Audrey Abi Akle

Stéphanie Minel

Audrey Abi

Bernard Yannou

DESIGN SPACE VISUALIZATION FOR EFFICIENCY IN KNOWLEDGE DISCOVERY LEADING TO AN INFORMED DECISION

Keywords: Visualization, Computer aided design (CAD), Decision making

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Design space visualization for efficiency in knowledge discovery leading to an informed decision

INTRODUCTION

There is a paradigm where designers shop for the best solution. It is called Design by Shopping and was coined by Balling [START_REF] Balling | Design by shopping: A new paradigm?[END_REF]. Indeed, Balling noted that the traditional optimization-based design process to "formulate the design problem, obtain analysis models and execute an optimization algorithm" leaves designers unsatisfied. Designers like consumers want to "shop" to gain an insight into trades, feasible and impractical solutions, and to learn about their alternatives before making decisions. Design by Shopping, firstly, allows designers to explore the design space, and, secondly, optimize and choose an optimal solution from a set of possible designs, and then develop realistic expectations with regard to what is possible. One embodiment of this paradigm is the Design Space Exploration [START_REF] Simpson | Visual steering commands and test problems to support research in trade space exploration[END_REF]. With Design Space Exploration (DSE), designers sample thousand (and more) design points iteratively, explore the design space that is a multidimensional set of data, gain insights and knowledge about the problem and make design decision. In DSE the design decision is performed following the discovery and evaluation of the elements in the design space in order to identify optimal solution by reducing the design space toward an area of performance. Based on the work of [START_REF] Miller | Preference Construction, Sequential Decision Making, and Trade Space Exploration[END_REF] we identify that exploring the design space consists of three main phases:

(1) Discovery: acquire knowledge and understanding of the problem, (2) Narrowing: active pursuit of a design by eliminating sets, exploring limits, highlighting preferences, etc. and (3) Selection: check satisfaction (see Figure 1). Considering the knowledge discovery in DSE, we find one point that challenges us. Indeed, some authors refer to informed decision making (Sulllivan et al., 2001), [START_REF] Mavris | Systems design and modeling: A visual analytics approach[END_REF], [START_REF] Chandrasegaran | The evolution, challenges, and future of knowledge representation in product design systems[END_REF]. From our literature review dealing with this decision type (see section 2); we particularly identify a need in information visualization for decision-makers. Graphs are useful for quickly visualizing the feasible solutions as opposed to impractical solutions, as well as those violating engineering constraints or client requirements. Three different situations can be represented with more or less data: representing the single vector of design parameters featuring the product solution, the single vector of solution performances for feasible solutions or two sets of design parameters and corresponding performances for feasible. There are already many works in visual design and it has already been shown that fast graphical design interfaces impact user performance in terms of design efficiency, design effectiveness and the design search process [START_REF] Ligetti | Assessing the impact of graphical design interfaces on design efficiency and effectiveness[END_REF]. We identify several tools for exploring the design space. For example, there are the ARL Trade Space Visualizer [START_REF] Stump | The ARL trade space visualizer: An engineering decision-making tool[END_REF], the VIDEO tool [START_REF] Kollat | A framework for visually interactive decision-making and design using evolutionary multi-objective optimization (VIDEO)[END_REF], the LIVE tool [START_REF] Yan | A Work-Centered Visual Analytics Model to Support Engineering Design with Interactive Visualization and Data-Mining[END_REF] and the Rave tool [START_REF] Daskilewicz | Rave: A computational framework to facilitate research in design decision support[END_REF]. Thus we find in these tools, various graphs: Scatter Plot Matrix, 2D or 3D Scatter Plot, Parallel Coordinate Plot, Bars Chart and Treemap. In addition, it has already been shown in a simplified framework that Parallel Coordinate Plot is the most suitable graph to selection in Design by Shopping [START_REF] Abi Akle | Graphical support adapted to designers for the selection of an optimal solution in design by shopping[END_REF][START_REF] Abi Akle | Information visualization for selection in Design by Shopping[END_REF]. It therefore appears both that knowledge discovery phase for insights gain and understanding of the design problem is a key element of the Design by Shopping. On the other hand, we observe that information visualization is an indispensable element to the practice of DSE. Thereby, our research is motivated by a question: What graph allows designers to be effective in the discovery phase and results in an informed decision?

We have thus identified three graphs useful for representing multidimensional sets of data (>3 variables) and with an unlimited number of design points: Simple Scatter Plot (SSP), Scatter Plot Matrix (SPM) and Parallel Coordinate Plot (PCP). We carried out experiments with 42 participants and designed a web platform with four design problem to solve. The platform allows to generate an unlimited number of design points (random or Pareto sampling), to reduce the design space with a range constraints controller, to visualize preferences, etc. in order to mimic the design activity. We identified a graph more suited to the discovery phase and to an informed decision making in design space exploration: the Scatter Plot Matrix (SPM).

