11institutetext:  Amazon.com, Inc. Seattle, WA, USA
11email<EMAIL_ADDRESS>
# Identifying Shopping Intent in Product QA for Proactive Recommendations

Besnik Fetahu  Nachshon Cohen  Elad Haramaty  Liane Lewin-Eytan  Oleg
Rokhlenko  Shervin Malmasi

###### Abstract

Voice assistants have become ubiquitous in smart devices allowing users to
instantly access information via voice questions. While extensive research has
been conducted in question answering for voice search, little attention has
been paid on how to enable proactive recommendations from a voice assistant to
its users. This is a highly challenging problem that often leads to user
friction, mainly due to recommendations provided to the users at the wrong
time. We focus on the domain of e-commerce, namely in identifying _Shopping
Product Questions_ (SPQs), where the user asking a product-related question
may have an underlying shopping need. Identifying a user’s shopping need
allows voice assistants to enhance shopping experience by determining _when_
to provide recommendations, such as _product_ or _deal_ recommendations, or
_proactive shopping actions recommendation_. Identifying SPQs is a challenging
problem and cannot be done from question text alone, and thus requires to
infer latent user behavior patterns inferred from user’s past shopping
history. We propose features that capture the user’s latent shopping behavior
from their purchase history, and combine them using a novel Mixture-of-Experts
(MoE) model. Our evaluation shows that the proposed approach is able to
identify SPQs with a high score of F1=0.91. Furthermore, based on an _online_
evaluation with real voice assistant users, we identify SPQs in real-time and
recommend shopping actions to users to add the queried product into their
shopping list. We demonstrate that we are able to accurately identify SPQs, as
indicated by the significantly higher rate of added products to users’
shopping lists when being prompted after SPQs vs random PQs.

##  1 Introduction

Voice assistants, like Alexa or Google Assistant provide ubiquitous services
through a variety of devices (e.g. smart speakers, phones, TVs etc.). Users
interact with voice-assistants for different purposes [42, 33] such as
question answering [11], task completion [9], conversational search [43],
entertainment, or control of smart devices. Determining the underlying _user
intent_ is an active field of research [47, 26, 53], given that new skills are
continuously added to such voice assistants.

In this context, making proactive follow-on suggestions is an active area of
research [12]. However, key challenges with voice-based conversational
recommender systems remain, such as their failure to adapt to evolving user
behavior [37]. Additionally, interactions are typically initiated by users,
and proactive system recommendations may lead to increased user friction.
Knowing _when_ to proactively recommend actions or items to users on the next
turn, such as suggesting the right product recommendations [8, 34, 36, 19, 28,
31] or next actions, is tightly dependent on accurately identifying the user’s
underlying intent. Correctly identifying this intent can avoid user
dissatisfaction by providing recommendations only when necessary. Our work
focuses on identifying the right time to make a proactive personalized
recommendation in a voice-based conversational system.

Figure 1: User A searches for ingredients for a _recipe_. Our approach
identifies the user having a shopping need, hence, triggering voice
assistant’s shopping recommendations. For User B’s (likely vegetarian) our
model predicts no shopping intent, hence, after answering, the turn is over.

E-commerce is an important functionality in voice assistants. In some
contexts, the shopping intent can be explicit, such as in product searches
(e.g. _"Buy me dog food"_ , or _"Search for an Apple watch stand"_). However,
intent cannot always be captured from the question alone, e.g., product-
related questions, such as: _"What can I use as a substitute for toilet
paper?"_ ; _"How long do you cook rice?"_ ; _"Are dogs allowed to eat tuna?"_.

The examples show that the user intent is often not clear from the Product
Question (PQ) itself. PQs may emerge from an underlying desire to potentially
purchase a product (_shopping need_), or only for a _general_ knowledge need.

Voice product search [5, 40] and question answering [16, 6, 39, 38] have
received attention, while understanding the reasons why users inquire
information about certain products, especially in terms of _shopping need_ ,
has seen very little progress. This is the main challenge that we tackle,
namely determine whether a user has a shopping need. This allows voice
assistants to determine _when_ to surface recommendations that improve user’s
shopping experience, e.g. after answering a product question, show product
recommendations, deals or promo suggestions, or offering additional product
details for further examination. In this work, we do not focus on _what_ to
recommend given the vast literature in product (sequential) recommendation [8,
34, 36, 19, 28, 31].

Figure 1 shows a hypothetical user interaction with a voice assistant.
Depending on the user’s interests, shopping recommendations may differ. For
instance, for _User A_ , based on their purchase history we can infer that
they may be interested in purchasing a queried product, recommending shopping
related actions, and thus ease the shopping process for the user. For _User B_
on the other hand, who has vegetarian preferences, the intent is general
knowledge need.

Identifying Shopping Product Questions (SPQ) in voice search poses several
challenges. Contrary to shopping intent prediction in e-commerce sites [10,
30, 15], signals such as click-rate, browse time, hardware related gestures
are not present. Furthermore, shopping need detection in voice search is
inherently harder given the multi-purpose use of voice assistants, compared to
a restricted use of e-commerce sites.

We propose an approach based on Graph Attention Networks (GAT) [49] to
identify shopping need. Questions are considered nodes and are connected if
they share the same product. As input features, we consider features that are
geared at capturing diverse and latent aspects of shopping need, such as
_product information_ , user _past purchasing behavior_ , and the _question_
itself. Finally, based on the mixture-of-experts approach (MoE) [44], we
propose a mechanism to compute a joint node representation from the diverse
features.

Experimental evaluation on more than 370k voice assistant user questions shows
that our proposed approach allows us to achieve highly accurate results in
distinguishing SPQs. Furthermore, we carry out an online experiment with real
voice assistant users, where we identify SPQs in real-time, and recommend
users to add the queried product into their _shopping lists_ following a SPQ,
identified with high accuracy.111As future work we plan to experiment with
different types of recommended shopping actions and assess the impact on the
user experience. The approach presented in this paper through detection of
SPQs allows voice assistants to recommend personalized responses according to
user’s interests and needs, such as “ _check price”_ , _“show deals”_ , _“add
to cart”_ , _“compare products”_ etc. Our main contributions in this work are
as follows:

  * •

A new problem definition for identifying SPQs that enable voice assistants to
determine when to recommend to users.

  * •

An approach to identify SPQs in voice search, using a novel way to combine
diverse features through Mixture of Experts stemming from various user signals
and containing different feature types.

  * •

As part of these experiments, we perform detailed ablations showing the impact
of the different features in identifying queries with shopping need,
highlighting the latent nature of users shopping behavior, and thus opening
directions for improving users shopping experience with voice assistants.

  * •

Detailed offline and online experimental evaluation on voice assistant users
and user queries.

##  2 Related Work

Users Needs in E-commerce. Informational need in e-commerce and voice shopping
is typically expressed by asking questions about products. This domain is also
referred to as Product Question Answering (PQA), and has attracted much
interest in the past years [6, 7, 23, 55, 14, 41]. Some research in the PQA
domain has also been conducted with respect to the voice medium [16, 6, 39,
38], where queries have different characteristics, and where cross-lingual and
speech-to-text components add another layer of complexity. Most of the works
in the PQA domain focus on different answering approaches and on different
types of questions [41, 52, 35, 54, 24, 25]. Here, we do not focus on the
answering side at all, but rather focus on the informational need behind
product questions.

Transactional need in e-commerce providers has been investigated, mainly
through purchase intent prediction [10, 30, 15, 2, 45], where behavior
features correspond to browse or hardware related (touch interaction on
smartphones) [15]. Esmeli et al. [10] extract features such as product click
rate, view time, number of visits, to predict purchase intent. Our key
differences with these works is that we tackle the problem of voice modality,
where voice assistants are used for highly diverse purposes [33, 42], which is
not the case of dedicated e-commerce sites, where the primary purpose is for
shopping. Secondly, features that can be extracted from Web sites are not
available for voice assistants, making the problem of predicting shopping
needs of voice users more challenging. The work of [18] explores the problem
of predicting the purchase rate of a ranking model in a voice setting, where
there is an extreme position bias towards the first offer. This work lies in
the domain of voice product search, where the user’s intent is clearly
transactional – unlike our work, which aims to identify an underlying shopping
need in the voice PQA domain.

Product Search and Recommendations. There has been a large body of research in
product search and product recommendations [3, 27, 29, 57, 50, 56], which are
crucial in the shopping domain, for providing relevant products, and
recommendations reflecting users’ preferences. The voice medium has also
attracted research in this context [5, 40], as shopping queries and behavioral
shopping patterns in voice are different than in web [20]. Learning product
embeddings is another challenge that has received much attention [17, 46, 48],
as having relevant product embeddings allows for more accurate recommendations
and better user experience. Our goal is different in that, the product of
interest is explicit in the user query. Our goal is to understand the
underlying user’s intent, rather than to provide the most relevant
recommendation. Nevertheless, advanced product embeddings used for product
recommendation might help in capturing the customer affinity to a specific
product and improve further the shopping need classification.

Behavioral Features in Search and Recommendation. The works in [1, 21] propose
the integration of implicit user feedback for ranking Web search results.
Although these works tackle a different problem and are not comparable to our
work, we share a similar goal in that we obtain user behavior features (e.g.
purchase history) that can serve to learn a latent user shopping intent
behavioral model.

Works on session based recommendation [36, 19, 28, 31] propose approaches for
incorporating diverse features to facilitate recommendation within a given
user session (e.g. music search). At its core it predicts the next item with
which the user will interact. These works are complementary to ours as they
address the question on _what_ to recommend, whereas we focus on determining
_when_ to trigger recommendations by the voice assistants recommender systems.

##  3 Data Analysis

###  3.1 Dataset and Labels

Since we cannot observe the true user intent, we rely on their purchase
behavior: if the user asked a question and then purchased the referred
product, we consider it as having a shopping need.

