Eun Kyoung Choe

Jaeyeon Jung
email: jjung@microsoft.com

Bongshin Lee
email: bongshin@microsoft.com

Kristie Fisher
email: kfisher@microsoft.com

Nudging People Away From Privacy-Invasive Mobile Apps Through Visual Framing

Keywords: Visual framing, privacy, privacy metrics, rating, nudge, framing effect, valence, positive framing, negative framing, Mechanical Turk

Smartphone users visit application marketplaces (or app stores) to search and install applications. However, these app stores are not free from privacy-invasive apps, which collect personal information without sufficient disclosure or people's consent. To nudge people away from privacy-invasive apps, we created a visual representation of the mobile app's privacy rating. Inspired by "Framing Effects," we designed semantically equivalent visuals that are framed in either a positive or negative way. We investigated the effect of the visual privacy rating, framing, and user rating on people's perception of an app (e.g., trustworthiness) through two experiments. In Study 1, participants were able to understand the intended meaning of the visual privacy ratings. In Study 2, we found a strong main effect for visual privacy rating on participants' perception of an app, and framing effects in a low privacy rating app. We discuss implications for designing visual privacy ratings, including the use of positive visual framing to nudge people away from privacy-invasive apps.

Introduction

Application marketplaces (a.k.a. app stores) have become the mainstream channels to distribute applications onto smartphones. Over 700,000 apps have been published both in the Apple App Store and in Google Play as of Dec. 2012 [START_REF]App Store Metrics[END_REF]4]. However, this abundance of choice comes with consequences; app stores are not free from privacyinvasive apps that collect personal information without sufficient disclosure or user consent. An investigation of 100 popular apps (for iPhone and Android) shows that many of these apps collect personal information such as location and the phone identification numbers, and some apps share these data with third parties without proper user consent [START_REF]What they know[END_REF]. For example, Path, a popular social media app, was recently found to transmit the user's contacts stored on the phone without explicit permission [START_REF] Wired | Apple says grabbing address book data is an iOS policy violation[END_REF].

When shopping for an app in the app store, people have an opportunity to compare different apps that provide similar functionality. For example, searching for "weather" returns more than 1,000 results in Google Play. Clicking an app of interest, people can view detailed information, such as description, screen shots, user ratings, and reviews of the app. In this detailed view, some app stores provide privacy-related information regarding which types of data the app may access; however, prior research has shown that people do not read the permission warnings, and even if they do, they do not understand what the information means, as the terms are vague and confusing [START_REF] Felt | Android permissions: user attention, comprehension, and behavior[END_REF][START_REF] Kelley | A conundrum of permissions: installing apps on an android smartphone[END_REF]. Nonetheless, a recent survey reports that 30% of survey respondents had uninstalled apps found to collect personal information that they did not want to share [START_REF] Boyles | Privacy and data management on mobile devices[END_REF].

The goal of this work was to explore novel ways to nudge people away from privacy-invasive apps when they search for and compare apps to install. Specifically, we created visual representations of an app's level of privacy protection and investigated how the visual representations influence people's perception of an app. Similar to a movie critics' rating [START_REF] Msn Movies | X-Men[END_REF], we created visuals for a privacy critics' rating of an app, conveying how privacy-preserving or privacy-invasive the app is (Figure 1).

To create influential, persuasive visuals for privacy ratings, we leveraged the wellknown "Framing Effects" [START_REF] Tversky | The framing of decisions and the psychology of choice[END_REF]. The key idea is that people's decisions, in part, depend on the way problems are stated (e.g., positively or negatively). A classic example is how a doctor describes the odds of a grueling operation: many would prefer to choose an operation of which an outcome is "90 out of 100 are alive after five years" than "10 out of 100 are dead after five years" [START_REF] Marteau | Framing of information: its influence upon decisions of doctors and patients[END_REF]. Even if these two phrases contain the same information, people-even experts (i.e., doctors)-are systematically subject to framing effects. In addition, framing effects occur without people knowing that they are being affected by it. People are susceptible to the framing as long as they understand the valence of an option-whether something is good or bad-without necessarily understanding what makes the option appealing.

Our goal was therefore not to make people understand the details of how an app's privacy rating is calculated or to help them make an informed decision. Rather, in this work, we investigated whether we can leverage framing effects to nudge people away from privacy invasive apps using visual representations. We used visual elements such as colors and symbols to make the valence information even more salient than text-only valence descriptions. To study the effect of the visual framing of a privacy rating, we first created semantically equivalent visuals for a privacy rating that highlight either the positivity or negativity of the rating, respectively. Then, we measured the effect of the positively-and negatively-framed visuals for privacy ratings on how people perceive smartphone apps with/without a user rating. Privacy rating is just one aspect among many other attributes of an app whereas the user rating reflects how general audience holistically thinks about the app based on their individual experiences. Furthermore, people self-reported [START_REF] Chin | Measuring user confidence in smartphone security and privacy[END_REF][START_REF] Egelman | Choice architecture and smartphone privacy: there's a price for that[END_REF][START_REF] Felt | Android permissions: user attention, comprehension, and behavior[END_REF] that they consider privacy-related information far less important than user ratings in making app choices. Therefore, we considered the effect of the privacy rating in conjunction with the user rating.

Our contributions are threefold. First, we detail visual attributes (e.g., color) and semantics (e.g., valence, sign) that contribute to visual framing. Second, we investigate if framing effects transfer to visual representations of a privacy rating and shift people's perceptions of apps. Third, based on the lessons learned from two experiments, we discuss design implications for visualizing privacy ratings of smartphone apps in such a way to nudge people to avoid privacy-invasive apps.

2

Related Work

Mobile App Privacy and Permissions

Recent studies reveal the limitations of existing approaches to presenting privacyrelated information on mobile phones. Looking into Android's permission interface, two studies show that people pay little attention to permission requests when installing an app and have a poor understanding of what each permission means [START_REF] Felt | Android permissions: user attention, comprehension, and behavior[END_REF][START_REF] Kelley | A conundrum of permissions: installing apps on an android smartphone[END_REF]. Semi-structured interviews of 20 Android users discover that participants do not understand Android permissions, which were described as "confusing, misleading, jargon-filled, and poorly grouped" [START_REF] Kelley | A conundrum of permissions: installing apps on an android smartphone[END_REF]. An online survey of 308 Android users reports that only 17.5% of respondents reported looking at permissions during their last app installation [START_REF] Felt | Android permissions: user attention, comprehension, and behavior[END_REF]. Moreover, only 3% of respondents could provide correct answers for permission comprehension questions [START_REF] Felt | Android permissions: user attention, comprehension, and behavior[END_REF]. On the contrary, the same study reports that 71% of respondents looked at some type of user reviews before installing an app. Similarly, Chin and colleagues reveal that participants often rely on user ratings when deciding an app to install [START_REF] Chin | Measuring user confidence in smartphone security and privacy[END_REF]. Realizing these limitations, researchers have made an effort to design simple and easy-to-understand visual representations of privacy-related information. For example, Kelley and colleagues show that presenting online privacy policies using standardized tables improves people' comprehension of the privacy policies [START_REF] Kelley | Standardizing privacy notices: an online study of the nutrition label approach[END_REF]. Cranor and colleagues use a symbol (called Privacy Bird) to indicate whether a website's privacy policy matches a user's preference [START_REF] Cranor | User interfaces for privacy agents[END_REF]. A follow-up study designs "privacy icons" and demonstrates that when the privacy icons were presented, participants were willing to pay a premium to purchase items from the websites that better protect users' privacy [START_REF] Tsai | The effect of online privacy information on purchasing behavior: an experimental study[END_REF]. Similar to these studies, we created simple visual representations for privacy information. However, our work is different from the previous work as our goal was not to improve people's comprehension of the exact details of privacy policies, but rather to create visual representations that can affect how people perceive an app and as a result can nudge them away from privacy-invasive apps.

One study by Lin and colleagues [START_REF] Lin | Expectation and purpose: understanding users' mental models of mobile app privacy through crowdsourcing[END_REF] addresses the limitations of Android's permission interface using crowdsourcing. The authors propose a privacy summary that highlights the use of permissions that did not match other people's expectations. To capture people's expectations, the authors conducted online surveys, recruiting respondents from Amazon Mechanical Turk. Such measured expectations can be useful for creating an app's privacy rating, which our work could leverage. In comparison, our work examines framing effects in visual representations for privacy information with the aim to identify what makes certain visuals more influential and persuasive than others and leverage those properties to influence user perceptions.

Theoretical Framework: Framing Effects

A few recent studies apply behavioral economics theories in designing persuasive technologies. Lee and colleagues [START_REF] Lee | Mining behavioral economics to design persuasive technology for healthy choices[END_REF] apply three persuasion techniques drawn from behavioral economics-the default option strategy, planning strategy, and asymmetric choice strategy-in designing choices for healthy eating. Inspired by their work, we explore a novel approach to employ framing effects to nudge people away from privacy-invasive apps. Humans have been thought to make rational choices, of which the core principles include invariance [START_REF] Tversky | Rational choice and the framing of decisions[END_REF]. Invariance requires that two versions of a choice that are recognized to be equivalent should yield the same preference regardless of the manner in which they are described. However, Tversky & Kahneman [START_REF] Tversky | The framing of decisions and the psychology of choice[END_REF] reveal that presenting the same option can alter people's decisions when varying the framing of acts, contingencies, or outcomes. Since Tversky & Kahneman first discussed framing effects explaining how valence influences people's willingness to take risks, framings have been studied using text descriptions in many domains including privacy domain [START_REF] Johnson | Defaults, framing and privacy: why opting in-opting out[END_REF]. To better understand when and why different types of framing will have an effect, Levin and colleagues develop a typology of framings, and distinguish between three different kinds of framings-risky choice, attribute, and goal framing [START_REF] Levin | All frames are not created equal: a typology and critical analysis of framing effects[END_REF]. Our work is particularly inspired by the attribute framing, which manipulates the valence of a single attribute within a given context. For example, an attribute of ground beef can be labeled as either "75% lean" or "25% fat," and a study shows that people favor the former even though the two labels convey the same information [START_REF] Levin | Framing of attribute information before and after consuming the product[END_REF]. Framing effects occur because people tend to be somewhat mindless when consuming information in daily life [START_REF] Thaler | Nudge: improving decisions about health, wealth, and happiness[END_REF]. Therefore, when applied properly, framing can be used as a powerful nudge to influence people to make a better decision [START_REF] Thaler | Nudge: improving decisions about health, wealth, and happiness[END_REF].

Nudging for Privacy Decision

Acquisti & Grossklags discuss the relevance of framing to privacy research [START_REF] Acquisti | What can behavioral economics teach us about privacy?[END_REF]. In [START_REF] Acquisti | Nudging privacy: behavioral economics of personal information[END_REF], Acquisti coins the term, "nudging privacy," and discusses how existing research in behavioral economics and psychology can help better understand privacy decision making. Furthermore, Balebako and colleagues broadly discuss how their ongoing work in location sharing can leverage nudging interventions to help people make better privacy decisions [START_REF] Balebako | Nudging users towards privacy on mobile devices[END_REF].