INFORMED DECISION

A notion used in several disciplines seems to be of importance to be in a situation to make a decision of quality, this is an informed decision. In monitoring & supervision field, Ireson states that "The management of this mass of information is crucial in aiding the decision-making process, ensuring, as far as possible, that the responders have full situational awareness to make informed decisions" [START_REF] Ireson | Local community situational awareness during an emergency[END_REF]. With the same idea, [START_REF] Bass | Intrusion detection systems and multisensor data fusion[END_REF] and [START_REF] Riveiro | Visual analytics for the detection of anomalous maritime behavior[END_REF] indicate the need for "situational awareness" for the formulation of an informed decision. Bass adds the need to "fusing data into information and knowledge, so network operators can make informed decisions" [START_REF] Bass | Intrusion detection systems and multisensor data fusion[END_REF]. In business / marketing field, [START_REF] Lurie | Visual representation: Implications for decision making[END_REF] suggest managing a large data set and the use of visualization tools could lead toward an informed decision. [START_REF] Glaser | Sense of sensing: from data to informed decisions for the built environment[END_REF] link the informed decision to the process of analysing large amounts of data, "tracking" of performance and detecting patterns and trends. Information systems field informs us that "the making of informed decisions requires the application of a variety of knowledge to information" [START_REF] Wiederhold | Mediators in the architecture of future information systems[END_REF]. In the building field, making informed design Abi Akle A., Minel S., Yannou B., 2017. Design space visualization for efficiency in knowledge discovery leading to an informed decision, In 21st International Conference on Engineering Design (ICED), August 21-25, Vancouver, Canada.

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decision needs to manage a large amount of information on the detailed design options and properties and to operate simulations of their performance. For them, the designer needs a large design space and an overview of parameter changes consequences to gain a deep understanding of the performance and so make an informed design decision [START_REF] Petersen | Method and simulation program informed decisions in the early stages of building design[END_REF]. [START_REF] Russell | Visual representation of construction management data[END_REF] consider that "Visual analytics, the science of analytical reasoning facilitated by interactive visual interfaces, has the potential to improve the construction management process through the enhanced understanding of project status and reasons for it, better informed decision making" [START_REF] Russell | Visual representation of construction management data[END_REF]. In information visualization field, [START_REF] Keim | Challenges in visual data analysis[END_REF] indicate that for an informed decision, "it is indispensable to include humans in the data analysis process to combine flexibility, creativity, and background knowledge with the enormous storage capacity and the computational power of today's computers". Later, Keim et al. (2008a[START_REF] Keim | Visual analytics: Scope and challenges[END_REF] state that "visual analytics" is the system to make an informed decision. It guarantees the full support of the user in navigating and analysing the data, memorizing insights and making informed decisions. Also, we find that "a tight coupling between cognition, interaction and visual analytics is necessary to enable the user to make informed decisions" [START_REF] Meyer | From Visualization to Visually Enabled Reasoning[END_REF]. Finally, we find the term informed decision in the design engineering field. [START_REF] Wood | Engineering design calculations with fuzzy parameters[END_REF], in preliminary design, model and manipulate such uncertainties in a computer-assisted environment, under the hypothesis that doing so will allow the designer to make faster and more-informed decisions. [START_REF] Sullivan | The structure and value of modularity in software design[END_REF] show, in particular, to make informed decisions about the choice of design rules and clustering of design parameters, designers needed to know how changes in the environment would affect them. [START_REF] Chandrasegaran | The evolution, challenges, and future of knowledge representation in product design systems[END_REF] indicate that "an effective computer support tool that helps the designer make better-informed decisions requires efficient knowledge representation schemes". [START_REF] Mavris | Systems design and modeling: A visual analytics approach[END_REF] argue that the integration of "visual analytics" in the design process provides designers the ability to gain knowledge and insights needed with the justified means of making an informed decision. The visualization seems essential to facilitate the generation of hypotheses and the formulation of an informed decision. They point out the data, knowledge, and insight necessary for the formulation of informed decisions is generated throughout the design process [START_REF] Mavris | Systems design and modeling: A visual analytics approach[END_REF]. From the literature, we identify eight themes that appear to contribute to the formulation of an informed decision (see Table 1). Furthermore, we identify four of these themes that are most widely used by the authors and seem essential in the definition of an informed decision: "Knowledge and insights gain", "Visualization", "Analysis and treatment" et "Manipulation and management".