A shopping intent does not often result in an instantaneous purchase.
Selecting the time window for the purchase information, from the time the user
issued a PQ up to the actual purchase, represents an interesting tradeoff.
Longer periods tend to better capture the hidden purchase intent, however at
the same time pose the risk of labeling unrelated utterances with shopping
intent. We analyzed different time windows, including one hour/day/week/two
weeks, and 28 days, and found that a period of 28 days increased coverage by
$187\%$ while increasing noise by $8\%$ only222We estimated noise by
scrambling the utterances and purchases and measure how often an utterance is
coincidentally followed by a purchase. compared to a period of one week. Given
a PQ, we checked the future user purchases. If the referred product – or a
similar product, sharing the same category – was purchased within 28 days, we
label the PQ as SPQ, otherwise NSPQ.

We collected a dataset of 374k PQs from a leading voice assistant, evenly
distributed among SPQs and NSPQs333 The sample was designed to have an even
distribution of NSPQs and SPQs, as such it is not representative of the
overall traffic and none of the results reported here should be extrapolated
to the entire traffic, split into 306k/34k/37k subsets for
train/validation/test set, respectively. For each question, we collected its
text, voice assistant’s response, the product and its category, and user past
purchases.

###  3.2 Data Insights

To understand correlated signals with SPQs, we perform a correlation analysis
(Pearson’s correlation coefficient), between the question type, their timing,
and the searched product.

####  3.2.1 Questions.

The relation between the user’s shopping need and the question text is
involved. We found that questions asked from users about vitamins or pet-food
have a high shopping intent (88%, 82%), while questions about televisions or
cellphones have low shopping intent (only 16%, 18%). Figure 2 shows the
histogram of the SPQ rate for the 50 most popular product categories. We see
that different product types have different SPQ rates compared to the average
50%. There are product types where most of the queries are general knowledge
need (4 categories with SPQ rate below 0.25%), while there are product types
where most questions have a shopping need (5 categories with SPQ rate above
0.75%). For the rest, more than half of the popular categories have SPQ rate
above 60%. This demonstrates that the product category, which can be inferred
from the text of the question, is indicative of SPQs.

Figure 2: SPQ rate per product type (out of 50 types with at least 1000 PQs,
which represent 29.7% of the entire dataset).

However, while a shopping need is correlated with some textual features of the
questions, this correlation is not natural nor intuitive. To demonstrate this,
we sampled 10K questions and presented them to annotators. The annotator was
presented with the question, the voice assistant response, and was required to
estimate whether the user will likely purchase the referred product. To
demonstrate this, we labelled 10K utterances, asking annotators to predict
whether the user will purchase the product. We saw a very low correlation of
0.05 between human evaluation and future purchases. This indicates that from
text alone, humans cannot predict whether a question has a shopping intent. An
example of this phenomena, e.g., _"“what do fish eat?”"_ , which to annotators
seems like a general knowledge need question, while later purchases of fishing
materials demonstrate the latent shopping need behind this question.

####  3.2.2 Users.

The user properties also have an involved relation with the shopping intent.
On the one hand, we found that shopping need is not a user property. We took
two random PQs from each user, and measured the correlation between the first
one being an SPQ and the second being a NSPQ. To avoid two questions referring
to the same product category, which could lead to a false correlation, we de-
duplicated PQs from the same user about the same product category. The
correlation was only $r=0.14$, showing that the same user may ask questions
that have or do not have a shopping need.

On the other hand, we found a relation between the user’s shopping history and
the shopping need. For each PQ, we checked if the referred product was
previously purchased by the user (28 days prior). This property has a strong
correlation of $r=0.67$ with the SPQ property, which is based on a future
purchase. This shows that when the user asks about a product category that was
previously purchased, they are more likely to purchase again.

While there is strong correlation between SPQs and preceding purchases, this
might simply reflect user tendency to repurchase products, and be unrelated to
the question. To refute this possibility, we measured the correlation of
purchasing the referred product during two consecutive 28-days periods, before
the actual user’s question (PQ). We found that the correlation is
significantly weaker when compared to consecutive periods coming before and
after the user’s PQ, showing that the PQ plays an important role. More
specifically, given a PQ referring to product $P$ from the user’s shopping
history, we consider four 28-days time periods $T_{0},T_{-1},T_{-2}$ and
$T_{-3}$, where $T_{0}$ is the time period coming after asking the question,
$T_{-1}$ is the time period coming before the question, and consecutively
$T_{-3}<T_{-2}<T_{-1}$. The correlation between purchasing product $P$ during
$T_{0}$ and purchasing $P$ during $T_{-1}$ is measured to be $r=0.67$. If this
correlation was due to the periodic nature of purchasing $P$, we would expect
a similar correlation of purchasing $P$ also during $T_{-2}$ and $T_{-3}$.
However, the correlation was $r=0.18$, showing that the high correlation of
purchasing $P$ between times $T_{0}$ and $T_{-1}$ is strongly connected to the
PQ, which indicates its shopping intent.

##  4 Identifying SPQs

To identify SPQs we present a model SPQI with three components: (a) features
of different types, (b) feature aggregation through mixture of experts [44],
and (c) a Graph Attention Network.

Textual Features. The PQ is an important intent indicator, containing key
aspects that are asked about a product (e.g., _price_ , _delivery time_ ,
etc.). Additionally, voice assistant’s response to the question contains
information if the question was understood and answered. Furthermore, when a
customer asks a question the voice assistant responds with either an _answer_
or indicates that it _does not understand_ the utterance. For cases where the
voice assistant provides an answer, the user can continue their conversation
by either asking additional questions regarding the product or make a shopping
decision. We encode textual features extracted with a RoBERTa model [32], and
use the [CLS] token for representation.

Product Features. An important aspect of a PQ is the queried product itself.
Products have a rich structure, and users may have shopping preferences
towards seemly unrelated products or specific categories, e.g., because they
share a similar price range or a similar theme, e.g., _“zucchini”_ and
_“broccoli”_ , or _“healthy food”_. We use the product and the product
category embeddings444Products are organized in a taxonomic product category
graph that are trained from scratch to capture shopping need related features.

We compute the product $p$, product category $c$, and parent category
$\hat{c}$ of $c$ embeddings. The features are considered independently from
each other555As future work we foresee integrating graph embeddings for
categories and products. and allow the model to learn how to leverage them
during training.

Behavioral Features. To identify shopping intent, the user’s shopping
behavioral patterns, extracted from the purchase history, are key. The
purchase history is a sparse feature, consisting of the tuples $\langle$user,
product$\rangle$. We pre-train a skip-gram based embeddings [4] for better
generalization. We use the tuple $\langle u,p\rangle$ to train a model that
predicts if the product $p$ will be purchased by user $u$, where
$u,p\in\mathbb{R}^{50}$ represent the product and user vectors. The resulting
embedding have two highly desirable properties: (1) create user vectors that
are similar according to purchasing patterns, and (2) product vectors that are
similar according to the users that co-purchased them.

Using the pre-trained model, for an input question, from the list of purchases
made by the user in previous 28 days $\mathbf{H}$, we compute: (i) average
dot/cosine similarity, maximum dot/cosine similarity, and the sum of
dot/cosine similarity.

###  4.1 Learning Joint Question Representations via MoE

An important question is how to combine the various feature types (textual,
numerical, categorical). Typically, such features are combined using either
max or average pooling mechanisms [13, 51]. One drawback of this is that they
do not allow the models to learn weights for each feature type according to
their impact on the classification task.

We propose the use of mixture-of-experts [44], which allows the dynamic mixing
of the different question feature types. Let $\mathbf{F}\in\mathbb{R}^{f\times
n}$ be the concatenated set of features. For each feature dimension in
$\mathbf{F}$, MoE weighs the importance of the $f$ individual feature types as
follows.

$\displaystyle\texttt{MoE}=\mathbf{W}^{\prime}_{e}\left(\mathbf{F}\mathbf{W}_{e}+c_{1}\right)^{T}+c_{2};\;\;\;\;\;\;\lambda_{f_{j},z}=\frac{\text{{MoE}}_{f_{j},z}}{\sum_{f^{\prime}\in|f|}\text{{MoE}}_{f^{\prime},z}}$
(1)

where $\mathbf{W}_{e}\in\mathbb{R}^{n\times i}$,
$\mathbf{W}^{\prime}_{e}\in\mathbb{R}^{n\times i}$, represent trainable
parameters used to compute feature type weights. Finally,
$\lambda_{f_{j},z}\in\mathbf{\Lambda}$ represents the feature weight for
dimension $z$ and feature type $f_{j}$, while
$\mathbf{\Lambda}\in\mathbb{R}^{f\times n}$ represents all the feature
weights.

The final joint question representation that is passed onto the GAT framework,
is computed by performing the Hadamard product between each feature vector and
the computed MoE weights:
$\mathbf{h}=\sum_{f^{\prime}\in|f|}\mathcal{F}_{f^{\prime}}\circ\mathbf{\Lambda_{f^{\prime}}}$.

###  4.2 Question Graph Construction & Classification

Considering questions as independent instances has the disadvantage of not
being able to leverage similar _questions_ or _users_ that may have similar
purchasing behavior. Hence, we formulate the SPQ identification problem as a
node classification task using GAT [49]. GAT allow to create soft dependencies
between questions (if they share the same product), and as such, influences
the prediction of shopping need for a question, by taking into account
questions about the same product from other users. An important aspect of
utilizing GATs, is the shape of the graph. As nodes are the user questions,
represented by the joint representation $\mathbf{h}$, and nodes are connected
if they share the same queried _product_.

We train a GAT model to compute the question representations:
$\mathbf{\widehat{h}}_{q}=\text{{GAT}}(\mathbf{h}_{q})$, which is used to
obtain the SPQ probability as
$\mathbf{s}_{q}=\sigma(\mathbf{\widehat{h}_{q}}\cdot\theta)$ to classify into
either an SPQ or a NSPQ.

##  5 Experiments

We discuss the offline and online evaluation results, and further introduce
baselines for the offline experimentation phase.

###  5.1 Baselines and Ablations

RoBERTa-Baseline. This baseline highlights the difficulty of identifying
shopping intent from text alone. We consider two variations: (1) RoBERTa-
Question, where only the question text is used for training and inference, and
(2) RoBERTa-Text, where additionally voice assistant’s answer is used for
training and inference.

MLP-Baseline. All the computed features are pushed through an MLP layer for
classification. We consider several ablations (MLP-Question and MLP-Text are
identical to the RoBERTa baselines) of this baseline.