In HCI research, the term "nudging" has been used in the broad sense outside of behavioral economics as a means to influence people's behaviors, such as shopping [START_REF] Kalnikaitė | How to nudge in situ: designing lambent devices to deliver information salience in supermarkets[END_REF] or exercise habits [START_REF] Rogers | Ambient influence: can twinkly lights lure and abstract representations trigger behavioral change?[END_REF]. When designing a system whose goal is to nudge people toward a certain direction, the system might undermine the decision making power of individuals. Thaler & Sunstein [START_REF] Thaler | Nudge: improving decisions about health, wealth, and happiness[END_REF] state that a nudge should be "easy and cheap to avoid." Thus, banning privacy-invasive apps entirely from app stores is not a nudge. We also note that the use of "framing" as a design approach for nudging might impose ethical concerns as discussed in [START_REF] Camerer | Regulation for conservatives: behavioral economics and the case for "asymmetric paternalism[END_REF]. We share the same vision with the prior research mentioned above in that leveraging framing can be a promising approach to nudge people toward better privacy decisions. However, these techniques should be applied with care so as not to limit people's freedom to choose.

Research Questions and Experimental Setups

We explore the following research questions (RQ) through two experiments:

RQ1. Can we create complementary visual framings that convey semantically equivalent privacy rating information? RQ2. Do complementary privacy ratings have similar influence on how people perceive an app? If not, is a negative visual framing more effective in nudging people away from privacy-invasive apps?

RQ3. Do people's perceptions of an app change if a privacy rating is accompanied by a user rating?

For each study, we created an online experimental setup and recruited participants using Amazon Mechanical Turk (MTurk). We made sure no one could participate more than once by setting a cookie in participants' browser and by removing answers from duplicate MTurk IDs. The studies were available only to U.S. and Canada residents with at least a 95% approval rate (through a screening option that MTurk provides). We compensated MTurk participants $0.50 USD per survey for Study 1, and $1 USD per survey for Study 2.

Study 1: Creating Complementary Visuals

As a first step toward identifying visuals that have higher nudging power, we investigated whether it is viable to create complementary visual framings that convey semantically equivalent privacy rating information. We designed two sets of icons: positively-framed (PF) icons using a green plus sign (+) and negatively-framed (NF) icons using a red minus sign (-) (Figure 1-a). Because most of the ratings (e.g., Amazon star rating) use PF icons (i.e., the more stars, the better), we speculated that people were already familiar with the PF icons but not with the NF icons. We measured the level of comprehension of the PF and NF icons after short training. Similar to prior framing studies, we conducted a between-subjects experiment with two groups: PF icon group and NF icon group.

Survey Content

We created online surveys for PF and NF conditions. The surveys consisted of a privacy rating description page and two sets of eight icon comparison questions (the first set was for training). On the first page, we introduced a hypothetical privacy metric called "Privacy Critics' Rating." To help participants understand valence of the metric, we provided the following explanation:

Imagine you are deciding whether to install an app on your smartphone. Privacy experts ran a series of tests to assess the app's level of privacy. These tests measured how likely the app is to capture your personal information and to share/sell your information to advertisers or other partners.

We manipulated the description and a screenshot of a fake game app, "Tic Tac Toe," in the following manner:

PF condition: We explained that the privacy rating scale represents the proportion of tests that the app passed. Therefore, if an app has more green-colored (PF) icons in the rating, it is more privacy-preserving. The screenshot included a privacy rating of 4 (out of 5).

NF condition: We explained that the privacy rating scale represents the proportion of tests that the app failed. Therefore, if an app has more red-colored (NF) icons in the rating, it is more privacy-invasive. The screenshot included a privacy rating of 1 (out of 5).

According to our description, a privacy rating of 4 (out of 5) in the PF condition is equivalent to a privacy rating of 1 (out of 5) in the NF condition. After the description, we asked participants to answer two sets of eight icon comparison questions as accurately and quickly as possible. We showed two different privacy ratings and asked which of the two privacy ratings is more privacy-invasive or privacy-preserving (Figure 2). These two questions appeared in a random order. We placed one question per page; participants had to click a "next" button in order to proceed to the next question, which allowed us to measure a task completion time for each question.

4.2

Measures

The dependent measures were accuracy (i.e., the number of correct responses) and task completion time (i.e., average response time per question). The first set of eight questions was a training set, thereby using only the second set of eight questions for the analysis. We also collected qualitative feedback in free-form text.

Participants

We recruited 129 participants and randomly assigned them to either the PF condition (N = 67) or NF condition (N = 62). We limited the study participants to mouse users because different input methods could influence task completion time significantly. Additionally, we filtered MTurkers who did not pay careful attention to the study instructions as indicated by the description reading time; we removed 47 participants who spent less than 20 seconds on the description page (it took the authors more than 20 seconds simply to read the description).

Results of Study 1

The independent samples t-test showed that in terms of accuracy, there was not enough evidence to suggest that PF icon group (N = 43, M = 7.81, SD = .59) differs from NF icon group (N = 39, M = 7.59, SD = 1.29), t(80) = 1.03, p = .31. To compare the task completion time (in second), we first removed outliers defined by the mean minus two standard deviations (M -2SD) (i.e., the number of correct responses < 5.7). Then we excluded incorrect responses in calculating the task completion time.

The independent samples t-test showed that in terms of task completion time, there was not enough evidence to suggest that PF icon group (N = 39, M = 3.29, SD = .92) differs from NF icon group (N = 36, M = 3.50, SD = 1.12), t(73) = -.89, p = .37.

Discussion of Study 1

After reading the description and solving eight icon comparison questions, the majority of participants in both groups were able to comprehend privacy ratings in a similar manner. The PF and NF icons resulted in the comparable level of comprehension and speed by survey participants. Thus, we chose to use this set of PF and NF icons (Figure 1-a) as stimuli for our subsequent framing study. We note, however, that even if there was no significant difference in the performance, 7 out of 36 (= 19%) participants in the NF icon group mentioned in their written comments that the NF icons were confusing. We suspect that this is due to a strong preconception of "the more, the better," which is what the prevalent PF rating scale conveys. At the same time, other participants commented that the red minus sign we used to convey the negativity was particularly helpful to mark privacy-invasive apps. In Study 2, we explored whether the complementary visuals have similar influence on how people perceive an app.

Study 2: Positive vs. Negative Framings

Given that people can comprehend the positive and negative visual framings in a similar manner with some training, we investigated whether there is any effect of framing on how people perceive apps. In prior framing studies using text descriptions, researchers report that negatively-framed information tends to influence more than the positively-framed information of the same magnitude [START_REF] Peeters | Positive-negative asymmetry in evaluations: the distinction between affective and informational negativity effects[END_REF][START_REF] Taylor | Asymmetrical effects of positive and negative events: the mobilization-minimization hypothesis[END_REF]. In Study 2, we tested which visual framing-positive or negative-was more effective in nudging people away from privacy-invasive apps using the icons we designed in Study 1. We also explored how people's perception of an app changes if a privacy rating of the app accompanies a user's overall rating (user rating). This resulted in a 2 (framing: PF icon; NF icon) x 3 (privacy rating: high; medium; low) x 2 (user rating: with a user rating of 3; without a user rating) mixed design with repeated measures; framing and user rating were between-subjects factors and privacy rating was a within-subjects factor, thereby forming four conditions: PF with & without a user rating of 3, and NF with & without a user rating of 3.

Survey Content

We created online surveys for the four conditions. The surveys consisted of three sections: (1) evaluating four apps (one dummy app followed by three apps of varying degrees of privacy ratings), (2) eight Privacy Critics' Rating icon comparison questions, and (3) demographic questions. At the beginning of the surveys, we introduced a hypothetical privacy metric called "Privacy Critics' Rating" using the same blurb from Study 1. Then, we showed a fake weather app with the following description and a screenshot.

Imagine that you need a weather app, so you are searching the marketplace on your phone. While looking around many weather apps, you come across the following app page:

The screenshot contained a visual privacy rating, icon, and app description (Figure 3). We manipulated the screenshot of the weather app in the following manner:

Privacy Rating: For each participant, we showed a series of four screenshots of a weather app: one dummy set (user rating: 3.5; privacy rating: 2) and three levels of privacy ratings (high; medium; low) in a random order (Figure 4). The dummy set was included at the beginning to account for a longer reading time and familiarity with the screenshot and the question types.

Positive Framing (PF) vs. Negative Framing (NF): We created a PF and NF condition by manipulating the description of the Privacy Critics' Rating and its icon design as in Study 1 (Figure 4).

With a User Rating of 3 (w/ UR) vs. Without a User Rating (w/o UR): We chose a user rating of three as a representative case for the w/ UR condition because most of the popular free apps in the marketplace have an equal or higher user rating than three. Three green-colored stars (out of five) appeared right above the privacy rating in the w/ UR conditions (Figures 3-b/d). No user rating was provided in the w/o UR conditions (Figures 3-a/c).

Measures

After showing each app, we measured people's perception toward each app by asking the following four questions: (1) trustworthiness of the app (TRUST), (2) likeability of the app (LIKE), (3) willingness to install the app (INST), and (4) willingness to recommend the app to a friend (RCMD). We measured TRUST because we suspected that people would associate an app's privacy rating with its trustworthiness. LIKE is a conventional measure in prior framing studies where a product's attribute is framed differently [START_REF] Levin | All frames are not created equal: a typology and critical analysis of framing effects[END_REF]. We measured INST and RCMD to gauge people's willingness to adopt an app. TRUST and LIKE were measured on a 7-point Likert scale, where 1 = not at all trustworthy / I strongly dislike this app, and 7 = very trustworthy / I strongly like this app. INST and RCMD were measured using Yes/No dichotomous questions.

In addition, we suspected that participants' interest level toward the weather app could be a factor related to other dependent measures. So we measured self-reported interest level (INTEREST) toward the weather app on a 7-point Likert scale, where 1 = not at all interested, and 7 = very interested, at the very beginning of the survey as we showed the screenshot without the privacy and user rating information. Within each condition, the privacy rating was the only factor that we varied. It was necessary for participants to understand how the privacy rating worked so that they could answer the evaluation questions based on correct understanding of the stimuli. Therefore, we measured accuracy and task completion time for the eight privacy rating icon comparison questions as in Study 1. However, this time, we placed these questions after the app evaluation questions to use them for filtering purpose rather than training purpose. We made this decision based on our pilot study and previous research indicating that framing effects are susceptible to rational thinking (e.g., asking people to provide a rationale for their choice eliminates or reduces the framing effects [START_REF] Miller | The effects of framing, problem variations, and providing rationale on choice[END_REF]).

Participants

We recruited 332 participants from Amazon Mechanical Turk and randomly assigned them to one of the four conditions: PF & w/o UR (N = 75); PF & w/ UR (N = 95); NF & w/o UR (N = 79); and NF & w/ UR (N = 83). We removed 18 duplicate participants (i.e., who participated in Study 1 or pilot studies). Using the same criteria from Study 1, we removed 79 participants who spent little time reading the descriptions. Among the remaining 235 participants, 55% of the participants (N = 129) were male, and 89% of the participants (N = 210) claimed that they own a smartphone.