Table 1. Themes that appear to contribute to the formulation of an informed decision

Themes

Contributors Situational awareness [START_REF] Ireson | Local community situational awareness during an emergency[END_REF], [START_REF] Bass | Intrusion detection systems and multisensor data fusion[END_REF] Knowledge and insights gain [START_REF] Sullivan | The structure and value of modularity in software design[END_REF], [START_REF] Keim | Challenges in visual data analysis[END_REF](Keim et al., , 2008a[START_REF] Keim | Visual analytics: Scope and challenges[END_REF], [START_REF] Glaser | Sense of sensing: from data to informed decisions for the built environment[END_REF], [START_REF] Petersen | Method and simulation program informed decisions in the early stages of building design[END_REF], [START_REF] Mavris | Systems design and modeling: A visual analytics approach[END_REF] Visualization [START_REF] Chandrasegaran | The evolution, challenges, and future of knowledge representation in product design systems[END_REF], [START_REF] Lurie | Visual representation: Implications for decision making[END_REF], [START_REF] Bass | Intrusion detection systems and multisensor data fusion[END_REF], [START_REF] Mavris | Systems design and modeling: A visual analytics approach[END_REF], [START_REF] Keim | Visual analytics: Scope and challenges[END_REF], [START_REF] Russell | Visual representation of construction management data[END_REF], [START_REF] Meyer | From Visualization to Visually Enabled Reasoning[END_REF], [START_REF] Riveiro | Visual analytics for the detection of anomalous maritime behavior[END_REF]) Manipulation and management [START_REF] Ireson | Local community situational awareness during an emergency[END_REF], [START_REF] Chandrasegaran | The evolution, challenges, and future of knowledge representation in product design systems[END_REF], [START_REF] Lurie | Visual representation: Implications for decision making[END_REF], [START_REF] Wood | Engineering design calculations with fuzzy parameters[END_REF], [START_REF] Petersen | Method and simulation program informed decisions in the early stages of building design[END_REF], [START_REF] Mavris | Systems design and modeling: A visual analytics approach[END_REF], [START_REF] Riveiro | Visual analytics for the detection of anomalous maritime behavior[END_REF], [START_REF] Meyer | From Visualization to Visually Enabled Reasoning[END_REF] Analysis and treatment [START_REF] Keim | Challenges in visual data analysis[END_REF](Keim et al., , 2008a[START_REF] Keim | Visual analytics: Scope and challenges[END_REF], [START_REF] Russell | Visual representation of construction management data[END_REF], [START_REF] Meyer | From Visualization to Visually Enabled Reasoning[END_REF], [START_REF] Riveiro | Visual analytics for the detection of anomalous maritime behavior[END_REF], [START_REF] Mavris | Systems design and modeling: A visual analytics approach[END_REF] Understanding [START_REF] Petersen | Method and simulation program informed decisions in the early stages of building design[END_REF], [START_REF] Russell | Visual representation of construction management data[END_REF], [START_REF] Mavris | Systems design and modeling: A visual analytics approach[END_REF]) Human and cognition [START_REF] Keim | Challenges in visual data analysis[END_REF], [START_REF] Wood | Engineering design calculations with fuzzy parameters[END_REF], [START_REF] Mavris | Systems design and modeling: A visual analytics approach[END_REF], [START_REF] Meyer | From Visualization to Visually Enabled Reasoning[END_REF]) Transformation [START_REF] Chandrasegaran | The evolution, challenges, and future of knowledge representation in product design systems[END_REF], [START_REF] Bass | Intrusion detection systems and multisensor data fusion[END_REF], [START_REF] Keim | Visual analytics: Scope and challenges[END_REF] We know that, in engineering design, once the design has been formalized, a necessary design task is to make a selection from amongst candidate designs or parametric values [START_REF] Otto | The method of imprecision compared to utility theory for design selection problems[END_REF]. The main challenge lies in resolving the inherent trade-offs that exist between the overall system and subsystems, and between conflicting and competing objectives [START_REF] Abi Akle | Graphical support adapted to designers for the selection of an optimal solution in design by shopping[END_REF][START_REF] Abi Akle | Information visualization for selection in Design by Shopping[END_REF]. Thus we define an informed decision in Design by Shopping as the selection of a design point, among several others that will achieve optimal benefits and minimum inconvenience, following an iterative and interactive treatment and analysis process in which designers are gaining understanding, knowledge and insights with visualization and manipulation of large sets and / or data model.

