i. we conclude that $\sum_{n\leq m}f(n)=0$, and thus $f$ must have
bounded partial sums. ∎

###  2.3. Proof of Theorems 1.3 and 1.4

######  Lemma 2.2.

Let $f=f_{1}\ast f_{2}$ where $f_{1}$ and $f_{2}$ are multiplicative functions
satisfying conditions i) ii) and iii) of Theorem 1.2. Let $m_{1}$ and $m_{2}$
be the periods of $f_{1}$ and $f_{2}$ respectively. Then $f=g\ast\tau$, where
$g$ satisifies the following properties.
a) $\sum_{n\leq x}|g(n)|=O_{\epsilon}(x^{\epsilon})$, for all $\epsilon>0$;
b) If $\gcd(n,m_{1}m_{2})=1$, then $g(n)=0$;
c) $\sum_{n=1}^{\infty}\frac{g(n)}{n}=\sum_{n=1}^{\infty}\frac{g(n)\log
n}{n}=0$.

###### Proof.

Let $Re(s)>1$. By the classical identity for the Dirichlet series of a
convolution and the Euler product formula, we have that

$F(s):=\sum_{n=1}^{\infty}\frac{f_{1}\ast
f_{2}(n)}{n^{s}}=\prod_{p\in\mathcal{P}}\left(\sum_{k=0}^{\infty}\frac{f_{1}(p^{k})}{p^{ks}}\right)\left(\sum_{k=0}^{\infty}\frac{f_{2}(p^{k})}{p^{ks}}\right).$

Now, by assumption iii., if $\gcd(p,m_{1}m_{2})=1$, then
$f_{1}(p^{k})=f_{2}(p^{k})=1$ for all powers $k\geq 1$. Therefore

$F(s)=\prod_{p|m_{1}m_{2}}\left(\sum_{k=0}^{\infty}\frac{f_{1}(p^{k})}{p^{ks}}\right)\left(\sum_{k=0}^{\infty}\frac{f_{2}(p^{k})}{p^{ks}}\right)\prod_{\begin{subarray}{c}p\in\mathcal{P}\\\
\gcd(p,m_{1}m_{2})=1\end{subarray}}\left(1-\frac{1}{p^{s}}\right)^{-2},$

and hence

(5)
$G(s):=\frac{F(s)}{\zeta(s)^{2}}=\prod_{p|m_{1}m_{2}}\left(\sum_{k=0}^{\infty}\frac{f_{1}(p^{k})}{p^{ks}}\right)\left(\sum_{k=0}^{\infty}\frac{f_{2}(p^{k})}{p^{ks}}\right)\left(1-\frac{1}{p^{s}}\right)^{2}.$

Recall that $\zeta(s)^{2}$ is the Dirichlet series of $\tau=1\ast 1$. Thus,
$G(s)$ is the Dirichlet series of $g:=f\ast\tau^{-1}=f\ast(\mu\ast\mu)$, where
$\mu$ is the classical Möbius function. Therefore, by the Euler product
formula for $G(s)$ above, we have that condition b) must be satisfied. Since
$f_{1}$ and $f_{2}$ are $O(1)$, we have that there exists a constant $c>0$,
such that for all primes $p$ and all powers $k\geq 1$, $|g(p^{k})|\leq ck$.
This implies that for each $\sigma>0$

$\sum_{p|m_{1}m_{2}}\sum_{k=1}^{\infty}\frac{|g(p^{k})|}{p^{k\sigma}}<\infty,$

and hence, by a classical result for Dirichlet series (see for instance [10],
pg. 188, Theorem 1.3), $G(s)=\sum_{n=1}^{\infty}\frac{g(n)}{n^{s}}$ converges
absolutely in the half plane $Re(s)>0$ and is given by (5) for each $s$ in
this half plane. In particular, for each $\epsilon>0$,
$\sum_{n=1}^{\infty}\frac{|g(n)|}{n^{\epsilon}}<\infty$, and hence, by
Kroenecker’s Lemma (see for instance [8], pg. 390, Lemma 2), we have that
condition a) is satisfied. Finally, by assumption i., there are primes
$q_{1}|m_{1}$ and $q_{2}|m_{2}$ such that