Results of Study 2

After removing 21 participants from the analysis whose number of correct responses to the eight filtering questions was less than M -2SD (i.e., the number of correct responses < 5.3), we observed that participants' initial interest level toward the weather app was highly related to how much they trust the weather app, F(1, 209) = 9.02, p = .003, and how much they like the weather app, F(1, 209) = 23.20, p < .001. Therefore, TRUST and LIKE were analyzed using a mixed-design analysis of covariance (ANCOVA) controlling for the INTEREST as the covariate (Table 1). INST and RCMD were analyzed using a Pearson chi-square test (Table 2).

Table 1. This table shows strong main effects of privacy rating on TRUST and LIKE, and the framing effect (significant interaction effect between privacy rating and framing) on TRUST. "TRUST" Question. After controlling for the effect of INTEREST, we found a significant main effect of privacy rating on TRUST, F(1.38, 289.13) = 55.55, p < .001. Planned contrasts revealed that a high privacy rating app (M = 5.15) was regarded as more trustworthy than a medium privacy rating app (M = 3.03), F(1, 209) = 50.39, p < .001. Also, a medium privacy rating app was regarded as more trustworthy than a low privacy rating app (M = 1.75), F(1, 209) = 29.60, p < .001. We found a significant interaction effect between privacy rating and framing, F(1.38, 289.13) = 7.48, p = .003 (Figure 5). To break down this interaction, planned contrasts were performed comparing each level of privacy rating to one another across PF and NF conditions. We found a significant interaction between a high privacy rating app and a low privacy rating app across PF and NF conditions, F(1, 209) = 9.09, p = .003, and between a medium privacy rating app and a low privacy rating app across PF and NF conditions, F(1, 209) = 8.89, p = .003. Also, there was a marginally significant interaction between a high privacy rating app and a medium privacy rating app across PF and NF conditions, F(1, 209) = 3.50, p = .06. As Figure 5 indicates, the difference between PF (M = 1.44) and NF (M = 2.05) was significant for a low privacy rating app, t(211) = 2.42, p = .02. No significant differences were found between PF (M = 5.27) and NF (M = 5.02) conditions in a high privacy rating app, t(211) = .34, p = .12, or between PF (M = 2.97) and NF (M = 3.10) conditions in a medium privacy rating app, t(211) = 1.58, p = .74.

We found a marginally significant interaction between privacy rating and presence of user rating, F(1.38, 289.13) = 3.19, p = .06. This indicates that user rating might have different effects on app's trustworthiness at different levels of privacy rating. To break down this interaction, planned contrasts were performed comparing each level of privacy rating to one another across w/ and w/o UR. We found a significant interaction when comparing a high privacy rating app to a medium privacy rating app across w/ and w/o UR, F(1, 209) = 4.43, p = .04. Also, we found a marginally significant interaction when comparing a high privacy rating app to a low privacy rating app across w/ and w/o UR, F(1, 209) = 3.44, p = .07. As Figure 6 indicates, when a user rating of 3 is shown, there is a decline in TRUST for a high privacy rating app while there is an increase in TRUST for a low and medium privacy rating app. Fig. 5. This shows a significant interaction between privacy rating and framing on TRUST. Participants interpreted the NF icons more positively than the PF icons of the equivalent rating.

"LIKE" Question. After controlling for the effect of INTEREST, we found a significant main effect for privacy rating on LIKE, F(1.46, 305.42) = 30.42, p < .001. This effect tells us that how much participants liked the weather app was different for high, medium, and low privacy rating apps. Planned contrasts revealed that participants liked the high privacy rating app (M = 5.01) significantly more than the medium privacy rating app (M = 3.45), F(1, 209) = 12.41, p = .001. Also, participants liked the medium privacy rating app significantly more than the low privacy rating app (M = 2.17), F(1, 209) = 48.01, p < .001.

We found a marginally significant main effect for user rating, F(1, 209) = 3.41, p = .07. Participants liked the app w/ UR of 3 (M = 3.66) more than the app w/o UR (M = 3.42).

We found a significant interaction between privacy rating and user rating, F(1.46, 305.42) = 12.02, p < .001 (Figure 7). This indicates that user rating had different effects on LIKE depending on different levels of privacy rating. To break down this interaction, planned contrasts were performed comparing each level of privacy rating to one another across w/ UR and w/o UR. We found a significant interaction when comparing a high privacy rating app to a medium privacy rating app across w/ UR and w/o UR, F(1, 209) = 15.45, p < .001. Also, we found a significant interaction when comparing a high privacy rating app to a low privacy rating app across w/ UR and w/o UR, F(1, 209) = 14.02, p < .001. The interaction graph (Figure 7) tells us that when a user rating of 3 is shown, there is a slight decline in LIKE for a high privacy rating app while there is an increase in LIKE for a low and medium privacy rating app. The remaining contrasts revealed no significant interaction when comparing a medium privacy rating app to a low privacy rating app in the w/o UR and w/ UR conditions, F(1, 209) = 2.59, p = .11.

"INSTALL" Question. We found a marginally significant association between framing and participants' choice of installing a low privacy rating app, χ 2 (1, N = 214) = Fig. 6. This shows a marginally significant interaction (p = .06) between privacy rating and user rating on TRUST.

3.42, p = .06 (Table 2). The odds ratio implies that the odds of participants installing a low privacy rating app were 3.36 times higher if the rating were negatively framed than positively framed.

We found a significant association between framing and participants' choice of installing a medium privacy rating app, χ 2 (1, N = 214) = 5.17, p = .02 (Table 2). The odds ratio implies that the odds of participants installing a medium privacy rating app were 2.21 times higher if the rating were negatively framed than positively framed.

We found a marginally significant association between a user rating and participants' choice to install a high privacy rating app, χ 2 (1, N = 214) = 3.20, p = .07 (Table 2). The odds ratio implies that the odds of participants installing a high privacy rating app were 1.80 times higher if there were no user rating than the app accompanying a user rating of 3. Fig. 7. This shows a significant interaction between privacy rating and user rating on LIKE. A user rating of 3 had a different effect on the high privacy rating app in comparison to medium/low privacy rating apps.

Table 2. More people answered that they would install / recommend a negatively-framed low privacy rating app than the same app that is positively-framed. The number of people who answered that they would install / recommend a high privacy rating app decreased when a user rating of 3 was shown than when no user rating was shown. "RECOMMEND" Question. We found a significant association between framing and participants' choice of recommending an app with low privacy rating, χ 2 (1, N = 214) = 5.97, p = .02 (Table 2). The odds ratio implies that the odds of participants recommending a medium privacy rating app to a friend were 5.53 times higher if the ratings were negatively framed than positively framed. We also found a significant association between user rating and participants' choice of recommending an app with high privacy rating, χ 2 (1, N = 214) = 4.45, p = .04. The odds ratio implies that the odds of participants recommending a high privacy rating app were 1.90 times higher if there were no user rating than a user rating of 3.

Discussion of Study 2

Study 2 results show a strong effect of privacy rating on all dependent measures. This indicates that when a privacy rating of a given app is disclosed visually, people are influenced by the privacy rating. The influence of the privacy rating appears to decline (although still significant) when we showed a user rating of 3 (Figures 6/7). This suggests that people are susceptible to both privacy rating and user rating. The effect of framing was subtle. First, we observed framing effects on TRUST in a low privacy rating app. Participants expressed a lower level of trustworthiness of an app when its privacy rating was positively framed than negatively framed. For medium and high privacy rating apps, framing effect did not occur. A similar trend was observed for INST and RCMD-a low privacy rating app was a common denominator for the framing effects to be observed, and when observed, it was always the negatively framed icons that people interpreted more positively. However, there was no framing effect on LIKE; after controlling for people's app interest level, privacy rating and user rating dominantly influenced LIKE. As we suspected, it appears that participants associated the privacy ratings with TRUST more than LIKE.

Prior framing studies using text descriptions consistently show that positive framing leads to more favorable evaluations than negative framing [e.g., [START_REF] Levin | Framing of attribute information before and after consuming the product[END_REF][START_REF] Marteau | Framing of information: its influence upon decisions of doctors and patients[END_REF]. Researchers demonstrate that describing an option in a negative light (e.g., "mortality rate") focuses attention on the unfavorable possibilities associated with this option, rendering it less acceptable to the decision-maker [START_REF] Levin | All frames are not created equal: a typology and critical analysis of framing effects[END_REF]. Therefore, we initially suspected that emphasizing negativity (e.g., privacy-invasiveness) would nudge people away from privacy invasive apps with a low privacy rating. However, our study results suggest this is not the case.

On the contrary, PF icons were more effective in making a low privacy rating app look more unfavorable. We suspect that people have strong connotations of "the more, the better" in the rating context. Because a negatively framed privacy invasive app has more signs in the rating than the equivalent PF icons (i.e., four minus signs vs. one plus sign), it is plausible that the higher number of ratings, regardless of its meaning, could have contributed to how people perceive the PF/NF icons.

Our results also suggest that there was no framing effect in the high and medium privacy rating apps. Therefore, the use of PF icons for depicting privacy ratings is a better choice for nudging people away from privacy invasive apps while not affecting high and medium privacy rating apps.

Discussion and Future Work

Our results suggest that visual representations of privacy information of apps can influence installation decisions by smartphone users. When disclosed visually on the detailed view of an app, the majority of participants commented that they found the privacy rating very helpful in deciding whether to install an app. In this section, we discuss design recommendations for a visual privacy rating as we answer our research questions. We also discuss limitations of this work and areas for future research.

Design Recommendations

To design complementary visuals that represent the same privacy rating in a positive or negative way (RQ1), we suggest leveraging visual attributes (e.g., colors) and semantics (e.g., valence, signs, symbols) that are prevalent throughout the culture for conveying valence. In text, people make favorable and unfavorable associations with positively and negatively phrased attribute labels (e.g., "fat" is associated with "bad" and "lean" with "good"). Similarly in visuals, we can take advantage of people's preconceptions built on their life experiences. In designing PF and NF privacy ratings, we tried out various symbolic figures in red and green and chose the most promising one based on pilot tests measuring task completion time and error rate. For example, plus and minus signs, or thumbs-up and thumbs-down are well-known symbols in the US (Figures 1-a/b). When colors (red for negative and green for positive) are added on top of these, the valence becomes more pronounced. However, we ruled out thumbs-up and thumbs-down icons because they are already being used to convey different meanings elsewhere (e.g., "thumbsup" is associated with "like" in Facebook). We also ruled out neutral figures (e.g., rectangles in Figure 1-c)-providing extra information (e.g., a legend) shall be necessary to frame neutral figures.

Study 2 results show that the complementary visuals do not have the same influence on how people perceive an app in certain conditions (RQ2). If the goal is to nudge people away from privacy-invasive apps, we suggest using a positive visual framing for privacy rating. However, this might not be the case outside the realm of "rating scale" design; when "the more, the better" connotation no longer exists, positive visual framing might lead to more favorable evaluations and vice versa, similar to prior framing effects shown in texts. Therefore, future research is warranted to study the influence of positive and negative visual framings in other contexts.