SPQI Setup and Ablations. We consider different model ablations using
different feature subsets, and we distinguish between SPQI-MoE and SPQI-
CONCAT, with the difference being on how the diverse features are aggregated
with MoE or simply concatenated. SPQI has four layers of graph convolutions
[22], and each node has 1024 dimensions.

##  6 Evaluation Results

###  6.1 Offline Evaluation

Model | Behavior | Product | Text+Product | Text+Behavior | Product+Behavior | Full
---|---|---|---|---|---|---
| P | R | F1 | P | R | F1 | P | R | F1 | P | R | F1 | P | R | F1 | P | R | F1
MLP-Concat | 0.827 | 0.910 | 0.866 | 0.758 | 0.755 | 0.757 | 0.758 | 0.766 | 0.762 | 0.856 | 0.916 | 0.885∗ | 0.876 | 0.881 | 0.878 | 0.875 | 0.881 | 0.878
MLP-MoE | 0.907 | 0.886 | 0.896 | 0.756 | 0.758 | 0.757 | 0.761 | 0.730 | 0.744 | 0.895 | 0.921 | 0.907∗ | 0.903 | 0.908 | 0.906 | 0.903 | 0.914 | 0.908
SPQI-Concat | 0.903 | 0.884 | 0.893∗ | 0.760 | 0.757 | 0.759 | 0.760 | 0.772 | 0.766 | 0.898 | 0.893 | 0.895 | 0.890 | 0.904 | 0.895∗ | 0.895 | 0.909 | 0.902∗
SPQI-MoE | 0.909 | 0.890 | 0.900∗ | 0.762 | 0.764 | 0.763∗ | 0.760 | 0.793 | 0.776∗ | 0.914 | 0.885 | 0.900 | 0.903 | 0.910 | 0.907 | 0.903 | 0.917 | 0.910∗

Table 1: SPQ detection performance of competing approaches. Significant
results (as per paired t-test) are marked with ∗ (_p-value_ $<0.05$). SPQI-MoE
ablations are compared to MLP-MoE, and SPQI-Concat vs. MLP-Concat. For the
sake of clarity, we omit from the table the results for RoBERTa-Question and
RoBERTa-Text, given that they are the lowest performing with F1=0.694 and
F1=0.699, respectively. On the same feature set, SPQI-MoE-Query and SPQI-MoE-
Text obtain significantly higher results with F1=0.777 and F1=0.774,
respectively.

##### Model Performance.

Table 1 shows the results of identifying SPQs. Our approach, SPQI-MoE achieves
the best results with $F1=0.91$. SPQI-Concat-Full has a 1% drop in F1. This
shows that the flexibility of Moe to dynamically decide per question, which
features are important in predicting user’s shopping need.

Comparing SPQI-Full-MoE against MLP-MoE-Full, the use of GATs provides an
additional advantage in terms of F1. This shows that even for the same feature
set and way to combine them, GATs ability to consider neighbouring nodes for
determining shopping need is helpful. This in itself is intuitive and has been
widely studied in recommender systems (e.g. collaborative filtering), where
tying users with similar search patterns can be helpful in recommendations. In
our case, according to how subgraphs are constructed for classification, our
models exploit information coming from users who share similar purchasing
patterns and ask similar questions. Finally, RoBERTa baselines obtain the
lowest performance, showing that identifying SPQs cannot be done from the
question text alone.

In nearly all the cases, combining the question features using the MoE
approach, allows the models to feature concatenation. Furthermore, computing
the final question representation using GATs, where neighbouring nodes can
have an impact on its representation, allows the model to obtain more accurate
intent classification results.

Although the SPQI models are larger (in parameters) than their MLP
counterparts, the performance decrease is not due to model size, as the
comparison between SPQI-MoE and SPQI-Concat shows that SPQI-MoE achieves
better performance.

Ablations. The user behavioral features achieve the biggest performance
improvement. SPQI-Text+Behavior is the most accurate model with $P=0.914$,
which is 1% higher than SPQI-Full-MoE. However, in terms of recall we get a
drop of 3%. The addition of product embedding features, allows the model to
improve its recall by covering cases of users that may have a sparse purchase
history or that few purchased products within the same category as the queried
product. Interestingly, none of the feature combinations manage to achieve the
full performance of SPQI-MoE, where we draw conclusions that while the
precision may be higher on certain feature subsets (e.g. SPQI-MoE-
Text+Behavior), the combination of all the features allows to obtain maximum
coverage.

Similarly, here too, the SPQ-MoE and SPQ-Concat ablations perform better than
most of their counterparts. In only one case the MLP obtain significantly
better performance.

In terms of architecture, SPQI-MoE-Text obtains an increase of $F1=+7\%$ over
RoBERTa. This improvement can be attributed to the use of GATs. Additionally,
we do not see any significant difference between the models when trained only
on the users questions, as opposed to both the question and the voice
assistant answers. One reason may be that answers are usually handled by Q&A
systems that do not provide personalized answers to the users.

Overall, only MLP-MoE-Text+Behavior achieves results that are significantly
better than those of SPQI-MoEText+Behavior. The performance decrease comes
from lower recall.

In summary, the ablations show that our contributions are twofold. First, the
proposed features that capture user’s shopping need are highly suitable for
voice search. Second, the proposed MoE for integrating the different features
into a joint question representation, and the use of GATs for the intent
classification task, with consistent improvement over the competing approaches
(e.g., MLP-Concat/MoE). Note that the MLP baseline represents a strong
baseline, since it is trained on the proposed features and uses the MoE.

###  6.2 Online Recommendation Evaluation

The SPQ definition in Section 3 is only a proxy of shopping intent. PQs can
theoretically be unrelated to the associated purchase. Our online evaluation,
with real voice assistant users, verifies that the identified SPQs by our
SPQI-MoE have a shopping intent. For this experiment, we split the users into
two groups: 1) $T1$ where the users, according to our model, are recommended a
shopping action to add a product into their shopping list, and 2) $C$ where
all the users issuing a PQ are prompted to add the product into their shopping
list.

$T1$ obtains 81.5% higher F1 score than $C$ in identifying SPQs, and
additionally users on $T1$ were 79.5% more likely to add the product into
their shopping list when compared to users in $C$. This experiment shows that
our approach is able to identify user’s shopping need, validated by users
adding the queried product into their shopping list. Note that, not adding a
product to the shopping list does not indicate the contrary, namely, the lack
of shopping intent.

##  7 Conclusion

We presented an approach to make proactive shopping recommendations in a
leading industry voice assistant. By identifying shopping need, we allow the
voice assistant to accurately recommend to its users relevant products or
proactively suggest shopping actions that enhance their shopping experience.

We proposed a set of features that capture shopping intent from various
perspectives, and relied on graph attention networks and on a novel mechanism
to combine features for classifying a PQ as an SPQ. Experimental evaluations
show that our proposed features and the way we encode user question through
GATs, yield significant improvements over text classification approaches,
achieving an increase in $F1$ of $+21\%$. The experiments confirm the highly
latent nature of shopping intent. This work, unlike others, explores users’
shopping intent in voice, not only in the context of product search, but also
in the context of product question answering. Furthermore, focusing on
informational intent and exploring whether the underlying need behind is
related to general knowledge or to shopping.

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# Aggregating Credences into Beliefs:
Agenda Conditions for Impossibility Results
(extended abstract)

Minkyung Wang CONCEPT
Cologne, GermanyDepartment of Philosophy
University of Cologne
Cologne, Germany<EMAIL_ADDRESS>Independent Researcher
Cologne, Germany Chisu Kim Independent Researcher
Cologne, Germany<EMAIL_ADDRESS>
###### Abstract

Binarizing belief aggregation addresses how to rationally aggregate individual
probabilistic beliefs into collective binary beliefs. Similar to the
development of judgment aggregation theory, formulating axiomatic
requirements, proving impossibility theorems, and identifying exact agenda
conditions of impossibility theorems are natural and important research topics
in binarizing belief aggregation. Building on our previous research on
impossibility theorems, we use an agenda-theoretic approach to generalize the
results and to determine the necessary and sufficient level of logical
interconnection between the issues in an agenda for the impossibility theorems
to arise. We demonstrate that (1) path-connectedness and even-negatability
constitute the exact agenda condition for the oligarchy result stating that
binarizing belief aggregation satisfying proposition-wise independence and
deductive closure of collective beliefs yields the oligarchies under minor
conditions; (2) negation-connectedness is the condition for the triviality
result obtained by adding anonymity to the oligarchy result; and (3)
blockedness is the condition for the impossibility result, which follows by
adding completeness and consistency of collective beliefs. Moreover, we
compare these novel findings with existing agenda-theoretic characterization
theorems in judgment aggregation and belief binarization.

##  1 Introduction

The question of how to rationally aggregate individual beliefs into collective
beliefs is important and ubiquitous in our society. In this regard, there has
been abundant literature on collective decision theory, judgment aggregation,
and probabilistic opinion pooling studies. One of the essential features of
belief is that there are different types of beliefs. For example, some beliefs
may be represented by traditional ”logical” languages—she believes that it is
raining outside—while other types of beliefs might be modeled by ”probability
functions”—she believes with 90 percent certainty that it is raining outside.
Logical languages are similar to our natural languages and are therefore
efficient for communicating with human agents, despite the fact that they
sometimes suffer from significant information reduction, as in the case of the
Lottery paradox. In contrast, probabilistic beliefs hold a fair amount of
information to deal with uncertain environments, although people usually do
not reach that level of precision. Considering these pros and cons of
different types of beliefs, it is not surprising that different types of
beliefs may be required at different stages of belief aggregation procedures
depending on situations. If objective chances of issues in question can be
given, it is epistemically preferable to report individual opinions in terms
of degrees of belief. If the conclusion of an epistemic collective decision
guides action (e.g., a jury verdict), it is practically better to report the
collective opinion by means of plain logic. Therefore, rational belief
aggregation should be able to deal with different types of beliefs. One
important topic in aggregating one type of belief into a different type of
belief is aggregating probabilistic beliefs into collective binary beliefs
(e.g., [12] [17]). We call this subject matter ”binarizing belief aggregation”
[17]. We can observe these belief aggregation problems in expert panels, the
scientific community, and political parties, whenever individuals’ opinions
can be encoded probabilistically, and the group’s beliefs should be more
decisive.