$\sum_{k=0}^{\infty}\frac{f_{1}(q_{1}^{k})}{q_{1}^{k}}=\sum_{k=0}^{\infty}\frac{f_{2}(q_{2}^{k})}{q_{2}^{k}}=0.$

Hence, by analyticity

$\sum_{k=0}^{\infty}\frac{f_{1}(q_{1}^{k})}{q_{1}^{ks}}=O(|s-1|),\;\sum_{k=0}^{\infty}\frac{f_{2}(q_{2}^{k})}{q_{2}^{ks}}=O(|s-1|),$

for all $s$ sufficiently close to $1$. This combined with (5) gives that
$G(s)=O(|s-1|^{2})$, for all $s$ sufficiently close to $1$, and since $G$ is
analytic, we have that $G(1)=G^{\prime}(1)=0$. But
$G(1)=\sum_{n=1}^{\infty}\frac{g(n)}{n}$ and
$G^{\prime}(1)=-\sum_{n=1}^{\infty}\frac{g(n)\log n}{n}$, and this completes
the proof. ∎

###### Proof of Theorem 1.3.

By the triangle inequality we have that for each positive integer $x$,

$|f(x)|\leq\left|\sum_{n\leq x-1}f(n)\right|+\left|\sum_{n\leq x}f(n)\right|.$

Therefore, by the pigeonhole principle, we have that at least one of the two
sums in the right-hand side above is at least $|f(x)|/2$. By Lemma 2.2, we
have that for each $\gcd(n,m_{1}m_{2})=1$, $f(n)=\tau(n)\geq 2^{\omega(n)}$,
where $\omega(n)=\sum_{p|n}1$. Since $\omega(n)$ can be as large as
$(1+o(1))\log n/\log\log n$ (see for instance [10] pg. 113, Theorem 5.4), we
complete the proof. ∎

###### Proof of Theorem 1.4.

By Lemma 2.2, we have that $f=g\ast\tau$. This combined with (2) gives that

$\displaystyle\sum_{n\leq x}f(n)$ $\displaystyle=\sum_{n\leq x}g(n)\sum_{m\leq
x/n}\tau(m)=\sum_{n\leq
x}g(n)\left(\frac{x}{n}\log(x/n)+(2\gamma-1)\frac{x}{n}+\Delta(x/n)\right)$
$\displaystyle=x\log x\sum_{n\leq x}\frac{g(n)}{n}-x\sum_{n\leq
x}\frac{g(n)\log n}{n}+(2\gamma-1)x\sum_{n\leq x}\frac{g(n)}{n}+\sum_{n\leq
x}g(n)\Delta(x/n).$

We will show that each of the first three sums in the right hand side above
vanishes for $x>m_{1}m_{2}$. This is the moment when we will use the condition
ii) of Theorem 1.2. The condition states that if $p^{k_{j}}\|m_{j}$ for
$j=1,2$, then $f_{j}(p^{t_{j}})=f_{j}(p^{k_{j}})$ for all $t_{j}\geq k_{j}$.
Here we allow that $k_{j}=0$. Thus
$f_{j}\ast\mu(p^{t_{j}})=f_{j}(p^{t_{j}})-f_{j}(p^{t_{j}-1})=0$, for all
$t_{j}\geq k_{j}+1$. Since $g=(f_{1}\ast\mu)\ast(f_{2}\ast\mu)$, we have that
$g(p^{t})=0$ for all $t\geq k_{1}+k_{2}+1$. This can be easily seen by the
fact that each Euler factor in the Euler product representation of
$\sum_{n=1}^{\infty}\frac{g(n)}{n^{s}}$ has the form

$\sum_{k=0}^{k_{1}}\frac{f_{1}\ast\mu(p^{k})}{p^{ks}}\sum_{l=0}^{k_{2}}\frac{f_{2}\ast\mu(p^{l})}{p^{ls}}.$