The results of Study 2 also show that people's perception of an app changes if a privacy rating is accompanied by a user rating but that a privacy rating still has a strong main effect (RQ3). This is promising in that people do understand and consider the level of privacy protection when it is presented to them visually. Prior research shows that textual privacy-related information is easily discarded [START_REF] Felt | Android permissions: user attention, comprehension, and behavior[END_REF][START_REF] Kelley | A conundrum of permissions: installing apps on an android smartphone[END_REF]. This again suggests the need for visual privacy ratings. This work, however, does not address whether the main effect of visual privacy rating is due to the privacy information shown in a rating scale, due to the visuals, or due to the combination of both. Therefore, future research is needed to tease out these factors.

Limitation and Future Work

In running online experiments with participants recruited from the MTurk site, we made a conservative assumption of filtering mindless Turkers defined by those who spent less than 20 seconds on the description page where we explained the privacy rating and what it represents. Because we introduced a privacy rating with which most people are unfamiliar, eliminating answers from an arbitrary guess was necessary. This is analogous to avoiding recruits who do not understand what "fat" and "lean" mean in the ground beef framing study [START_REF] Levin | Framing of attribute information before and after consuming the product[END_REF]. By the same token, we eliminated outliers whose number of correct responses to the icon comparison questions was less than M -2SD. However, this does not guarantee that those who spent longer than 20 seconds on the description page correctly interpreted the privacy rating, which is beyond our control when running an experimental study online.

In this work, it was not our main interest to investigate the relative influence of privacy-related information in comparison to user ratings (although we touched upon the interaction between a privacy rating and user rating in Study 2). However, our study results suggest that understanding this topic would inform how much emphasis should be placed visually on the privacy ratings in comparison to user ratings. To answer this question, we would need to understand how people's perceptions of an app change depending on different designs and the relative weight of a visual privacy rating in return.

We would also like to further investigate how visual framings can be applied outside the privacy domain as a method for nudging. Health information feedback of consumer health devices, for example, is designed with a specific goal in mindpromoting health-enhancing behaviors. One promising direction is to design visual framings that represent the outcome of a healthy or unhealthy behavior stressing either the positive or negative consequences as we provide health information feedback. Framing research in texts has addressed when framing effects are reduced or eliminated [START_REF] Druckman | Using credible advice to overcome framing effects[END_REF][START_REF] Miller | The effects of framing, problem variations, and providing rationale on choice[END_REF], which helps HCI researchers select application areas for employing the visual framing.

Conclusion

We created visuals for a mobile app's privacy rating by leveraging the well-known "Framing Effects." In Study 1, we showed that it is viable to create semantically equivalent privacy ratings framed in either a positive or negative light. In Study 2, we showed a strong main effect for visual privacy rating on people's perception of an app. Our study results suggest that when an app's privacy property is provided in the form of a visual privacy rating, people are able to understand and are heavily influenced by it. We also examined how the visual framings shift people's perception of an app, and observed the occurrence of framing effects in a low privacy rating app. In designing a visual privacy rating to nudge people away from privacy-invasive apps, we recommend using positive visual framing, and leveraging visual attributes and semantics that are prevalent throughout the culture for conveying valence. Investigating the relative influence of a visual privacy rating in comparison to a user rating warrants future research efforts. In closing, this work provides empirical guidance for creating influential, persuasive visual framing that can nudge people away from privacy invasive apps.

Fig. 1 .

 1 Fig. 1. Visual Framing. Positively-framed visuals of a rating of 3 (left column) is semantically equivalent to negatively-framed visuals of a rating of 2 (right column).

Fig. 2 .

 2 Fig. 2. An example of the icon comparison questions and the complementary visuals for the PF (a) and NF (b) conditions.

Fig. 3 .

 3 Fig. 3. Screenshots of the detailed view. Each screenshot represents the following: (a)-PF & w/o UR, (b)-PF & w/ UR, (c)-NF & w/o UR, and (d)-NF & w/ UR. Below the privacy rating, we provided descriptions of an app.

Fig. 4 .

 4 Fig. 4. This illustrates how we manipulated the privacy rating (3 levels) and framing (2 levels).

Acknowledgment

We would like to thank Tim Paek (Microsoft Research) for his thoughtful comments on earlier drafts of this paper. We also thank Stuart Schechter (Microsoft Research) for providing suggestions on the study design.


Olivier Lablée

Trace formulas and Zeta functions in Spectral Geometry

Keywords: Trace formulas, Spectral geometry, Lenght spectrum, Geodesic flow, Zeta functions, Hyperbolic surfaces AMS Classification : 11F72, 11M36, 35P05, 35P20, 35K08, 37D20, 53D25, 58J50, 58J53, 58E10

research documents, whether they are published or not. The documents may come

Trace formulas and Zeta functions in Spectral Geometry

Introduction

The Laplacian in our three-dimensional Euclidean space R 3 is the usual linear differential operator defined by ∆ := ∂ 2

x + ∂ 2 y + ∂ 2 z . This operator can be generalized to a compact Riemannian manifold (M, g) : this generalization is called the Laplace-Beltrami operator (or just Laplacian) and denoted by the symbol ∆ g . The spectral theory of this operator is a central object in differential geometry. Indeed, the spectrum of ∆ g contains vast information about the geometry and the topology of the associated manifold. The study of this operator and in particular the study of its spectrum is called Spectral Geometry. Spectral geometry have numerous connections with others fields of mathematicals or physics, e.g. for understanding the relationships between the formalism of classical mechanics and quantum mechanics : this is semi-classical analysis. Spectral geometry is also very important in the study of dynamical systems, for example in classical and quantum chaos. To understand the fundamental relationships between the sequence of eigenvalues of -∆ g and the geometrical properties of the manifold (M, g), one of the most powerfull tools in this way is the principle of trace formulas. More generally, traces formulas are also useful techniques in analysis of PDE, in spectral theory, physics mathematics, in number theory, etc ...

1

In this article we focus on trace formulas applied to spectral geometry and to dynamical systems on a hyperbolic surface via some special zeta functions. First, we present a simple exact computation of trace formulas on a particular manifold : a flat torus. This example of trace formula have a fundamental application in spectral geometry : the famous Milnor Theorem. Next, we interesset with in the powerfull notion of heat kernel, indeed, using trace formulas, we can establish the Minakshisundaram-Pleijel expansion and the asymptotic Weyl law for eigenvalues. Then, we finish with trace formulas on hyperbolic manifolds : in a first time we present the relationships between spectrum of the manifold and between the lenght spectrum (i.e. the set of lengths of closed geodesics). And, in a second time, we present Selberg formula and its applications in dynamical system, in particular for counting periodic geodesics on hyperbolic surface with an adapted zeta function.

Background on spectral geometry

Start here by some basics notions concerning the spectrum of the Laplacian on a compact Riemannian manifold, for more details classical references on this subject are the book of M. Berger, P. Gauduchon and E. Mazet [BGM], the book of P. Bérard [Bér4], of I. Chavel [Cha], the book of S. Rosenberg [Ros], see also the book [Lab].

Formal principle of trace formulas

The formal principle of trace formulas is the following : let us consider an unbounded self-adjoint linear operator H on a Hilbert space, and suppose that the spectrum of H is discrete :

Spec(H) = {λ k , k ≥ 0} .
This means that for all integer k ≥ 0, there exists a vector ϕ k = 0 such that : H ϕ k = λ k ϕ k . Let us also consider f a "nice" function. The fundamental principle of trace formulas is to compute the trace of the operator f (H) in two ways :

• the first way : with the eigenvalues of the linear operator f (H) :

Trace( f (H)) = +∞ ∑ k=0 f (λ k ).
• The second way : with the integral kernel of f (H)

: if for all x ∈ M f (H)ϕ(x) = M K f (x, y)ϕ(y) dy then Trace( f (H)) = M K f (x, x) dx.

2

In consequence we get the following equality :

+∞ ∑ k=0 f (λ k ) = M K f (x, x) dx.
In pratice, the main difficulty is to find a "good" choice for the function f . The usuals choices are :

• f (x) = e -xt where t ≥ 0 (heat function),

• f (x) = 1

x s , where s ∈ C such that Re(s) > 1 (zeta Riemann function),

• f (x) = e -itx h where t ≥ 0 (Schrödinger function)

• etc ...

The Laplacian operator

Let (M, g) be a compact Riemannian manifold of dimension n ≥ 1. The Laplacian on (M, g) is the linear operator defined by f → ∆ g ( f ) := div(∇ f ). In local coordinates we have : for any C 2 real valued function f on M and for any local chart φ : U ⊂ M → R of M, the Laplacian ∆ g is given by the local expression :

∆ g f = 1 √ g n ∑ j,k=1 ∂ ∂x j √ gg jk ∂( f • φ -1 ) ∂x k
where g = det(g ij ) and g jk = (g jk ) -1 . This operator appears in many diffusion equations, for example :

• The heat equation : Consider here a domain Ω ⊂ R 3 with a boundary Γ := ∂Ω. The heat equation on Ω describe the heat diffusion process on this domain. The heat equation is the following linear equation :

∂u ∂t (x, t) = ∆u(x, t).
Here u(x, t) is the temperature at the time t of the point x ∈ Ω.

• The Schrödinger equation : in Quantum Mechanics, a physical particle on a manifold M in a time intervall I is describe by a wave function, that is a function (x, t) ∈ M × I → ψ(x, t), where the quantity Ω ψ(x, t) dx represents the probability to find the particle into the domain Ω ⊂ M at the time t. In particular, this requires that

ψ 2 L 2 (M) = Ω ψ(x, t) dx = 1.
The quantum dynamics is governed by the famous Schrödinger equation :

ih ∂ψ ∂t (x, t) = -∆ g ψ(x, t) + V(x)ψ(x, t).
Here V is a function of L ∞ loc (M) such that lim |x|→∞ V(x) = +∞ : this is the potential.

The spectrum of a compact manifold

For simplify we suppose here that the manifold (M, g) is a closed manifold, i.e. M is compact and without boundary : for example the sphere, the torus... The closed spectral problem is : find all real numbers λ such that there exists a function u ∈ C ∞ (M) with u = 0 such that :

-∆ g u = λu.

From a spectral point of view we want to find the (ponctual) spectrum of the unbounded operator -∆ g on the domain D = C ∞ (M). One of the pillars of spectral geometry is the following fact (see for example [Bér4], [Cha], [Lab]) : the spectrum and the ponctual spectrum of the operaror -∆ g coincides and it consists of a real infinite sequence

0 = λ 0 < λ 1 ≤ λ 2 ≤ • • • ≤ λ k ≤ • • • such that λ k → +∞ as k → +∞.
Moreover, each eigenvalue has a finite multiplicity, recall that the multiplicity of an eigenvalue λ k is the dimension of the vector space ker λ k I d + ∆ g . In the notation (λ k ) k≥0 eigenvalues are counting with their multiplicities : in particular the notation 0 = λ 0 < λ 1 means that the first eigenvalue is simple and equal to zero. As it turns out, this result is also available for manifolds with boundary (but we have not always

λ 0 = 0). The incovenient of the notation 0 = λ 0 < λ 1 ≤ λ 2 ≤ • • • ≤ λ k ≤ • • • is relative
to concerning the multiplicity of each eigenvalues, we cant see explicitely the multiplicity of eigenvalues : so for some contexts we also use the following alternative notation : let us denote the distincts eigenvalues of (M, g) by :

0 = λ 0 , m 1 = 1 < λ 1 , m 1 < λ 2 , m 2 • • • < λ k , m k < • • •
where the integer m i denote the multiplicity of the eigenvalue λ i . For sumarize, for any closed (or more generally for any compact) Riemannian manifold there exists an unique sequence of reals numbers

(λ k ) k≥0 such that Spec(-∆ g ) = {λ k , k ≥ 0} .
The previous sequence is called the spectrum of the manifold (M, g) and denoted by Spec(M, g). We said that two (compact) Riemannian manifolds (M, g) and

(M ′ , g ′ ) are isospectral if and only if Spec(M, g) = Spec(M ′ , g ′ ).
A spectral invariant on (M, g) is a quantity which is determined by Spec(M, g).