Similar to the development of judgment aggregation theory (e.g., [7] [16]),
formulating axiomatic requirements, proving impossibility theorems, and
identifying exact agenda conditions of impossibility theorems are natural and
important research topics in binarizing belief aggregation. Building on our
previous research on impossibility theorems, this paper uses an agenda-
theoretic approach to determine which level of logical interconnection between
the issues in an agenda is necessary and sufficient for the impossibility
theorems to arise. Indeed, our previous paper assumed the agenda to be an
algebra, which is the most typical when dealing with probabilistic beliefs.
However, in practice, the agenda being an algebra might be quite demanding
because we might not be interested in, for example, the conjunction of two
propositions when making a collective decision on the two propositions.
Besides the literature on judgment aggregation, agenda-theoretic approaches
can be found in other fields as well. In probabilistic opinion pooling,
general agendas were investigated to characterize linear pooling (e.g., [3]
[4]). In the belief binarization problem, general agendas were studied to
characterize impossibility theorems (e.g., [5] [6]).

In this study, we demonstrate that (1) path-connectedness and even-
negatability constitute the exact agenda condition for the oligarchy result,
which states that binarizing belief aggregation satisfying proposition-wise
independence and deductive closure of collective beliefs yields the
oligarchies under certain conditions; (2) negation-connectedness is the
condition for the triviality result obtained by adding anonymity to the
oligarchy result; and (3) blockedness is the condition for the impossibility
result, which follows by adding completeness and consistency of collective
beliefs. Moreover, we compare these novel findings with existing agenda-
theoretic characterization theorems in judgment aggregation and belief
binarization. All proofs of lemmas and theorems are provided in the full
paper.

##  2 Binarizing Belief Aggregation and the impossibility results

We begin by introducing notations and definitions we will use throughout this
paper. Let $W$ be a finite non-empty set of possible worlds. An agenda
$\mathcal{A}$ is a non-empty set of subsets of $W$ that is closed under
complement. Let $N:=\\{1,...,n\\}$($n\geq 2$) be the set of individuals. For
each $i\in N$, an individual $i$’s probabilistic belief $P_{i}$ is a function
extendable to a probability function on the smallest algebra that includes
$\mathcal{A}$. We denote by $\vec{P}:=(P_{1},...,P_{n})=(P_{i})_{i\in N}$ a
profile of $n$ individuals’ probabilistic beliefs. Binarizing belief
aggregation deals with individuals’ probabilistic beliefs and the group’s
binary beliefs. Binary beliefs are represented by a function
$Bel:\mathcal{A}\rightarrow\\{0,1\\}$. Sometimes, we abuse the notation and
denote by $Bel$ the belief set $\\{A\in\mathcal{A}|\mbox{ }Bel(A)=1\\}$, and
$BelA$ is a shorthand for $A\in Bel$ or $Bel(A)=1$. A binarizing aggregator
(BA) $F$ is a function that takes a profile $\vec{P}$ of $n$ probabilistic
beliefs in a given domain and returns a binary belief $F(\vec{P})$.

Now, let us define the axiomatic requirements on BA that are needed to
formulate our impossibility results. First, we need the following rationality
requirements on the domain and codomain of a BA.

$\bullet$ Universal Domain (UD): the domain of $F$ is the set of all profiles
$\vec{P}$ of $n$ probabilistic beliefs

$\bullet$ Collective Deductive Closure (CDC)/Consistency (CCS)/Completeness
(CCP): for all $\vec{P}$ in the domain, the resulting collective beliefs
$F(\vec{P})$ is deductively closed/consistent/complete, respectively

Note that a binary belief $Bel$ is deductively closed iff it holds that, if
$Bel\vDash A(i.e.,\bigcap Bel\subseteq A)$, then $BelA$ for all
$A\in\mathcal{A}$. Moreover, $Bel$ is consistent if $Bel\nvDash\emptyset$, and
$Bel$ is complete if $BelA$ or $Bel\overline{A}$ for all $A\in\mathcal{A}$
where $\overline{A}$ is the complement of $A$. Second, we enlist different
rationality requirements on BAs themselves.

$\bullet$ Certainty Preservation (CP)/Zero Preservation (ZP): for all
$A\in\mathcal{A}$, if
$\vec{P}(A)(:=(P_{1}(A),\cdots,P_{n}(A)))=(1,...,1)$/$\vec{P}(A)=(0,...,0)$,
then $F(\vec{P})(A)=1$/$F(\vec{P})(A)=0$, respectively, for all $\vec{P}$ in
the domain of $F$.

$\bullet$ Anonymity (AN): $F((P_{\pi(i)})_{i\in N})$ $=F((P_{i})_{i\in N})$
for all $\vec{P}$ in the domain of $F$ and all permutation $\pi$ on $N$.

$\bullet$ Independence (IND): for all $A\in\mathcal{A}$, there exists a
function $G_{A}$ such that $F(\vec{P})(A)=G_{A}(\vec{P}(A))$ for all $\vec{P}$
in the domain of $F$.

$\bullet$ Systematicity (SYS): there exists a function $G$ such that
$F(\vec{P})(A)=G(\vec{P}(A))$ for all $A\in\mathcal{A}$ and for all $\vec{P}$
in the domain of $F$.

Our previous paper [17] proved the following theorems under the assumption
that $\mathcal{A}$ is an algebra with at least three elements besides the
empty set and W, which we call a non-trivial algebra. We aim to relax this in
this study.

1\. (The Oligarchy Result) The only BAs satisfying UD, CP, ZP, IND, and CDC
are the following oligarchies: there is a non-empty subset $M$ of $N$ such
that

$F(\vec{P})(A)=\left\\{\begin{array}[]{rl}1&\mbox{if }P_{i}(A)=1\mbox{ for all
}i\in M\\\ 0&\mbox{otherwise}\par\end{array}\right.$

for all $A\in\mathcal{A}$.

2\. (The Triviality Result) The only BAs satisfying UD, CP, ZP, IND, CDC and
AN are the oligarchy with $M=N$, which we call the trivial rule.

3\. (The Impossibility Result) There is no BA satisfying UD, CP, IND, CCP, and
CCS.

##  3 The Agenda Condition for the Oligarchy Result

This section presents and proves our first main result: the agenda condition
for the oligarchy result. The following two agenda conditions have been
extensively studied, as they characterize the most famous impossibility
agendas in judgment aggregation.

######  Definition 1 (Path-connected and Even-negatable Agenda).

(1) For any $A,B\in\mathcal{A}$, we say that $A$ conditionally entails $B$
($A\vDash^{*}B$) if there is a subset $\mathcal{Y}\subseteq\mathcal{A}$ that
is consistent with $A$ and $\overline{B}$111That is,
$\mathcal{Y}\cup\\{A\\}\nvDash\emptyset$ and
$\mathcal{Y}\cup\\{\overline{B}\\}\nvDash\emptyset$ such that
$\\{A\\}\cup\mathcal{Y}\vDash B$ (i.e.,
$\bigcap(\\{A\\}\cup\mathcal{Y})\subseteq B$ and we write this as
$A\vDash_{\mathcal{Y}}^{*}B$). An agenda $\mathcal{A}$ is path-connected (PC)
if $A\vDash^{**}B$ for all contingent issues $A,B\in\mathcal{A}$, where
$\vDash^{**}$ is the transitive closure of $\vDash^{*}$.
(2) An agenda $\mathcal{A}$ is even-negatable (EN) iff there is a minimally
inconsistent set $\mathcal{Y}\subseteq\mathcal{A}$ such that
$\mathcal{Y}_{\neg\mathcal{Z}}:=(\mathcal{Y}\setminus\mathcal{Z})\cup\\{\overline{A}|\mbox{
}A\in\mathcal{Z}\\}$ is consistent for some subset
$\mathcal{Z}\subseteq\mathcal{Y}$ of even size.

Path-connectedness means that every two issues are connected by a path, i.e.,
a chain of conditional entailment relations. Regarding conditional entailment
relation, let us mention a useful fact. If $A\vDash^{*}_{\mathcal{Y}}B$, it
also holds that $\overline{B}\vDash^{*}_{\mathcal{Y}}\overline{A}$, and thus
if $A\vDash^{**}B$, then $\overline{B}\vDash^{**}\overline{A}$. And even-
negatability says that a minimally inconsistent subset of the agenda can be
made consistent by negating some even number of its element. It is well-known
that an agenda is even-negatable unless the propositions in the agenda are
composed only with negation and biconditional from some logically independent
propositions. Note that these two conditions are weaker than the agenda being
a non-trivial algebra, which is the assumption on the agenda in [17].

######  Lemma 1.

Every non-trivial algebra is path-connected and even-negatable.

From now on, we add one more assumption on $\mathcal{A}$ that
$\emptyset\notin\mathcal{A}$(and thereby $W\notin\mathcal{A}$).222In the
following, especially in Theorem 2 and Theorem 3, we will use some results of
Nehring & Puppe (2010), where the agenda consists of contingent issues. To
describe our proof more simply, we adopt that assumption.  Thus, our agenda
$\mathcal{A}$ is a complement-closed finite non-empty set of some contingent
subsets of the underlying set $W$. The following lemma shows that path-
connectedness is sufficient to obtain what is called the contagion lemma.

######  Lemma 2 (Agenda Condition for the Contagion Lemma).

Let $\mathcal{A}$ be path-connected. If a BA $F$ with UD satisfies CDC, CP,
and IND, then it satisfies SYS.

This lemma parallels the one in generalized opinion pooling of Dietrich & List
(2017a): path-connectedness characterizes that if generalized OP satisfies CP
and IND, then it satisfies SYS. In our lemma as well, its converse—if
$\mathcal{A}$ is not path-connected, then there is a BA F on $\mathcal{A}$
satisfying CDC, CP, and IND but not SYS—also holds. The counterexample will be
indicated in the proof of Theorem 1.

The following definition and lemma will be needed to prove our succeeding main
theorem.