Therefore, by b) of Lemma 2.2, if $n>m_{1}m_{2}$, we have that $g(n)=0$, and
by c) of the same Lemma we obtain the desired claim. ∎

######  Remark 2.1.

We observe that without ii. of Theorem 1.2, we could prove a slighty more
general statement but with weaker conclusions in comparison with the one
obtained in Theorem 1.4. Indeed, this was done in a preprint version of this
paper (see arXiv:2110.03401, v3). There we show that

$\sum_{n\leq x}f_{1}\ast f_{2}(n)=\sum_{n\leq
x}g(n)\Delta(x/n)+O_{\epsilon}(x^{\epsilon}),$

for any $\epsilon>0$. To establish this, one should impose a growth condition
on each $f_{j}$ so that one could prove a) from Lemma 2.2.

###### Proof of Corollary 1.1.

By Theorem 1.4, we have that $\sum_{n\leq x}f(n)$ can be expressed as a finite
linear combination of the functions $(\Delta(x/n))_{n}$. The proof is then an
immediate consequence of the triangle inequality. ∎

##  3\. Discussion on $\Omega$ bounds for $f_{1}\ast f_{2}$

As we point out in the introduction, there is a considerable gap between our
omega bound (Theorem 1.3) and our upper bound (Corollary 1.1). Here we propose
an approach to prove our conjecture that $\sum_{n\leq x}f_{1}\ast
f_{2}(n)=\Omega(x^{1/4})$, where each $f_{j}$ is periodic with bounded partial
sums that satisfy i., ii. and iii. of Theorem 1.2.

A nice result proved by Tong [11] states that

$\int_{1}^{X}\Delta(x)^{2}dx=(A+o(1))X^{3/2},$

where $A$ is the constant given by

(6) $A=\frac{1}{6\pi^{2}}\sum_{n=1}^{\infty}\frac{\tau(n)^{2}}{n^{3/2}}.$

Observe that if a function $\lambda(x)$ is $o(x^{1/4})$, then
$\int_{1}^{X}|\lambda(x)|^{2}dx=o(X^{3/2})$. Therefore, this combined with
Tong’s result gives a second proof (different from Hardy’s) that
$\Delta(x)=\Omega(x^{1/4})$, and also allows us to prove the following result.

######  Proposition 3.1.

Let $q$ be a prime number and $f$ be the unique $q$-periodic multiplicative
function with bounded partial sums, and such that $f(q)\neq 0$. Assume that
$q\geq 5$. Then

$\sum_{n\leq x}f\ast f(n)=\Omega(x^{1/4}).$

###### Proof.

By Theorem 1.2 we have that $f(q^{k})=f(q)$ for all $k\geq 1$. Then, condition
i. determines uniquely the value of $f(q)$:

$1+f(q)\sum_{k=1}^{\infty}\frac{1}{q^{k}}=0.$

Therefore, $f(q)=-(q-1)$. Now, for $k\geq 1$, by condition iii. of Theorem
1.2, we have that $f\ast\mu(p^{k})$ is $0$ unless $p=q$. In this case we have
that $f\ast\mu(q^{k})$ is: $0$ if $k\geq 2$, and $f(q)-1=-q$, if $k=1$. The
Euler factor corresponding to $q$ in the Euler product representation of the
Dirichlet series of $g:=(f\ast\mu)\ast(f\ast\mu)$ is

$\left(1-\frac{q}{q^{s}}\right)^{2}=1-\frac{2q}{q^{s}}+\frac{q^{2}}{q^{2s}}.$

This immediately implies that $g(q)=-2q$ and $g(q^{2})=q^{2}$. Hence, by
Theorem 1.4, for $x>q^{2}$

$\sum_{n\leq x}f\ast f(n)=\Delta(x)-2q\Delta(x/q)+q^{2}\Delta(x/q^{2}).$

Now, by combining Tong’s result (6) with a simple change of variables, for any
$n>0$ we obtain that