The spectral partition function Z M

Definition 2.1. Let (M, g) be a compact Riemannian manifold, we define the spectral partition function by the following serie of functions

Z M (t) := +∞ ∑ k=0 e -tλ k .
We will see, using Weyl asymptotics (section 4.4) that the previous serie converge uniformly on R ⋆ + . Using this uniform convergence it is obvious that function t → Z M (t) is continous, non-increasing on R ⋆ + and we have also lim t→+∞ Z M (t) = 0 (because 0 ∈ Spec(M, g) in the closed case). With the notation

0 = λ 0 , m 1 = 1 < λ 1 , m 1 = 1 < λ 2 , m 2 < • • • < λ k , m k < • • •
we have also the expression :

Z M (t) = +∞ ∑ k=1 m k e -t λ k .
The main interest of the spectral partition function lies in the fact that this function Z M determine the spectrum of the manifold (M, g). Indeed, for any real number µ > 0 consider the function

e µt Z M (t) -e µt = +∞ ∑ k=0 m k e (µ-λ k )t -e µt = +∞ ∑ k=1 m k e (µ-λ k )t therefore, if µ < λ 1 then lim t→+∞ e µt Z M (t) -e µt = 0; else if µ = λ 1 then lim t→+∞ e µt Z M (t) - e µt = m 1 ; else if µ > λ 1 then lim t→+∞ e µt Z M (t) -e µt = +∞.
Thus λ 1 is the unique real number µ > 0 such that the function t → e µt Z M (t)e µt admits a finite limit as t tends to infinity. Consequently the function Z M determine the first non-null eigenvalue λ 1 . By induction, with the function

e µt Z M (t) - i-1 ∑ j=1 m j e (µ-λ j )t = +∞ ∑ j=i m j e (µ-λ j )t
it is obvious to check that the function Z M determine all eigenvalues λ k k ≥ 0. For finish, observe also that the function Z M is associated to the spectral density distribution :

D M (t) := +∞ ∑ k=0 δ λ k
here δ λ k denotes the Dirac mass distribution at the point λ k . Indeed, using the Fourier Transform map (in the sense of distributions) we have :

D M (t) := +∞ ∑ k=0 δ λ k = 1 √ 2π +∞ ∑ k=0 e -itλ k = Z M (it)
with the following Fourier Transform convention :

f (x) := 1 √ 2πh R f (t)e -ixt h dt.
3 A simple example of trace formula : the case of flat torus

The Poisson summation formula for a lattice

The simplest example of exact trace formula is the Poisson summation formula : let f ∈ S(R) and consider the function :

F(x) := ∑ k∈Z f (x + k).
Since the function f belongs to the Schwartz's space S(R) it is obvious that the serie F is absolutely convergent and is 1-periodic. Thanks to the Fourier's series theory we have the following equality

F(x) = ∑ n∈Z c n (F)e 2iπnx with c n (F) = 1 0 F(t)e -2iπnt dt.
Therefore we have

c n (F) = 1 0 ∑ k∈Z f (t + k) e -2iπnt dt = ∑ k∈Z 1 0 f (t + k)e -2iπnt dt = ∑ k∈Z k+1 k f (u)e -2iπnu du = f (n)
with the convention :

f (x) = +∞ -∞ f (t)e -2iπyt dt.
Consequently, for all x ∈ R, we get the Poisson summation formula :

∑ k∈Z f (x + k) = ∑ n∈Z f (n)e 2iπnx ;
in particular with x = 0 we obtain the usual classical form :

∑ k∈Z f (k) = ∑ ℓ∈Z f (ℓ).
More generally, let a := (a 1 , a 2 , ..., a n ) ∈ R n such that for all i, a i = 0; consider the associated lattice :

Γ := a 1 Z + a 2 Z + • • • + a n Z
(this is a sub-lattice in R n of rank n ≥ 1). The Poisson summation formula for this lattice reads :

∑ k∈Γ f (k) = 1 Vol(Γ) ∑ ℓ∈Γ ⋆ f (ℓ)
with the notation :

f (y) := R n f (t)e -2iπ y,t dt
and Γ ⋆ is the dual lattice of Γ :

Γ ⋆ := {x ∈ R n ; ∀y ∈ Γ, x, y ∈ Z} .
Since (a 1 , a 2 , ..., a n ) is a Z-basis of Γ, the dual lattice Γ ⋆ is given by the equality

Γ ⋆ = A -1 T Z n where A :=       a 1 0 • • • 0 0 a 2 0 . . . . . . 0 • • • 0 0 • • • 0 a n       . Consequently 1 a 1 , 1 a 2 , ..., 1 a n is Z-basis of the dual lattice Γ ⋆ ; it follows that Vol(Γ ⋆ ) = 1
Vol(Γ) . Moreover it is easy to show that Γ ⋆⋆ = Γ (we have not

Γ ⋆ = Γ).
A classical calculus of Fourier transform show that the Fourier transform of the function

f (x) = e -α x 2 ∈ S(R n )
where α > 0 is given for all y ∈ R n by :

f (y) = π α n 2 e -π
∑ x∈Γ ⋆ e -α x 2 = π n 2 Vol(Γ) α n 2 ∑ y∈Γ e -π 2 y 2 α (3.1) because Γ ⋆⋆ = Γ and Vol(Γ ⋆ ) = 1 Vol(Γ) .
In particular, with α = 4π 2 t, t > 0, we get the following formula :

∑ x∈Γ ⋆ e -4π 2 t x 2 = Vol(Γ) (4πt) n 2 ∑ y∈Γ e -y 2 4t . (3.2)

Trace formula on a flat torus

Now, let us interpret this last formula in term of spectral geometry datas. For this we need to compute the spectrum of the torus associated to the lattice Γ.

Start with the case of dimension one, i.e. the manifold the one dimensional torus T a of length a > 0, in others words the manifold id the circle of radius a. We will show briefly that spectrum of the Laplaciand 2 dx 2 with closed conditions on this circle is given by :

Spec(T a ) = 4π 2 n 2 a 2 , n ∈ Z with associated eigenfunctions : x → e n (x) = e 2iπnx a , n ∈ Z. Indeed, if the scalar number λ is in the spectrum of -d 2 dx 2 , there exists a non trivial function u ∈ L 2 (T a ) such that -u ′′ = λu.
Thanks to the L 2 Fourier's series theory the sequence of functions

x → e n (x) = e 2iπnx a n∈Z
is a Hilbert basis of L 2 (T a ), hence there exists an unique sequence (c n ) n∈Z ∈ C Z such that :

u = ∑ n∈Z c n e n .
So we can rewrite the differential equation -u ′′λu = 0 as :

∑ n∈Z c n 4π 2 n 2 a 2 -λ e n = 0.
Since u = 0, there exists n 0 ∈ Z such that c n 0 = 0, hence because (e n ) n∈Z is a basis we get :

-u ′′ -λu = 0 ⇒ λ = 4π 2 n 2 0 a 2 .
Thus we have the inclusion Spec(T a ) ⊂ 4π 2 n 2 a 2 , n ∈ Z . Conversely, it is clear that for any integer n ∈ Z we haved 2 dx 2 e n = 4π 2 n 2 a 2 e n , so finally we get

Spec(T a ) = 4π 2 n 2 a 2 , n ∈ Z . Now, let us come back to the lattice Γ := a 1 Z + a 2 Z + • • • a n Z.
The associated flat torus is the following quotient space :

T a := R n /Γ.
Using multidimensional Fourier's series theory, the sequence of functions :

x → e 2iπ ω,x ω∈Γ ⋆ = (x 1 , x 2 , . . . , x n ) → e 2iπk 1 x 1 a 1 + 2iπk 2 x 2 a 2 +•••+ 2iπknxn an (k 1 ,k 2 ,...,k n) ∈Z n
is a Hilbert basis of L 2 (T a ), so the same argument as in dimension one shows that :

Spec(T a ) = 4π 2 k 2 1 a 2 1 + k 2 2 a 2 2 + • • • + k 2 n a 2 n , (k 1 , k 2 , . . . , k n ) ∈ Z n .
In other words : the spectrum of the Laplacian (with closed conditions) on a flat torus R n /Γ is given by :

Spec(R n /Γ) = 4π 2 γ R n , γ * ∈ Γ * .
We will interpret this equality using the spectral function Z M and the lenght spectrum of the flat torus R n /Γ. In the one-dimensional case, the lenght spectrum of the circle T a is obviousely given by :

Σ = {|n| a, n ∈ Z} ,
and in the multidimensional case we have

Σ = n ∑ i=1 (k i a 1 ) 2 , (k 1 , k 2 , ..., k n ) ∈ Z .
Since (see section 3.2) we have the equality :

∑ x∈Γ ⋆ e -4π 2 t x 2 = Vol(Γ) (4πt) n 2 ∑ y∈Γ e -y 2 4t
and because

Z Γ (t) = ∑ x∈Γ ⋆ e -4π 2 t x 2
holds for all t ≥ 0, we obtain the following trace formula :

Z Γ (t) = ∑ λ∈Spec(Γ) e -λt = Vol(Γ) (4πt) n 2 ∑ ℓ∈Σ e -ℓ 2 4t . (3.3)
In the previous equality on the right-hand side we have the geometrical informations of the manifold : dimension, volume,... and on the left-hand side we have spectral informations. This formula have a very important application : The Milnor counter-example of isospectrality.

A fundamental application : Milnor's counterexample of isospectrality

In 1966, M. Kac [Kac] in his famous article "Can one hear the shape of a drum?" investigate the following question : given a Riemannian manifold (the membrane of a drum), is the spectrum of ∆ g (the harmonics of the drum) determine geometrically, up to an isometry, the manifold (M, g) ? The exact formulation of isospectrality question is : if two Riemannian manifolds (M, g) and (M ′ , g ′ ) are isospectral, are they isometric ? This question may be traced back to H. Weyl in 1911-1912 and became popularized thanks to M. Kac's article. Before 1964, this problem was stay quite mysterious and J. Milnor [Mil] give the first counter-example : a pair of 16-dimensional flat torus which are isospectral and nonisometric. The proof of Milnor is based on the previous trace formula.

Theorem 3.1. [START_REF] Milnor | Eingenvalues of the Laplace operator on certain manifolds[END_REF]. There exists two lattices Γ and Γ ′ of R 16 such that the associated tori T 16 (Γ) and T 16 (Γ ′ ) are isospectral but not isometric.