######  Definition 2 (Non-simple Agenda and Pair-negatable Agenda).

(1) An agenda $\mathcal{A}$ is non-simple(NS) iff there is a minimally
inconsistent subset $\mathcal{Y}\subseteq\mathcal{A}$ with $|\mathcal{Y}|\geq
3$.

(2) An agenda $\mathcal{A}$ is pair-negatable iff there is a minimally
inconsistent set $\mathcal{Y}\subseteq\mathcal{A}$ such that
$\mathcal{Y}_{\neg Z}$ is consistent for some subset
$\mathcal{Z}\subseteq\mathcal{Y}$ with $|\mathcal{Z}|=2$.

Non-simple agendas can be used as a criterion for determining whether a given
agenda has minimal complexity. Pair-negatable agendas are a special case of
even-negatable agendas. The following lemma shows that a pair-negatable agenda
is sufficient to be an even-negatable agenda, and a path-connected agenda
already has a fairly complex structure.

######  Lemma 3.

(1) An agenda $\mathcal{A}$ is even-negatable iff $\mathcal{A}$ is pair-
negatable.

(2) If an agenda $\mathcal{A}$ is path-connected, then it is non-simple.

Now we prove that the agenda being path-connected and even-negatable is the
sufficient and necessary condition for the oligarchy result.

######  Theorem 1 (Agenda Condition for the Oligarchy Result).

Let $|N|\geq 3$. An agenda $\mathcal{A}$ is path-connected and even-negatable
iff the only BAs on $\mathcal{A}$ satisfying UD, ZP, CP, IND, and CDC are the
oligarchies.

The only-if direction of the theorem generalizes the oligarchy result and
shows that even if an agenda satisfies a weaker condition—path-connectedness
and even-negatability—than a non-trivial algebra, the oligarchy result holds.
If we examine the proof of the oligarchy result in [17] in detail, we can
observe that the agenda condition was used solely to establish the following
two facts:
(Fact 1) if $\vec{a}\leq\vec{b}$ and if $G(\vec{a})=1$, then $G(\vec{b})=1$
where $G$ is a function satisfying $F(\vec{P})(A)=G(\vec{P}(A))$.
(Fact 2) if $\vec{a}+\vec{b}-\vec{1}\geq\vec{0}$ and if $G(\vec{a})=1$ and
$G(\vec{b})=1$, then $G(\vec{a}+\vec{b}-\vec{1})=1$.
Therefore, to prove the only-if direction, it is enough to derive (Fact 1)
from even-negatability and (Fact 2) from path-connectedness. The agenda
conditions are only relevant to (Fact 1) and (Fact 2), and once we see that
they hold then we can apply the proof of the oligarchy result in [17].

Our proof also reveals that if we assume the stronger property of SYS instead
of IND, then Lemma 1 is not needed, and non-simplicity (NS) is sufficient to
obtain the oligarchy result. This observation indicates that stronger
properties of a BA lead to weaker agenda conditions for achieving the
oligarchy result. To provide additional agenda conditions for the oligarchy
result, let us introduce the concept of monotonicity (MON) for a BA as
follows:

(MON) If $\vec{P}(A)\leq\vec{P}^{\prime}(A)$ and $F(\vec{P})(A)=1$, then
$F(\vec{P}^{\prime})(A)=1$

where $\leq$ is applied to each component of two vectors. If we assume MON, we
can bypass the need to prove (Fact 1), thereby eliminating the requirement for
the agenda to be even-negatable (EN). This is because (Fact 1) is already
implied by SYS and MON. The following table illustrates the agenda conditions
that are sufficient to achieve the oligarchy result based on different
properties of BA:

| IND | SYS
---|---|---
without MON | PC, EN | NS, EN
with MON | PC | NS

It is noteworthy that the agenda condition required for our oligarchy result
is the same as the one for the dictatorship and oligarchy results in judgment
aggregation (e.g., [7] [2]). In our proof of the if-direction, we extend their
counterexamples to our domain in a manner that satisfies UD, ZP, CP, IND, CDC,
and CCS: the counterexample for a non-path-connected agenda is a minimal
extension satisfying MON, as we do not exclude even-negatablility; the one for
a non-even-negatable agenda is an extension satisfying not MON but SYS, as we
do not exclude path-connectedness. So the proof follows a similar structure to
those in judgment aggregation, but the ways of extension to construct
counterexamples are not trivial—particularly the counterexample for not even-
negatable agendas—, and so our proof includes novel ideas that are needed due
to the difference between binary and probabilistic beliefs.

##  4 The Agenda Condition for the Triviality Result

This section presents and proves our second main result: the agenda condition
for the triviality result. Stronger properties of a BA yield weak agenda
conditions. Thus, one might ask whether the agenda condition for the oligarchy
result can be weakened, if we add AN. We will demonstrate that the agendas
that yield the triviality result can be characterized by negation-
connectedness, which is also the agenda condition for an impossibility result
of belief binarization methods as shown in [6].

######  Definition 3 (Negation-connected Agenda).

An agenda $\mathcal{A}$ is negation-connected (NC) iff for every contingent
issue $A\in\mathcal{A}$ it holds that $A\vDash^{**}\overline{A}$.

So negation-connectedness means that every issue has a path to its complement.
According to Proposition 1 in Dietrich & List (2021), the agenda being
negation-connected is equivalent to the agenda being partitioned into
subagendas each of which is path-connected, where a subagenda is a non-empty
subset of the agenda that is closed under complementation.

The following lemma will be needed for the proof of the first part of the
succeeding theorem. Part (1) allows us to consider the stronger condition,
namely path-connectedness, than negation-connectedness to prove the triviality
result. Part (2) will be used when the agenda is path-connected and not even-
negatable.

######  Lemma 4.

(1) If the triviality result holds—i.e., the only BA on $\mathcal{A}$
satisfying UD, CDC, ZP, CP, IND, and AN is the trivial one—for any path-
connected agenda $\mathcal{A}$, then the same holds for any negation-connected
agenda.

(2) If an agenda $\mathcal{A}$ is not even-negatable, then for any minimally
inconsistent subset $\mathcal{Y}\subseteq\mathcal{A}$ and any even-sized
subset $\mathcal{Z}\subseteq\mathcal{Y}$ it holds that
$\mathcal{Y}_{\neg\mathcal{Z}}$ is also minimally inconsistent.

The following lemma will be needed for the proof of the second part of the
succeeding theorem. This lemma looks technical but it is closely related to
the notion of median point in the next section. Indeed, if $\mathcal{H}_{0}$
is the empty set, then $\bigcap\mathcal{M}$ is the set of all median points
where $\mathcal{H}_{0}$ and $\mathcal{M}$ are defined in the following lemma.

######  Lemma 5.

Let $\mathcal{H}_{0}$ be the set $\\{A\in\mathcal{A}|\mbox{
}A\vDash^{**}\overline{A}\mbox{ and }\overline{A}\vDash^{**}A\\}$. If
$\mathcal{A}$ is not negation-connected, then there is a non empty subset
$\mathcal{M}\subseteq\mathcal{A}\setminus\mathcal{H}_{0}$ such that for any
minimally inconsistent set $\mathcal{Y}\subseteq\mathcal{A}$ it holds that
$|\mathcal{Y}\cap\mathcal{M}|\leq 1$. Furthermore, for any minimally
inconsistent set $\mathcal{Y}\subseteq\mathcal{A}$ intersecting
$\mathcal{H}_{0}$ it holds that $|\mathcal{Y}\cap\mathcal{M}|=0$. In addition,
for $B\in\mathcal{A}\setminus\mathcal{H}_{0}$, it holds that $B\in\mathcal{M}$
iff $\overline{B}\notin\mathcal{M}$.

Now let us prove the theorem that negation-connectedness is the sufficient and
necessary condition for the triviality result.

######  Theorem 2 (Agenda Condition for the Triviality Result).

An agenda $\mathcal{A}$ is negation-connected iff the only BA on $\mathcal{A}$
satisfying UD, ZP, CP, CDC, IND, and AN is the trivial one.

The only-if direction of the theorem shows that the triviality result holds if
the agenda is negation-connected, which is a generalization of the triviality
result. The proof suggests further that, if we assume SYS, then non-simplicity
(NS) becomes the sufficient condition to obtain the triviality result. In this
case, neither EN nor MON is needed, unlike in the case of Theorem 1, as
illustrated in the following table:

| IND | SYS
---|---|---
with or without MON | NC | NS

Compared to the case of the oligarchy result, when we add AN, we obtain the
triviality result even under a weaker agenda condition: (i) instead of
requiring path-connectedness (PC), negation-connectedness (NC) is sufficient,
and (ii) the triviality result holds even when the agenda is not even-
negatable (EN). The difference mentioned in (i) does not play a role in
finding the sufficient condition according to Lemma 4. However, the necessary
condition is not path-connectedness but negation-connectedness. In cases where
the agenda is PC and EN, we can apply Theorem 1 since the oligarchy satisfying
AN is the trivial one (i.e., the oligarchy with $M=N$). Thus, we only need to
focus on the cases where the agenda is PC and not EN. When the agenda is
assumed to be not EN, we encounter the following difficulty: to show the
triviality result, we used (Fact 1), which could be proved if the agenda was
assumed to be EN. Our strategy here is to prove a weaker claim than (Fact 1):
(Fact 1′′) If $G(\vec{a})=1$, then $G(\vec{c})=1$ for all
$\vec{c}\geq|2\vec{a}-\vec{1}|$.
The new claim (Fact 1′′) is weaker than (Fact 1), as it only guarantees that
vectors greater than $|2\vec{a}-\vec{1}|$ are mapped to 1, rather than all
vectors greater than $\vec{a}$.

One might ask whether we can apply the proof presented in Dietrich & List
(2021) to our theorem, or vice versa. However, there are differences between
the two proofs. On the one hand, we cannot use their proof because, while they
deal with probabilistic beliefs, we are dealing with profiles of probabilistic
beliefs: in particular, for negation-connected agendas in our framework, we
can only show (Fact 1′′) instead of (Fact 1). On the other hand, since we have
not relied on the assumption that $|N|\geq 2$, our proof can be applied to the
context of belief binarization, where $|N|=1$, and so we can recover their
results.