$\|\Delta(x/n)\|_{L^{2}[1,X]}:=\left(\int_{1}^{X}\Delta(x/n)^{2}dx\right)^{1/2}=\frac{\sqrt{A+o(1)}}{n^{1/4}}X^{3/4}.$

The Cauchy-Schwarz and Minkowski inequalities for $L^{2}$ spaces imply that

$\displaystyle\|\Delta(x)-2q\Delta(x/q)$
$\displaystyle+q^{2}\Delta(x/q^{2})\|_{L^{2}[1,X]}$ $\displaystyle\geq
q^{2}\|\Delta(x/q^{2})\|_{L^{2}[1,X]}-2q\|\Delta(x/q)\|_{L^{2}[1,X]}-\|\Delta(x)\|_{L^{2}[1,X]}$
$\displaystyle\geq\sqrt{A+o(1)}X^{3/4}\left(\frac{q^{2}}{q^{2/4}}-\frac{2q}{q^{1/4}}-1\right)$
$\displaystyle\geq\sqrt{A+o(1)}X^{3/4}\left(q^{3/2}-2q^{3/4}-1\right).$

With standard calculus we can check that the function
$\lambda(q):=q^{3/2}-2q^{3/4}-1$ is increasing for $q\geq 1$, and since
$\lambda(5)=3.4929$, we have that $\lambda(q)>0$ for all $q\geq 5$. This shows
that in this range of $q$,

$\bigg{\|}\sum_{n\leq x}f\ast
f(n)\bigg{\|}_{L^{2}[1,X]}\geq\sqrt{A+o(1)}\lambda(q)X^{3/4},$

which gives the desired omega bound. ∎

In our proof above, we see that this method does not work in the case that
$q\in\\{2,3\\}$, since in these cases $\lambda(q)<0$. In particular, in the
case $q=2$, the associated periodic multiplicative function is the classical
$f(n)=(-1)^{n+1}$. For this particular case, in the next figure we plot the
partial sums of $f\ast f$, and the numerics are in agreement with our
conjecture.

Figure 1. The dashed curves are given by $x\mapsto\pm 4x^{1/4}$, and the
continuous line is given by $x\mapsto\sum_{n\leq x}f\ast f(n)$, where
$f(n)=(-1)^{n+1}$.

We conclude by mentioning a possible approach to the conjectured omega bound.
Since $\sum_{n\leq x}f_{1}\ast f_{2}(n)$ can be expressed as $\sum_{n\leq
T}c_{n}\Delta(x/n)$, where $T$ is a positive integer and $c_{n}$ are complex
numbers, one could approach the conjectured omega bound by studying the
quadratic form obtained from the squared $L^{2}[1,X]$ norm of $\sum_{n\leq
T}c_{n}\Delta(x/n)$. The conjectured omega bound would follow if, for
instance, one could prove that the eigenvalues of the symmetric matrix
$(a_{n,m})_{n,m\leq T}$ are positive, where

$a_{n,m}=\lim_{X\to\infty}\frac{1}{X^{3/2}}\int_{1}^{X}\Delta(x/n)\Delta(x/m)dx.$

To prove this, our preliminary calculations show that, by using the classical
Voronoï’s formula for $\Delta(x)$ (see Lemma 1 of [7]), firstly we need to
understand the effect of positive integers $a$ and $b$ in the correlations

$\sum_{n\leq x}\tau(an)\tau(bn).$

Acknowledgements. I am thankful to Oleksiy Klurman for a fruitful discussion
that resulted in this work. This was in February of 2020 while I was visisting
MPIM (Bonn), for which I am also thankful for its warm hospitality. Also, I
would like to thank the anonymous referee for several remarks and corrections
that improved the exposition of this paper, and to my students Caio Bueno and
Kevin Medeiros for their comments on a revised version of it. I am supported
by CNPq - grant Universal number 403037/2021-2.

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Departamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio
Carlos, 6627, CEP 31270-901, Belo Horizonte, MG, Brazil.
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