Indeed, let us consider the lattice Γ(n) generated by the lattice γ n ⊂ Z n defined by :

γ n := (x 1 , x 2 , . . . , x n ) ∈ Z n ; n ∑ j=1 x j ∈ 2Z
with n = 8 or n = 16. One checks immedialely that :

Γ ⋆ (n) = Γ(n); in conse- quence the volume of Γ(n) is equal to 1.
The first step of the proof consist to show with basic arguments that the lattices Γ( 16) and Γ(8) ⊕ Γ(8) are non-isometric. Then, it follows immediately that their associated tori are also non-isometric. The second step of this proof is based on trace formula. In this way, we introduce the lattice Γ t (n) := √ tΓ(n) with t > 0 a parameter. We have :

Γ ⋆ t (n) = 1 √ t Γ ⋆ (n) = 1 √ t Γ(n).
On one hand, we have :

∑ x∈Γ(n) e -πt x 2 = ∑ x∈Γ t (n) e -π x 2
and, on the other hand, with the formula (3.1)

∑ x∈Γ t (n) e -π x 2 = 1 Vol (Γ t (n)) ∑ ℓ∈Γ ⋆ t (n) e -π ℓ 2 = 1 t n 2 Vol(Γ(n)) ∑ ℓ∈Γ ⋆ t (n) e -π ℓ 2 = 1 t n 2 Vol(Γ(n)) ∑ ℓ∈Γ(n) e -π ℓ 2 t .
In other words, if we introduce the following theta function :

θ Γ (t) := ∑ x∈Γ e -πt x 2 ;
then, we obtain for all t > 0

θ Γ(n) (t) = 1 t n 2 θ Γ(n) 1 t .
Moreover, observe that for all t > 0

Z Γ(n) t 4π = ∑ λ∈Spec(Γ(n)) e -λ t 4π = 1 t n 2 ∑ y∈Γ(n) e -π y 2 t = 1 t n 2 θ Γ(n) 1 t ;
therefore, finally, for all t > 0, we have

Z Γ(n) t 4π = 1 t n 2 θ Γ(n) 1 t = θ Γ(n) (t).
The last part of the proof may be summarized in : first, to observe that the theta function

z → θ Γ(n) (z) - 1 z n 2 θ Γ(n) 1
z admits a holomorphic expansion on {z ∈ C; Re(z) > 0}, consequently, for all z ∈ C; Re(z) > 0 we have

θ Γ(n) (z) = 1 z n 2 θ Γ(n) 1 z .
And the last part arguments of the proof consists to show that θ Γ( 16) and θ Γ( 8)⊕Γ( 8) are modular forms of weight equal to 4 such that

θ Γ(16) (∞) = θ Γ(8)⊕Γ(8) (∞) = 1,
and it is possible to conclude that

θ Γ(16) = θ Γ(8)⊕Γ(8) .
Finally, the tori Γ( 16) and Γ(8) ⊕ Γ( 8

Heat kernel and trace formulas

In this section we present the fundamental notion of heat kernel and one of the most important result in geometry and topology : the Minakshisundaram-Pleijel expansion. Let (M, g) be a closed Riemannian manifold and let us denotes by (λ k ) k≥0 the spectrum of (M, g).

The heat kernel

The Cauchy problem for the heat equation on a closed Riemannian manifold (M, g) is the following equation : find a function u :

M × R * + → R such that :      ∂u ∂t (x, t) = ∆ g u(x, t) u(x, 0) = u 0 (x) ∀x ∈ M.
where u 0 is a given smooth function on M. Physically, for every point x ∈ M the number u 0 (x) is an initial temperature data on the manifold M and the solution (x, t) → u(x, t) describe the evolution of the temperature for the point x ∈ M with the time t.

The first way to solve this equation is to use the spectral theory of ∆ g and the bounded functional calculus of operator. Indeed, we can consider the bounded one-parameter semi-group {U(t)} t≥0 defined for all t ≥ 0 by

U(t) := e t∆ g .
The generator of this semi-group is the operator ∆ g . One sees immedialely that the function u(x, t) = e t∆ g u 0 (x) is a solution of heat equation. By the bounded functional calculus, for any t ≥ 0 and for any integer k we also have

e t∆ g e k = e -tλ k e k
where (e k ) is a Hilbert basis of eigenvectors. So, for any initial condition u 0 = +∞ ∑ k=0 a k e k ∈ L 2 (M), we have :

u(x, t) = U(t)u 0 (x) = e t∆ g +∞ ∑ k=0 a k e k (x) = +∞ ∑ k=0
a k e -tλ k e k (x).

In conclusion, for all x ∈ M and for all t ≥ 0,

u(x, t) = +∞ ∑ k=0 u 0 , e k L 2 e -tλ k e k (x).
Another way to solve heat equation is to use the heat kernel of the manifold. The theory of heat kernel on Riemannian manifolds is a deep and a fundamental theory providing many powerful tools for understanding the relationship between geometry and diffusion processes. The diffusion operator related to the Laplacian on a compact Riemannian manifold is the heat operator. This operator is an operator acts on smooth functions and admits a fundamental solution which is called the heat kernel. Indeed, the heat kernel is the fundamental solution of the heat process diffusion equation ∂ ∂t u = ∆u. The heat kernel is a function ,y,t) such that for all y ∈ M the function (x, t) → E(x, y, t) is a solution of the heat equation and for all x ∈ M and every test function ϕ ∈ D(M) we have the initial data condition lim

E : (x, y, t) ∈ M × M × R * + -→ E(x
t→0 + M E(x, y, t)ϕ(y) dV g (y) = ϕ(x).
For a more complete introduction to heat kernel theory, see for example [Dav], [Ya-Sc], [BGV], [Gri], [Dod] and for some application see also the article [SaC].

Definition 4.1. The heat kernel (or fundamental solution) of the heat equation on a closed Riemannian manifold (M, g) is a function :

E :    M × M × R * + -→ R (x, y, t) -→ E(x, y, t)
such that : (i) E is C 0 in the three variables (x, y, t); E is C 2 in the second variable y and C 1 in the third variable t.

(ii) For all (x, y, t) ∈ M 2 × R * + we have

∂E ∂t (x, y, t) = ∆ g,y E(x, y, t),
where ∆ g,y is the Laplacian for the second variable y; in other words, for all fixed x ∈ M, the function (y, t) → E(x, y, t) is a solution of the heat equation.

(iii) For all x ∈ M and for all test function ϕ ∈ D(M)

lim t→0 + M E(x, y, t)ϕ(y) dV g (y) = ϕ(x).
Since the manifold is compact, we have existence and uniqueness of a such function E (for the construction see [BGM], [Dod] or [Cha2]). We call this solution the heat kernel of M.

Example 4.2. In the Eucliden case M = R n with g = can, heat kernel is given by the expression

E(x, y, t) = 1 (4πt) n 2 e -x-y 2 4t ,
This exemple is very important because it can be used to it for construct the heat kernel in the general case.

Let us introduce the one parameter family integral operator :

P t :    L 2 (M) → L 2 (M) ϕ → M E(x, y, t)ϕ(y) dV g (y)
where t > 0 is the time parameter, in other words for all x ∈ M we have

P t (ϕ)(x) = M E(x, y, t)ϕ(y) dV g (y).
For all t > 0, the associated integral kernel is (x, y) → E(x, y, t). An important fact is that for all ϕ ∈ L 2 (M), the function P t (ϕ) is a solution of the heat equation. As it turns out, the family {P t } t≥0 is a non-negative semi-group of bounded operator and this semi-group coincide with U(t) := e t∆ g . Now, let us focus on the link between the heat kernel and the spectrum of the Laplacian. First, observe that we have the fundamental equality :

E(x, y, t) = +∞ ∑ k=0 e -tλ k e k (x)e k (y),
in the sense that the serie converges uniformly and absolutely on M × M × [ε, +∞[ and is limit is equal to E(x, y, t). Indeed, let us consider the function On the other hand,

E : M 2 × R + → R defined by E (x,
∂E ∂t (x, y, t) = +∞ ∑ k=0 -λ k e -tλ k e k (x)e k (y)
hence for all (x, y, t) ∈ M 2 × R * + we obtain the equality

∂E ∂t (x, y, t) = ∆ g,y E (x, y, t).
Moreover, for all ϕ ∈ D(M) we have

M E (x, y, t)ϕ(y) dV g (y) = +∞ ∑ k=0 e -tλ k e k (x) M e k (y)ϕ(y) dV g (y) = +∞ ∑ k=0 e -tλ k e k (x) ϕ, e k L 2 whence lim t→0 + M E (x, y, t)ϕ(y) dV g (y) = +∞ ∑ k=0 e k (x) ϕ, e k L 2 = ϕ(x).
Hence, E is a fundamental solution of heat equation which can be expressed as

E(x, y, t) := +∞ ∑ k=0 e -tλ k e k (x)e k (y).

The spectral partition function Z M as a trace

Next, let us examine what happens with the heat kernel on the diagonal y = x. For a fixed ε > 0, for all t ≥ ε and for all x ∈ M the serie

(x, t) -→ ∑ k≥1 e -tλ k e 2 k (x)
is convergent and its limit is equal to E(x, x, t). Consequently, for all t ∈ [ε, +∞[

M E(x, x, t) dV g (x) = M +∞ ∑ k=1 e -tλ k e 2 k (x) dV g (x) = +∞ ∑ k=1 e -tλ k M e 2 k (x) dV g (x) = +∞ ∑ k=1 e -tλ k e k 2 L 2 = +∞ ∑ k=1 e -tλ k .
So, we can define the trace of the operator e t∆ g by

Z M (t) := trace e t∆ g = M E(x, x, t) dV g (x) = +∞ ∑ k=1 e -tλ k . (4.1)
We conclude that the serie of functions t → +∞ ∑ k=1 e -tλ k is uniformly and absolutely convergent on [ε, +∞[.

From Minakshisundaram-Pleijel expansion ...

A very important corollary of the of heat kernel construction is the Minakshisundaram-Pleijel expansion:

Theorem 4.3. (Minakshisundaram-Pleijel expansion). Let (M, g) be a compact Riemannian manifold of dimension n, and let us denote by E the heat kernel of (M, g).

We have the expansion

E(x, x, t) ∼ t→0 + 1 (4πt) n 2 u 0 (x, x) + u 1 (x, x)t + • • • + u k (x, x)t k + . . .
where the functions x → u k (x, x) are smooths on M and depend only on the curvature tensor and its covariant derivatives.

If for any integer k, we denote

a k := M u k (x, x) dV g (x),
then we obtain the nice equality

Z M (t) = +∞ ∑ k=1 e -tλ k ∼ t→0 + 1 (4πt) n 2 a 0 + a 1 t + • • • + a k t k + • • • (4.2)
i.e., for any integer N > 0 and for all t > 0,

Z M (t) = 1 (4πt) n 2 N ∑ k=0 a k t k + O t N-n 2 +1 .
The computation of the coefficients a k is difficult. Nevertheless, we have :

a 0 = Vol(M, g)
and

a 1 = 1 6 M Scal g dV g .
The expression of the term a 2 is also know:

a 2 = 1 360 M 2 |R| 2 -2 |Ric| 2 + 5Scal 2 g dV g ,
where R is the Riemann curvature tensor. The others coefficients a k for k ≥ 2 are very complicated to compute and they are quite futile, more details can be found in the books [BGM], [Ber]. Note that, in the case of surfaces (i.e.dim(M) = 2) the Gauss-Bonnet formula yields

a 1 = π 3 χ(M)
where χ(M) is the Euler-Poincaré characteristic of the surface M. In consequence for any closed surface we have :

Z M (t) = Vol(M, g) 4πt + χ(M) 12 + πt 60 M K 2 dV g +P(t)t 2
where t → P(t) is a bounded function on M.