The if-direction gives a counterexample of the triviality result when an
agenda is not negation-connected, which implies the agenda being not path-
connected. The counterexample presented in Theorem 1 is not applicable in this
case because it does not satisfy AN. Moreover, there would be no
counterexample if we only assumed an agenda to be not path-connected. This is
the reason why we need to weaken path-connectedness to negation-connectedness,
even though they fulfill the same role concerning the sufficient agenda
condition for the triviality result.

Our counterexample for a non-negation-connected agenda is an extension of the
belief binarization rule proposed in Dietrich & List (2021). We extend the
rule while maintaining MON, but not minimally, which differs from the way of
the extension in Theorem 1.

##  5 The Agenda Condition for the Impossibility Result

Now we will show that the agendas for the impossibility result can be
characterized by blocked agendas.

######  Definition 4 (Blocked Agenda).

An agenda $\mathcal{A}$ is blocked iff there is an issue $A\in\mathcal{A}$
such that $A\vDash^{**}\overline{A}$ and $\overline{A}\vDash^{**}A$.

So a blocked agenda contains an issue that has a path to its complement.
Recall that $\mathcal{H}_{0}$ is defined by the set $\\{A\in\mathcal{A}|\mbox{
}A\vDash^{**}\overline{A}\mbox{ and }\overline{A}\vDash^{**}A\\}$. Then
$\mathcal{A}$ is negation-connected iff $\mathcal{H}_{0}=\mathcal{A}$, and
$\mathcal{A}$ is blocked iff $\mathcal{H}_{0}\neq\emptyset$. If $\mathcal{A}$
is negation-connected, then it is blocked. The following definition and lemma
will be needed for the succeeding theorem.

######  Definition 5 (Median Point).

Let $\mathcal{A}$ be an agenda on the set $W$ of possible worlds. A possible
world $m\in W$ is a median point iff for any minimally inconsistent subset
$\mathcal{Y}\subseteq\mathcal{A}$, it holds that $|\\{A\in\mathcal{Y}|\mbox{
}m\in A\\}|\leq 1$.

So a median point is a possible world that is contained in at most one issue
in every minimally inconsistent set. It is well-known in judgment aggregation
that if a median point is guaranteed to exist, then we can easily construct an
anonymous, complete, and consistent judgment aggregator where a median point
is thought of as a default collective judgment unless everybody believes the
issue being true/false at the median point to be false/true [15]. The
following lemma states that the agenda not being blocked is the necessary and
sufficient condition for the existence of a median point.

######  Lemma 6.

An agenda $\mathcal{A}$ is not blocked iff there is a median point.

Now let us formulate and prove our last theorem.

######  Theorem 3 (Agenda Condition for the Impossibility Result).

An agenda $\mathcal{A}$ is blocked iff there is no BA on $\mathcal{A}$
satisfying UD, CP, IND, CCP, and CCS.

Indeed, CCS and CCP together are stronger assumptions than CDC. As a result,
we obtain the impossibility result more easily, without assuming AN and non-
dictatorship, and with a more relaxed agenda condition. The proof demonstrates
that by adding SYS, the impossibility result still holds even without CP and
even when no agenda condition is assumed—e.g., even when
$\mathcal{A}=\\{A,\overline{A}\\}$.

The blocked agenda is also the agenda condition for the impossibility results
on judgment aggregation with AN in [16] and belief binarization in [2]. Our
counterexample for non-blocked agenda is an extension of the counterexample in
Dietrich & List (2018). It is an extension that satisfies MON, but not
minimally so. This is the same as the extension in Theorem 2, but different
from the one in Theorem 1. Note that the median point $m$ in the proof of this
theorem plays the same role as $\mathcal{M}$ in the proof of Theorem 2. The
only difference is that $m$ is a possible world and $\mathcal{M}$ is a set of
issues. This difference arises from assuming CDC versus assuming CCS and CCP.

##  6 Discussion

There is no BA satisfying … | Agenda Condition
---|---
(1) UD, ZP, CP and IND + CDC + Non-oligarchy | path-connected, even-negatable
(2) UD, ZP, CP and IND + CDC + AN + Non-triviality | negation-connected
(3) UD, CP and IND + CCS and CCP | blocked
There is no judgment aggregator satisfying … | Agenda Condition
(1′) UD, ZP, CP and IND + CDC + Non-oligarchy | path-connected, even-negatable
(2′) UD, ZP, CP and IND + CDC + AN + Non-triviality | negation-connected
(3′) UD, CP and IND + CCS and CCP + non-dictatorship | path-connected, even-negatable
(4′) UD, CP and IND + CCS and CCP + AN | blocked
There is no belief binarization rule satisfying … | Agenda Condition
(2′′) UD, CCS , CP and IND + CDC + Non-triviality | negation-connected
(4′′) UD, CCS, CP and IND + CCP | blocked
Table 1: Classification of Agenda Conditions for Impossibility Results

All the results in this paper are stated in Table 1: (1) path-connectedness
and even-negatability constitute the exact agenda condition for the oligarchy
result; (2) negation-connectedness is for the triviality result; and (3)
blockedness is for the impossibility result. These new findings can be
compared to the existing characterization theorems in judgment aggregation and
belief binarization. Regarding (1), it has the same agenda condition as (1′)
[2] and (3′) [7] in judgment aggregation. For (2), it is similar to (2′′) [6]
in belief binarization, with the difference being the use of ZP instead of CCS
for for (2′′). Since applying our proofs can weaken CCS to ZP, the agenda
condition for (2′), which has not been discussed in the literature, is also
negation-connected because an anonymous and independent judgment aggregator
can be viewed as a belief binarization function. As for (3), it is similar to
(4′) [16] in judgment aggregation and (4′′) [5] in belief binarization.

Let us mention some further research topics. One might think that the
rationality norms for collective binary beliefs could be weakened since
adhering to deductive closure might be too demanding for group agents.
Instead, we could focus on requiring group beliefs to respect consistency or
pairwise consistency. By exploring these weaker norms, we can investigate
stronger impossibility results. Furthermore, let us discuss how to obtain
possibility results. For this purpose, it is advantageous that binarizing
belief aggregation provides a framework that generalizes the problem of
judgment aggregation or belief binarization. As in judgment aggregation, we
can employ and study premise-based binarizing belief aggregation methods.
Alternatively, we can combine an individual belief binarization procedure with
judgment aggregation. If we assume that linear or geometric pooling methods
are very natural given individual credences, we can apply belief binarization
methods to the pooled group credence. Of course, we can also come up with new
procedures that cannot be reduced to existing methods. Ultimately, we should
keep in mind that binarizing belief aggregation is an epistemic collective
decision problem. Therefore, we should be concerned about which methods
accurately track the truth. One natural approach would be to investigate
belief binarization methods that minimize the expected distance from the truth
in light of the group’s pooled credence. In conclusion, binarizing belief
aggregation opens a new research area in which various procedures of belief
aggregation, different studies on the relation between credences and beliefs,
and epistemic decision theory can be combined and explored.

### Acknowledgements

We would like to express our sincere gratitude to Hannes Leitgeb and Christian
List for their invaluable feedback and profound insights. The research
conducted by the first author benefited from the generous support of the
German Academic Scholarship Foundation and the Alexander von Humboldt
Foundation.

## References

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  * [3] Franz Dietrich & Christian List (2017): _Probabilistic opinion pooling generalized. Part one: general agendas_.  Social Choice and Welfare 48, pp. 747–786, 10.1007/s00355-017-1034-z.
  * [4] Franz Dietrich & Christian List (2017): _Probabilistic opinion pooling generalized. Part two: the premise-based approach_.  Social Choice and Welfare 48, pp. 787–814, 10.1007/s00355-017-1035-y.
  * [5] Franz Dietrich & Christian List (2018): _From degrees of belief to binary beliefs: Lessons from judgement-aggregation theory_.  Journal of Philosophy 115(5), pp. 787–814, 10.5840/jphil2018115516.
  * [6] Franz Dietrich & Christian List (2021): _The relation between degrees of belief and binary beliefs: a general impossibility theorem_.  In Igor Douven, editor: Lotteries, Knowledge, and Rational Belief. Essays on the Lottery Paradox, Cambridge University Press, pp. 224–254, 10.1017/9781108379755.012.
  * [7] Elad Dokow & Ron Holzman (2018): _Aggregation of binary evaluations_.  Journal of Economic Theory 145, p. 495–511, 10.1016/j.jet.2007.10.004.
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  * [9] Christian Genest & James V. Zidek (1986): _Combining probability distributions: a critique and annotated bibliography_.  Statistical Science 1(1), pp. 114–135, 10.1214/ss/1177013831.
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  * [12] Magdalena Ivanovska & Marija Slavkovik (2019): _Aggregating Probabilistic Judgments_.  In: Proceedings Seventeenth Conference on Theoretical Aspects of Rationality and Knowledge, EPCTS 297, pp. 273–292, 10.4204/EPTCS.297.18.
  * [13] Christian List & Philip Pettit (2002): _Aggregating Sets of Judgments: An Impossibility Result_.  Economics and Philosophy 18(1), pp. 89–110, 10.1017/S0266267102001098.
  * [14] Kevin J. McConway (1981): _Marginalization and linear opinion pools_.  Journal of the American Statistical Association 76, pp. 410–414, 10.1080/01621459.1981.10477661.
  * [15] Klaus Nehring & Clemens Puppe (2007): _The structure of strategy-proof social choice - Part I: General characterization and possibility results on median spaces_.  Journal of Economic Theory 135, pp. 269–305, 10.1016/j.jet.2006.04.008.
  * [16] Klaus Nehring & Clemens Puppe (2010): _Abstract Arrovian aggregation_.  Journal of Economic Theory 145, p. 467–494, 10.1016/j.jet.2010.01.010.
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# Zeta function regularization technique in the electrostatics context for
discrete charge distributions

F. Escalante  Departamento de Física, Universidad Católica del Norte, Avenida
Angamos 0610, Casilla 1280, Antofagasta, Chile<EMAIL_ADDRESS>
###### Abstract

Spectral functions, such as the zeta functions, are widely used in Quantum
Field Theory to calculate physical quantities. In this work, we compute the
electrostatic potential and field due to an infinite discrete distribution of
point charges, using the zeta function regularization technique. This method
allows us to remove the infinities that appear in the resulting expression. We
found that the asymptotic behavior dependence of the potential and field is
similar to the cases of continuous charge distribution. Finally, this exercise
can be useful for graduate students to explore spectral and special functions.