A fondamental corollary of the Minakshisundaram-Pleijel expansion is the following fact: if we know the spectrum of (M, g), then in particular we know

• the dimension of the manifold;

• the volume of the manifold;

• the integral of the scalar curvature Scal g over the manifold.

More precisely :

Corollary 4.4. If two Riemannian manifolds (M, g) and (M ′ , g ′ ) are isospectral then : (i) the dimensions of M and M ′ are equal.

(ii) The volumes of (M, g) and (M ′ , g ′ ) are equal.

(iii) The integrals of the scalar curvature over (M, g) and (M ′ , g ′ ) are equal. In particular, if M and M ′ are surfaces, then χ(M) = χ(M ′ ).

In the case of surfaces, the dimension and the Euler-Poincaré characteristic are topological invariants.

... to Weyl asymptotic formula

To finish, let us present a very famous corollary : that is the Weyl asymptotic formula. This formula is based on the Karamata Tauberian Theorem, this theorem reads that for any positive measure µ on R + if +∞ 0 e -tx dµ(x) ∼ t→0 a t α holds for any α, a > 0, then

λ 0 dµ(x) = µ ([0, λ]) ∼ λ→∞ a Γ(α + 1) λ α .
Let us apply this result to the following positive measure µ :

= +∞ ∑ k=1 δ λ k , since Z M (t) ∼ t→0 Vol(M, g) (4πt) n 2
, and using the Karamata Tauberian theorem, we obtain

λ 0 dµ(x) ∼ λ→∞ Vol(M, g) (4π) n 2 1 Γ n 2 + 1 λ n 2 = B n Vol(M, g) (2π) n λ n 2 with B n := π n 2 Γ( n 2 +1)
. And, on the other hand, the counting eigenvalues function satisfy :

# ({k ∈ N, λ k ≤ λ}) = λ 0 dµ(x);
consequently, we deduce the : Theorem 4.5. (Weyl asymptotic formula). Let (M, g) be a compact Riemannian manifold of dimension n, and let us denote by (λ k ) k≥0 the eigenvalues of -∆ g on (M, g). Then

# ({k ∈ N, λ k ≤ λ}) ∼ λ→+∞ B n Vol(M, g) (2π) n λ n 2 where B n := π n 2 Γ( n 2 +1)
is the volume of the closed unit ball in (R n , can).

An obvious consequence of this asymptotics is the equivalent one

λ k ∼ k→+∞ (2π) n B n Vol(M, g) 2 n k 2 n .

Spectral zeta function and uniformization theorem

In this subsection we present the relationship between the eigenvalues of a closed surface and it associated conformal geometry. For more details on this amazing subject see for example [Cha] or [Gur].

The spectral zeta function associed to a surface (M, g 0 ) is given by the formula :

ζ g 0 (s) := +∞ ∑ n=1 1 λ s n
where s ∈ C such that Re(s) > 1. For all s ∈ C such that Re(s) > 1 we have : e -t t s-1 dt converge for all s ∈ C such that Re(s) > 0, moreover for all s ∈ C such that Re(s) > 0 we have the relation :

ζ g 0 (s) = +∞ ∑ n=1 λ -s n = +∞ ∑
1 Γ(s) = lim n→+∞ n!n s s(s + 1)(s + 2) • • • (s + n)
the function Γ admit a meromorphic continuation to C -{0, -1, -2, ...} with simple poles at s = 0, -1, -2, ... . By changing variables we have for all x > 0 1

x s = 1 Γ(s) +∞ 0 e -xt t s-1 dt
therefore, for all s ∈ C such that Re(s) > 1 we get

ζ g 0 (s) = +∞ ∑ n=1 1 Γ(s) +∞ 0 e -λ n t t s-1 dt = 1 Γ(s) +∞ 0 +∞ ∑ n=1 e -λ n t t s-1 dt = 1 Γ(s) +∞ 0 (Z M (t) -1) t s-1 dt because λ 0 = 0 thus ζ g 0 (s) = 1 Γ(s) 1 0 (Z M (t) -1) t s-1 dt + 1 Γ(s) +∞ 1 (Z M (t) -1) t s-1 dt.
Start by studying the first integral : using the remark 7.6.2, we have for all

s ∈ C such that Re(s) > 1 1 Γ(s) 1 0 (Z M (t) -1) t s-1 dt = 1 Γ(s) 1 0 Vol(M, g) 4πt + χ(M) 12 + πt 60 M K 2 dV g +P(t)t 2 -1 t s-1 dt = 1 Γ(s) 1 0 Vol(M, g) 4πt + χ(M) 12 + πt 60 M K 2 dV g -1 t s-1 dt + 1 Γ(s) 1 0 P(t)t s+1 dt. The function s → 1 Γ(s) 1 0 P(t)t s+1 dt is holomorphic for Re(s) > -2 and 1 Γ(s) 1 0 Vol(M, g) 4πt + χ(M) 12 + πt 60 M K 2 dV g -1 t s-1 dt = 1 Γ(s) t s-1 s -1 Vol(M, g) 4π + χ(M)t s 12s + πt s+1 60(s + 1) M K 2 dV g - t s s t=1 t=0
this quantity converges for all s ∈ C such that Re(s) > 1 and has a meromorphic continuation to C with a simple pole in s = 1 (because

Γ(s) = lim n→+∞ s(s+1)(s+2)•••(s+n) n!n s
).

For finish : let us study the second integral : for all s ∈ C such that Re(s

) > 1 1 Γ(s) +∞ 1 (Z M (t) -1) t s-1 dt = 1 Γ(s) +∞ 1 +∞ ∑ n=1 e -λ n t t s-1 dt
and for all t ≥ 1, e -λ n t = e -λ 1 t e (λ 1 -λ n )t , so since the eigenvalues λ n grows like n as n tends to infinity (Weyl asymptotic formula) and since 0

< λ 1 ≤ λ 2 • • • ≤ λ n for t large enough we have +∞ ∑ n=1 e -λ n t ≤ Ce -λ 1 t
consequently using the holomorphic theorem under integral sign we get easily that the function s → 1

Γ(s) +∞ 1 +∞ ∑ n=1
e -λ n t t s-1 dt is holomorphic on C. Finally the function ζ g 0 has a meromorphic expansion to C with a simple pole in s = 1, in particular ζ g 0 is analytic in 0, hence the quantity ζ ′ g 0 (0) exist and

ζ ′ g 0 (0) = lim s→0 ζ g 0 (s) -ζ g 0 (0) s -0 .
Next, the derivative of

ζ g 0 is ζ ′ g 0 (s) = +∞ ∑ i=1 -ln(λ i ) 1 λ s i hence for s = 0 we get ζ ′ g 0 (0) = - +∞ ∑ i=1 ln(λ i ) = -ln n ∏ i=1 λ i
therefore we obtain :

e -ζ ′ g 0 (0) = +∞ ∏ i=1 λ i = det -∆ g 0 .
Thus we can define the determinant of ∆ g 0 by the formula : 0) . Now, let g ∈ [g 0 ], so there exists u ∈ C ∞ (M) such that g = e 2u g 0 , the Polyakov formula (see for example [Cha]) is :

det -∆ g 0 = e -ζ ′ g 0 (
ln det -∆ g det -∆ g 0 = - 1 12π M ∇u 2 + K g 0 u dV g 0
we can consider the following functional :

F :        C ∞ (M) → R u → ln det(-∆ g) det(-∆ g 0 ) .
Since the determinant is not a scale invariant we consider the normalized functional determinant S :

       C ∞ (M) → R u → -12π ln det(-∆ g) det(-∆ g 0 ) + 2πχ(M) ln (Vol(M, g))
where χ(M) is the Euler's characteristic of M. The critical functions of S are functions u such that

-∆ g 0 u + K g 0 = ce 2u
where c is a real constant. Using Gaussian curvature equation we have : u is a critical function of S if and only if the surface M with the conformal metric g := e 2u g 0 admits a constant Gaussian curvature K g . In 1998, B. Osgood, R. Phillips and P. Sarnak S(u) of the previous functional exists and it is a maximum : there exists u 0 ∈ C ∞ (M) such that Gaussian curvature K g (where g := e 2u g 0 ) is constant on M.

It is turn out that each surface can be endowed with a metric conformally equivalent to a metric with constant Gaussian curvature. The main consequence is the famous Poincaré-Klein-Koebe uniformization theorem : Theorem 4.7. (Uniformization theorem). Let (M, g 0 ) be a closed surface with a Riemannian metric g 0 . Then there exists an unique metric 1 g conformal to g 0 on M with a constant Gaussian curvature K g ∈ {+1, 0, -1}.

The values of the constant K g ∈ {+1, 0, -1} depends on the topology of the surface M. Recall that the Gauss-Bonnet formula for a closed surface is :

M K dV g = 2πχ(M)
and for a surface with boundary

M K dV g + M K g dS g = 2πχ(M)
where χ(M) is the Euler-Poincaré characteristic of M and K g is the geodesic curvature on the boundary. According to the Gauss-Bonnet formula the sign of the curvature is determined by the Euler-Poincaré characteristic 2 χ(M) of M. Therefore the universal covering of every closed surface can be isometrically embedded onto :

1. the round sphere S 2 for genus zero ; χ(M) = 2 > 0, and in this case K g ≡ +1.

2. The Euclidian plane R 2 for genus one : χ(M) = 0, in this case K g ≡ 0.

3. The hyperbolic space H 2 for genus g ≥ 2:

χ(M) = -2 < 0, in this case K g ≡ -1.
1 called a uniformization metric.

2 For M a closed orientable surface χ(M) = 2 -2g where g is the genus of the surface.

Trace formulas on hyperbolic manifolds

In this section we investigate trace formulas on closed hyperbolic manifolds, in particular the case of surfaces.

Lenght spectrum versus spectrum

The lenght spectrum of a Riemannian manifold (M, g) is the set of lengths of closed geodesics on (M, g) counted with multiplicites (the multiplicity of a lenght is the number of free homotopy classes of closed curves containing a geodesic of the given lenght). In 1973, Y. Colin de Verdière [Col1], [START_REF] De Verdière | Spectre du Laplacien et longueurs des géodésiques périodiques II[END_REF] showed that, in the compact case, up to a generic hypothesis (which it is always satisfied in the case of negative sectional curvature), the spectrum of the Laplacian determines the lenght spectrum. Colin de Verdière's proof is based on trace formulas. More precisely, in the first article [Col1], the author focus on closed hyperbolic surface and he use the heat kernel of the surface via the following trace formula (see formula 4.2) :

+∞ ∑ k=0 e -λ k (M) z = M E x, x, 1 z dV g (x)
with a complex time t = 1 z ∈ C, and he used an approximation of the heat kernel E x, x,

+ O + (1)
with u γ are real numbers such that u γ > 0 and O + (1) is a bounded function on the set of complex numbers z ∈ C such that Re(z) ≥ ξ 0 > 0.

One consequence of this result is :

Corollary 5.2. For any compact hyperbolic surface M with a constant sectional curvature : K ≡ -1, the spectrum of the surface M determine the lenght spectrum of M.

In the second article [Col2], Y. Colin de Verdière gave a generalisation in the case of hyperbolic manifolds of dimension n ≥ 2 with a constant sectional curvature : K ≡ -1, for more recent approach and details see also [START_REF] De Verdière | Spectrum of the Laplace operator and periodic geodesics : thirty years after[END_REF].

(M,g), K<0

Fig. 2. A closed hyperbolic surface with some periodic geodesics.

The Selberg formula for hyperbolic surface

Now let us present some points from the fascinating story of Selberg formula on hyperbolic surfaces. Main application of this formula concerning dynamical system, in particular for understand trajectories and flows on negatively curved surfaces. The proof of this formula is based on many technical computations (see [Sel], [Mar] or [Vor]). The statement of this formula is the following : let (M, g) be a compact hyperbolic surface, and consider a test function h : C → C such that :

• h is analytic for z such that |Im(z)| ≤ σ with σ > 1 2 ;

• h is even;

• for all z in the strip |Im(z

)| ≤ σ, we have h(z) = O 1 (1+|Re(z)|) N .
Under this previous hypothesis, the Selberg formula read :

Theorem 5.3. (Selberg Formula). Using the previous hypothesis, we have : The first application of this formula is to a Weyl law for surface :

+∞ ∑ k=0 h λ k - 1 4 = Vol(Γ) 4π
# ({k ∈ N, λ k ≤ λ}) ∼ λ→+∞ Vol(Γ) 4π λ.

The zeta Selberg function

One other important application of Selberg formula concerning the density of closed geodesics on a compact hyperbolic surface : let us introduce the following counting function :

Π(L) := # ({γ ∈ Γ, |γ| ≤ L}) .
One's important hope is to understand this function, and more precisely to give a precise estimate of Π(L) for L → +∞. In the goal to estimate the function Π(L) for L → +∞ in the case of a compact hyperbolic surface M with a constant sectional curvature : We have an important analogy with the zeta Riemann function. Before to study the function ζ Selberg , we briefly recall some classical elements about the classical Riemann zeta function. The Riemann zeta function is defined for all s ∈ C,Re(s) > 1 by the serie :

K ≡ -1, A. Selberg,
ζ(s) := +∞ ∑ n=1 1 n s .
This function admits a holomorphic expansion to C -{1}. Moreover, for all s ∈ C such that Re(s) > 1 we have we have also the Euler prodcut :

ζ(s) = ∏ p∈P 1 1 -e -s ln p
here P is the set of prime numbers. For all x > 0, we define the counting prime number functions π(x) := # ({p ∈ P, p ≤ x}) . The statement of the prime number theorem is : for x → +∞ we have

π(x) = ℓ i (x) (1 + O(1))
where ℓ i is the log-integral function :

ℓ i (x) := x 2 du ln u
. The actual description of zeros for ζ is the following :

• we have trivial zeros : {-2k, k ≥ 1};

• there exists no zeros in the half plane Re(z) ≥ 1;

• and there exists an infinity of non-trivial zeros on the vertical line Re(z) =

{-n, n ≥ 0} 1 2 ± 1 4 -λ k , k ≥ 0
where (λ k ) k≥0 are the eigenvalues of the closed surface M. (iv) Moreover, for all s ∈ C, we have the relation of duplication :

ζ Selberg (1 -s) = ζ Selberg (s) exp 2(g -1) s-1 2 0
πx tan(πx) dx here g ≥ 2 is the genus of the surface M.

Let us examine the item (iii) : we have two types of zeros :

• trivial zeros : -n, n ≥ 0.

• and non-trivial zeros :

s k := 1 2 ± 1 4 -λ k , k ≥ 0. Since λ 0 = 0 < λ 1 ≤ λ 2 ≤ • • • ≤ λ k ≤ • • • with λ k → +∞ as k → +∞, it is obvious that there exists an integer N > 0 such that λ 0 < λ 1 ≤ λ 2 ≤ • • • ≤ λ N ≤ 1 4 < λ N+1
thus, for all integer k ∈ {0, ..., N}

s k = 1 2 ± 1 4 -λ k ∈ [0, 1];
and for all integer k ≥ N + 1

s k = 1 2 ± 1 4 -λ k ∈ z ∈ C, Re(z) = 1 2 .
Observe that since λ 0 = 0 then 0 and 1 are zeros of ζ Selberg . Moreover, the repartition of the real zeros {s k , 0 ≤ k ≤ 1} are symmetric with respect to the vertical line Re(z) = 1 2 ; and the complex zeros {s k , k ≥ N + 1} are symmetric with respect to the horizontal axe Re(z). For summarize : without a finite number, all non trivial zeros of the zeta function ζ Selberg are localised on the line Re(z) = 1 2 . In some sense the Riemann Hypothesis is true on every compact hyperbolic surface with a constant sectional curvature. A consequence of the previous theorem 5.4 is concerning the counting function Π(L) : Theorem 5.5. There exists ε > 0 such that :

× × × × × × × × × × × × × × × × × × × × × × × × × Re(z)=1/2 -1 -3 -2 1 Re(z) Im(z)
Π(L) = # ({γ ∈ Γ, |γ| ≤ L}) = ℓ i e L + O L→+∞ 1 (1-ε)L where L i is the log-integral function L i (x) := x 2 du ln u .
This theorem is a called prime geodesics theorem. In particular, we deduce the equivalent :

# ({γ ∈ Γ, |γ| ≤ L}) ∼ L→+∞ e L L .
Note that again the analogy with the Riemann zeta function ζ and the prime number theorem. The definition of the function ζ Selberg can be define to any compact Riemannian manifold and the problem of counting closed geodesics can be generalized to counting orbit associated to a flow, indeed the case of geodesics appears as trajectories associated to the dynamics of classical Hamiltonian flow of H = -∆ g (geodesic flow) in the phase plane, identified with the cotangent bundle T ⋆ M.

In 1967, in a fundamenal paper, S. Smale [Sma] has the idea to use the function ζ Selberg in a more general context : the case of of Anosov flows. Note that a geodesic flow on a compact manifold with a negative curvature is a particular case of Anosov flow. Unfortunately, since the crux of the proof for theorem 5.4 concerning the zeta function ζ Selberg is based on a trace formula for the heat kernel of a surface with a constant sectional curvature, then it was not clear that this function was very usefull for understand general Anosov flows on manifold with no constant curavture. More precisely, S. Smale ask himself the question "Does the item (i) of the theorem 5.4 is always true for Anosov Flow ? Or on a compact hyperbolic surface with a non constant sectional curvature K < 0 ? : The author wrote : "Does ζ Selberg have a meromorphic continuation to all C ? An affirmative answer would be roughly necessary and sufficient condition for ζ Selberg to be useful. I must admit a positive answer would be a little shocking!"

More recent results

Later, in 2013, P. Giuletti, C. Liverani, M. Policott [GLP], and in 2016, S. Dyatlov and M. Zworski [Dy-Zw] shows thats the items (i) and (ii) on the function ζ Selberg holds for C ∞ Anosov flows (note that a geodesic flow on a compact manifold with a negative curvature is a Anosov flow). But what happens for item (iii) ? Let us present a very recent result in this way. We have seen that the fundamental hypothesis of the works of Selberg is that the sectional curvature of the surface is constant equal to -1. In the case of hyperbolic surface with a variable sectional curvature K < 0, the situation is quite delicate ... As it turns out the "good" zeta function in the case of variable curvature is the following Semi-Classical zeta function (for a details on this fact see [Fa-Ts2]) : The huge paper [Fa-Ts2] is a very deep and technical article; it is no reasonable to explain here the mains arguments of the main theorem. However, let us just mention that the authors use several complex techniques of semi-classical analysis. In particular they consider and study spectral properties and trace formulas applied to some transfer operators, theses operators are associated to a "prequantum" Anosov flow.

  ) are isospectral (because theta function determine the spectrum). As it turns out, in 1984 and 1985, respectively C. Gordon, E.N. Wilson [Go-Wi] and T. Sunada [Sun] gave a systematic construction for counter-example. In 1992, C. Gordon, D. Webb and S. Wolpert [GWW1], [GWW2] gave the first planar counter-example. See also the articles of P. Bérard [Bér1], [Bér2], [Bér3].

  k e k (x)e k (y). We have∆ g,y E (x, y, t) = +∞ ∑ k=0 e -tλ k e k (x)∆ g,y e k (y) = +∞ ∑ k=0-λ k e -tλ k e k (x)e k (y).

  the usual Riemman function which it is convergent for all s ∈ C such that Re(s) > 1. So the serie ζ g 0 (s) and convergent for all s ∈ C such that Re(s) > 1. Now we shall prove that the function ζ g 0 admit a meromorphic continuation to C with a simple pole at s = 1 : first recall that Γ the Euler Gamma function Γ(s) := +∞ 0

Fig. 1 .

 1 Fig. 1. Three geometries on closed surfaces.

  the set of prime closed oriented geodesics of the surface M. In the formula we use the convention :f (y) := +∞ -∞ f (t)e -2iπyt dt.

  in a famous paper of 1956 [Sel] introduce a special zeta function : s ∈ C → ζ Selberg (s) for understand the dynamic of geodesics on this surface. This function zeta is defined for all s ∈ C by the double product formula : ζ Selberg (s) := ∏ γ∈Γ +∞ ∏ k=0 1e -(s+k)|γ| . An another formulation of ζ Selberg is done by : ζ Selberg (s) = exp -

  . and we have the famous Riemann hypothesis : all non-trivials zeros are on the vertical line Re(z) = 1 2 ? Now, we come back to the function ζ Selberg , main results concerning this function are summarized in : Theorem 5.4. (Selberg, 1956). Let M be a compact hyperbolic surface M with a constant sectional curvature : K ≡ -1 and let us denotes by Γ the set of prime closed oriented geodesics of M. Then : (i) The previous function ζ Selberg is absolutely convergent for s such that Re(s) > 1. (ii) This function admits an analytic extansion to C. (iii) The zeros of the previous extansion on C are :

Fig. 3 .

 3 Fig. 3. The zeros of zeta function ζ Selberg in the complex plane.

Fig. 4 .

 4 Fig. 4. Poles and zeros of semi-classical zeta function ζ SC in the complex plane. In the article [Fa-Ts2], the authors gave also the following exciting open question : Conjecture 5.7. For all ε > 0, without a finite number, all zeros of the function ζ SC are in {z ∈ C, |Re(z)| ≤ ε} {z ∈ C, |Im(z)| ≤ C} here C is a constant.

  Let us denote by L the set of free homotopy classes of M, and if a belongs to L then |a| denote the lenght of the unique periodic geodesic of a. Then, for any z ∈ C such that Re(z) ≥ ξ 0 > 0 we have :

	1 z . The main result of [Col1] is : Theorem 5.1. (Colin de Verdière, 1973). Let M be a compact hyperbolic surface with a constant sectional curvature : K ≡ -1. +∞ ∑ k=0 e -λ k z = z 4π vol(M) + z a∈L, γ =e 4π +∞ ∑ u γ e -z|a| 2 4