##  1 Introduction

The mathematician Leonard Euler introduced a version of the zeta function in
1737. However, it was extensively studied by the German mathematician Bernard
Riemann, who, in his article “On the Number of Primes Less Than a Given
Magnitude”, published in 1859, took as a starting point the Euler’s theorem
(1737), where he proved that the sum of the reciprocals of the prime numbers
is a divergent series [1]. Riemann defined a function in the complex plane
$\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$, where $s\in\mathbb{C}$, and he proved
that this sum converges when $\operatorname{Re}(s)>1$ and diverges when
$\operatorname{Re}(s)=1$, which means that the $\zeta$-function has a simple
pole.

The zeta functions properties, like Hurwitz, Barnes, and Epstein zeta
functions, have been well-studied [2], and it is of our particular interest
the Epstein zeta function [3, 4], associated with sums of squares of integers.
Besides, the zeta function has proved to be a very powerful tool in Quantum
Field Theory (QFT), e.g. the Casimir Effect or theory of strong interactions
[5, 6, 7].

One of the fundamental aspects of QFT is the task to extract results
physically significant from quantities that, in principle, are ill-defined.
Usually, these quantities are related to sequences of numbers $\lambda_{k}$
where $k\in\mathbb{N}$, which, for most applications, are eigenvalues of
certain operators Laplace like that grow without bound when $k\to\infty$. In
principle, the divergences that appear may be related to measurable physical
quantities. Therefore, one must make sense of the calculations of those
quantities. This can be achieved by a method called “regularization”, where it
extracts the relevant information the calculations made. This procedure
generally leads to the appearance of a scale parameter, say $\Lambda$, which
we can roughly say, allows giving a finite value to originally divergent
expressions. This parameter is usually called “cutoff” or “regulator”.

The original procedure in QFT consists of taking the limit $\Lambda\to\infty$
when the problematic (divergent) terms are isolated and suppressed from the
desired expression. So the regulator has no relevant meaning and is just part
of a mathematical artifact to make sense of certain formal expressions. It is
important to note that there are various regularization methods, such as
Pauli-Villars, dimensional regularization, cut-off regularization, among
others [8].

However, these regularization schemes are not QFT exclusively. In the context
of electrostatics, we can find situations where we have to deal with diverging
quantities. Some authors have been studied the calculation of the
electrostatic potential and field in continuous charge distributions, such as
the infinite charged wire and the infinite charged plate, using cut-off and
dimensional regularizations and renormalization schemes[9, 10, 11].

This work aims to calculate the electrostatic potential and electric field of
infinite one and two-dimensional lattices as an example of the application of
the zeta function regularization technique, in the context of electrostatics.

##  2 Point charges distribution

The electrostatic potential due to a set of point charges is given by [12]

$V=\frac{1}{4\pi\epsilon_{0}}\sum_{i}\frac{q_{i}}{r_{i}}$ (1)

where $r_{i}$ is the distance form the $i$th charge, $q_{i}$, to the point at
which $V$ is evaluated.

We consider a lattice of infinite identical point charges, located on the $xy$
plane, separated by a distance $a$ from each other in the $x$ and $y$
coordinates, as shown in figure (1).

Figure 1: Graphic representation of the two dimensional point charge
distribution.

The origin of the coordinate system, $O$, is at the center of the lattice,
where a point charge is located. The electrostatic potential on the $+z$ axis
can be written as

$V(z)=\frac{q}{4\pi\epsilon_{0}}\sum_{n,m=-\infty}^{\infty}\frac{1}{\sqrt{x_{n}^{2}+y_{m}^{2}+z^{2}}},$
(2)

where $x_{n}=an$ and $y_{m}=am$, $n$ and $m$ being integers, represent,
respectively, the $x$ and $y$ coordinates of point charges from the lattice.
So we can rewrite the electrostatic potential as

$V(\xi)=\frac{1}{4\pi\epsilon_{0}}\frac{q}{a}\sum_{n,m=-\infty}^{\infty}\frac{1}{\sqrt{n^{2}+m^{2}+\xi^{2}}},$
(3)

with $\xi=z/a$ as a dimensionless parameter, related to the $z$ coordinate. To
obtain a dimensionless expression, we normalize equation (3) to the potential
of a point charge at a distance a from the origin along the $+z$ axis,
$V_{0}=q/4\pi\epsilon_{0}a$. Thus, we get the expression

$\tilde{V}(\xi)=\sum_{n,m=-\infty}^{\infty}\frac{1}{\sqrt{n^{2}+m^{2}+\xi^{2}}}.$
(4)

The sum in equation (4) diverges due to the infinite summation range, and thus
this divergence is analogous to the ultraviolet divergence in QFT. On the
other hand, if $\xi\to 0$, the sum also diverges at the origine (when
$n=m=0$), and such kind of divergence is analogous to the infrared divergence
in QFT [13]. However, in this work we are interested in the large $\xi$
regime. Therefore, in order to calculate the infinite sum, we must use an
appropriate regularization method. For this purpose, we will use the zeta
function method.

##  3 Zeta function regularization method

The multidimensional Epstein zeta function [14] is defined by

$\zeta_{E}(s,d;\xi^{2})=\sum_{n_{d}\in\mathbb{Z}}(n_{1}^{2}+n_{2}^{2}+\cdots+n_{d}^{2}+\xi^{2})^{-s}$
(5)

where $d$ is the dimension of the function and $s\in\mathbb{C}$. By means of
the Mellin transform, the multidimensional Epstein zeta function is given by

$\begin{split}\zeta_{E}(s,d;\xi^{2})&=\frac{1}{\Gamma(s)}\int_{0}^{\infty}dt\;t^{s-1}\sum_{n_{d}\in\mathbb{Z}}e^{-t(n_{1}^{2}+n_{2}^{2}+\cdots+n_{d}^{2}+\xi^{2})}\\\
&=\frac{1}{\Gamma(s)}\int_{0}^{\infty}dt\;t^{s-1}e^{-t\xi^{2}}\sum_{n_{d}\in\mathbb{Z}}e^{-t(n_{1}^{2}+n_{2}^{2}+\cdots+n_{d}^{2})}.\end{split}$
(6)

valid for $\operatorname{Re}(s)>d/2$. However, we are interested in the values
where $\operatorname{Re}(s)<d/2$, so we need to make an analytical
continuation in the complex plane of $s$ to include the value $s=1/2$. Using
the transformation formula of the Jacobi theta function of a lattice [15]

$\Theta(\omega)=\left(\frac{i}{\omega}\right)^{d/2}\Theta\left(-\frac{1}{\omega}\right),$
(7)

where $\Theta(\omega)=\sum_{n\in\mathbb{Z}^{d}}e^{-i\pi\omega(n_{j}\cdot
n_{j})}$, and with $\omega=it/\pi$, we have

$\sum_{n\in\mathbb{Z}^{d}}e^{-t(n_{j}\cdot
n_{j})}=\left(\frac{\pi}{t}\right)^{d/2}\sum_{n\in\mathbb{Z}^{d}}e^{-\frac{\pi^{2}}{t}(n_{j}\cdot
n_{j})}.$ (8)

With the equation (8), the equation (6) can be written as it follows

$\zeta_{E}(s,d;\xi^{2})=\frac{1}{\Gamma(s)}\int_{0}^{\infty}dt\;t^{s-1}e^{-t\xi^{2}}\left(\frac{\pi}{t}\right)^{d}\sum_{n\in\mathbb{Z}^{d}}e^{-\frac{\pi^{2}}{t}(n_{1}^{2}+n_{2}^{2}+\cdots+n_{d}^{2})}.$
(9)

The term $n=0$ can be treated separately from the sum, so we obtain

$\begin{split}\zeta_{E}(s,d;\xi^{2})=\frac{1}{\Gamma(s)}&\left\\{\int_{0}^{\infty}dt\;t^{s-1}e^{-t\xi^{2}}\left(\frac{\pi}{t}\right)^{d}+\sum_{n\in\mathbb{Z}^{d}-\\{0\\}}\int_{0}^{\infty}dt\;t^{s-1}e^{-t\xi^{2}}\left(\frac{\pi}{t}\right)^{d}\right.\\\
&\left.\times
e^{-\frac{\pi^{2}}{t}(n_{1}^{2}+n_{2}^{2}+\cdots+n_{d}^{2})}\right\\}.\end{split}$

From the first integral we can recognize the Gamma function

$\int_{0}^{\infty}d\tau\;\tau^{\alpha-1}e^{-A\tau}=\Gamma(\alpha)A^{-\alpha},$
(10)

in the second term, we use the integral representation for the modified Bessel
function of second kind

$\int_{0}^{\infty}d\tau\tau^{\alpha-1}e^{-\beta/\tau-\gamma\tau}=2\left(\frac{\beta}{\gamma}\right)^{\alpha/2}K_{\alpha}(2\sqrt{\beta\gamma}).$
(11)

Finally, we obtain an regularized expression for the multidimensional Epstein
zeta function valid in whole the complex plane [2]

$\begin{split}\zeta_{E}(s,d;\xi^{2})&=\frac{\pi^{d/2}}{\Gamma(s)}\frac{\Gamma(s-d/2)}{\xi^{2(s-d/2)}}+2\frac{\pi^{s}}{\Gamma(s)}\frac{1}{\xi^{s-d/2}}\sum_{n\in\mathbb{Z}^{d}-\\{0\\}}(n_{1}^{2}+n_{2}^{2}+\cdots+n_{d}^{2})^{(s-d/2)/2}\\\
&\times
K_{s-d/2}\left(2\pi\xi\sqrt{n_{1}^{2}+n_{2}^{2}+\cdots+n_{d}^{2}}\right).\end{split}$
(12)

The expression (12) is analytic in the whole complex plane except for certain
values for $d$. In general, when $d$ is even, the poles are located at
$s=\\{\frac{d}{2},\frac{d}{2}-1,...,1\\}$, whereas when $d$ is odd, the poles
are located at
$s=\\{\frac{d}{2},\frac{d}{2}-1,...,\frac{1}{2},-\frac{2l+1}{2}\\}$ where
$l\in\mathbb{N}_{0}$. In other words, the poles are determined by the poles of
the Gamma function of the first term of the expression (12), where the zeta
function is infinite and coincides with the original sum (3).

##  4 Electrostatic potential and the field of a one-dimensional lattice

Let’s consider an infinite system of positive point charges along the $x$
axis, separated a distance $a$ from each other. The point charges are evenly
distributed on both sides of the coordinate origin $O$, where a point charge
is located, as shown in Figure 2.

Figure 2: Graphic representation of the considered situation.

The electrostatic potential at a distance $z$ from the origin, on the $+z$
axis, due to a one dimensional lattice ($d=1$ and $y_{n}=0$) is given by the
equation (4)

$\tilde{V}_{1}(\xi)=\sum_{n=-\infty}^{\infty}(n^{2}+\xi^{2})^{-1/2},$ (13)

where the sub-index 1 refers to the electrostatic potential in the one
dimensional case.

In the above sum we can recognize the Epstein zeta function (5) (with $d=1$),
if we rewrite the sum in terms of a parameter $s$:

$\zeta_{E}(s,1;\xi^{2})=\sum_{n=-\infty}^{\infty}(n^{2}+\xi^{2})^{-s}.$ (14)

This infinite sum can be regularized through an analytic continuation of the
Epstein zeta function. According to the equation (12), we obtain:

$\zeta_{E}(s,1;\xi^{2})=\frac{\sqrt{\pi}}{\Gamma(s)}\frac{\Gamma(s-1/2)}{\xi^{2(s-1/2)}}+4\frac{\pi^{s}}{\xi^{s-1/2}}\sum_{n=1}^{\infty}n^{s-1/2}K_{s-1/2}(2\pi
n\xi).$ (15)

Since the term $n=0$ it is excluded from the above sum, we can use the fact
that $\sum_{n-\\{0\\}}f(n)=2\sum_{n=1}^{\infty}f(n)$. However, we must note
that if $s=1/2$, the Gamma function in the first term of the equation (15) has
a simple pole. Through change of variable $s=\epsilon+1/2$, we can compute a
series expansion as $\epsilon\to 0$, obtaining:

$\frac{\Gamma(\epsilon)}{\Gamma(1/2+\epsilon)}\frac{1}{\xi^{2\epsilon}}\sim\frac{1}{\sqrt{\pi}}\left(\frac{1}{\epsilon}+2\ln\left(\frac{2}{\xi}\right)\right)+O(\epsilon),$
(16)

neglecting the $O(\epsilon)$ terms, the Epstein zeta function (15) can be
expressed as

$\zeta_{E}(\epsilon,1;\xi^{2})=\frac{1}{\epsilon}+2\ln\left(\frac{2}{\xi}\right)+4\sum_{n=1}^{\infty}K_{0}(2\pi
n\xi).$ (17)

The expression (17) contains a diverging term, $1/\epsilon$, that can be
isolated by means of the principal part prescription [7], which means taking
the finite part, in order to obtain well defined and finite result for the
electrostatic potential in the one-dimensional case

$\tilde{V}_{1}(\xi)=2\ln\left(\frac{2}{\xi}\right)+4\sum_{n=1}^{\infty}K_{0}(2\pi
n\xi).$ (18)

Since the Bessel function $K_{0}(x)$ tends to zero in the limit of large $x$
(see the Appendix), the asymptotic behavior of the electrostatic potential is
given by the dominant first term of equation (18)

$\tilde{V}_{1,as}(\xi)\sim 2\ln\left(\frac{2}{\xi}\right).$ (19)

As is known, we can obtain the dimensionless electric field form the potential
gradient, so, the electrostatic field in the $+z$ axis, in terms of the $\xi$
coordinate, is given by

$\tilde{E}(\xi)=-\frac{\partial\tilde{V}(\xi)}{\partial\xi}.$ (20)

Using the general expression for the electrostatic potential (18) and noting
that

$\frac{d}{dx}K_{0}(nx)=-nK_{1}(nx),$ (21)

we can obtain an analytic expression for the electric field in the one
dimensional case:

$\tilde{E}_{1}(\xi)=\frac{2}{\xi}-8\pi\sum_{n=1}^{\infty}n\;K_{1}(2\pi n\xi).$
(22)

Therefore, the asymptotic behavior of the electric field in the large $\xi$
regime is given by

$\tilde{E}_{1,as}(\xi)=\frac{2}{\xi}.$ (23)

##  5 Electrostatic potential and the field of a two-dimensional lattice

Now, we consider a point charge lattice in the $xy$ plane, as it shows in
figure (1). The electrostatic potential at a distance $z$ of the plane $xy$,
on the $+z$ axis, is given by the equation (4). In terms of the Epstein zeta
function (12), with $d=2$, we have

$\zeta_{E}(s,2;\xi^{2})=\frac{\pi}{\Gamma(s)}\frac{\Gamma(s-1)}{\xi^{2s-2}}+4\frac{\pi^{s}}{\Gamma(s)}\frac{1}{\xi^{s-1}}\sum_{n,m=1}^{\infty}(n^{2}+m^{2})^{(s-1)/2}K_{s-1}\left(2\pi\xi\sqrt{n^{2}+m^{2}}\right).$
(24)

Since the above expression contains a double sum, we have used that
$\sum_{n,m-\\{0\\}}f(n,m)=4\sum_{n,m=1}^{\infty}f(n,m).$

The equation (24) has no poles when $s=1/2$, so we can evaluate directly to
obtain an expression for the electrostatic potential due to this
configuration, obtaining the following expression:

$\tilde{V}_{2}(\xi)=-2\pi\xi+8\sqrt{\xi}\sum_{n,m=1}^{\infty}\frac{1}{(n^{2}+m^{2})^{1/4}}K_{1/2}\left(2\pi\xi\sqrt{n^{2}+m^{2}}\right),$
(25)

where the sub-index 2 refers to the two-dimensional lattice. Note that
$K_{-1/2}(x)=K_{1/2}(x)$.

The asymptotic behavior, $\xi\to\infty$, of the electrostatic potential is
given by the first term, since the second term in the equation (25) tends to
zero (see the Appendix), so

$\tilde{V}_{2,as}(\xi)\sim-2\pi\xi.$ (26)

In order to obtain and analytic expression for the electric field of a two-
dimensional lattice, we calculate the potential gradient of equation (25). By
means of equation (30), we can express the electric field for large $\xi$ by

$\tilde{E}_{2}(\xi)\approx
2\pi+8\pi\sum_{n,m=1}^{\infty}e^{-2\pi\sqrt{n^{2}+m^{2}}\xi},$ (27)

where clearly $e^{-2\pi\sqrt{n^{2}+m^{2}}\xi}\to 0$ in the $\xi\to\infty$
regime. So, the asymptotic behavior of the electric field is given by

$\tilde{E}_{2,as}(\xi)=2\pi.$ (28)

##  6 Conclusions

We have calculated the electrostatic potential and the field of one and two-
dimensional infinite lattice of point charges, using the zeta function
regularization method. Through the analytic extension of the Epstein zeta
function, we obtained expressions that only in the one-dimensional case was
necessary to remove $1/\epsilon$ singularity using the principal prescription
part. In the two-dimensional case, no such singularity appeared, and the
resulting expressions become very useful for numerical evaluations. We
explored the asymptotic behavior of the electrostatic potential and field in
one and two-dimensional distributions. In the one-dimensional case, we found
that the dominant terms of the potential and field behave like an infinite
charged line, where $V(r)\sim\ln(1/r)$ and $E(r)\sim 1/r$, respectively. On
the other hand, for the two-dimensional case, the asymptotic behavior of the
electrostatic potential and field, are similar to the infinite charge plate,
where $V(r)\sim-r$ and $E(r)=\text{constant}$.

Finally, this example shows some features of the zeta function regularization
applied to an electrostatic problem, which can be useful as a pedagogical
exercise for graduate students studying spectral functions

## Acknowledgment

The author thanks Professors Julio M Yáñez, Juan C Rojas and Roberto A Lineros
(Universidad Católica del Norte, Antofagasta, Chile) for valuable suggestions
and time spent in many discussions.

## Appendix

The asymptotic expansion of the Bessel function of second kind is given by
[16]

$K_{\nu}(x)\sim\sqrt{\frac{\pi}{2x}}e^{-x}$ (29)

For $x\to\infty$ with $\nu$ fixed. In the special case $\nu=1/2$ we have

$K_{1/2}(x)=\sqrt{\frac{\pi}{2x}}e^{-x}.$ (30)

The Bessel function of second kind obtained in equations (18), (25), (22) and
(27), has the form

$f(x)=\sum_{|\textbf{n}|\in\mathbb{Z}^{d}-\\{0\\}}K_{\nu}(|\textbf{n}|x)\sim\sum_{|\textbf{n}|\in\mathbb{Z}^{d}-\\{0\\}}\sqrt{\frac{\pi}{2|\textbf{n}|x}}e^{-|\textbf{n}|x}$
(31)

where $|\textbf{n}|=\sqrt{n_{1}^{2}+n_{2}^{2}+\cdots+n_{d}^{2}}$ represents a
sequence of real numbers. Since we are interested in the asymptotic behavior
of $f(x)$, this expression is very useful for numerical evaluations. However,
in the $x\to\infty$ regime and considering that $|\textbf{n}|$ is an
increasing sequence of numbers, we can see that the numerical value of $f(x)$
goes to zero. So we can consider that $f(x)$ in $x\to\infty$ regime can not be
considered as a dominant term.

###  6.1 List of symbols

$\operatorname{Re}$ | set of all real numbers
---|---
$\mathbb{Z}$ | set of all integers (whether positive, negative or zero)
$\mathbb{C}$ | set of all complex numbers
$\mathbb{N}$ | set of all natural numbers
$\mathbb{N}_{0}$ | set of all natural numbers including zero
$\in$ | is member of
$\\{\;\\}$ | set
$\epsilon_{0}$ | permittivity of free space
$K_{\nu}(x)$ | Modified Bessel function of second kind
$\Gamma(x)$ | Gamma function

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