GRAPHS ADAPTED

In our context of design space exploration in order to make an informed decision several graphs (design space representations) are available to us. So, we are in a case-representation of multidimensional sets of data with an unlimited number of alternatives (design points). Based on the work of [START_REF] Miettinen | Survey of methods to visualize alternatives in multiple criteria decision making problems[END_REF], [START_REF] Wegman | Hyperdimensional data analysis using parallel coordinates[END_REF] and [START_REF] Keim | Designing pixel-oriented visualization techniques: Theory and applications[END_REF] amongst others about graph characteristics, we identify the Simple Scatter Plot (SSP), the Scatter Plot Matrix (SPM) and the Parallel Coordinate Plot (PCP) (see Figure 2). Thus, we propose comparing these tree graphs under the ability to make an informed decision in design space exploration. Scatterplot is a conventional method to visualize the relationship between two variables. As a visual structure, Scatterplot uses the position to encode the values of two variables and their relationships. This is a projection of the data (represented by the design points) in a 2D space. [START_REF] Siirtola | Interactive visualization of multidimensional data[END_REF] considers the Scatterplot is useful for easily detect non-linear patterns and positive or negative correlations between variables. The Scatterplots are Cartesian representations and therefore have a long history with, for example, the learning of linear algebra at school. Due to this training, it results in a strong development of intuitions about the appearance of this type of representation [START_REF] Wegman | Hyperdimensional data analysis using parallel coordinates[END_REF]. The Scatterplot Matrix is a collection of Scatterplots Simples (x-y) ordered by pairs. This representation provides an overview of data. The size of the matrix depends on the number of visualized variables. The Scatterplots are duplicated in the matrix relative to the diagonal. It is possible to use colour glyphs, shape, or add a size marker to add a supplementary variable. [START_REF] Ware | Information Visualization: Perception for Design (Interactive Technologies)[END_REF] highlights that, although the colour used with Scatterplots allows the identification of clusters and patterns, interpretation could be difficult. Parallel Coordinates Plot is defined as a graph displaying multiple criteria without drastically increasing the complexity of the display [START_REF] Inselberg | Parallel Coordinates: VISUAL Multidimensional Geometry and its Applications[END_REF]. In PCP, variable values are displayed on separate axes laid out in parallel. The design points (or alternatives) are depicted as profile lines that connect points on the respective axes. According to [START_REF] Gettinger | A comparison of representations for discrete multi-criteria decision problems[END_REF], this representation can be readily interpreted and provides a good overview. Furthermore, patterns such as positive, negative and non-trivial (multiple) correlations, may quickly be identified at a glance.

EXPERIMENTAL DESIGN

To answer our research question "What graph allows designers to be effective in the discovery phase and result in an informed decision?" we conducted a controlled experiment that adopted a between subject approach. Each participant performed the experiment on one graph and resolved 4 design optimization problems. They are classic problems of mechanical design from literature: two-member Truss design (Truss), gear train design (Gear), multiple-disk clutch brake design (Disk) and pressure vessel design (Vessel). The first three problems come from the work of [START_REF] Deb | Innovization: Innovating design principles through optimization[END_REF] and the fourth problem comes from [START_REF] Canbaz | Preventing and resolving design conflicts for a collaborative convergence in distributed setbased design[END_REF]. To realize the experimentation we have developed a web "platform" (http://these.aaa.alwaysdata.net/expe2/) where problems and the three graphs are available. This platform allows, among other the generation of design points (random or non-Abi Akle A., Minel S., Yannou B., 2017. Design space visualization for efficiency in knowledge discovery leading to an informed decision, In 21st International Conference on Engineering Design (ICED), August 21-25, Vancouver, Canada.

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dominated Pareto solutions sampling), glyph colouring according to the designer preferences and reducing the design space.

Procedure

The experiment is divided into two main phases: (1) training part and (2) test part. Moreover, we incorporate a type of milestone during the session: multiple choice forms. We use three forms, one at the beginning, another between the training part and the test part and one at the end of the session. The experimentation is sized to be limited to a two-hour session. The training part (1) is used to upgrade the level of knowledge of participants. It is divided in three steps: a crash course, a "getting started" step with a tutorial to resolve the Truss problem and a practice phase with the platform guide where participants resolve the Gear problem. The test part (2) is the phase where the graphs are tested and we realize our measurements. The tests are performed on two design problems consecutively without help supports: the Vessel problem followed by the Disk problem. The instruction for both problems is to solve the bi-objective optimization problem using the method of Design Space Exploration to select an optimal solution with a justified means. The optimization problems are biobjectives that is to say there are two antagonistic performance variables to maximize or minimize. For each of the two problems, the participants have a 10-minute time limit and after each problem, a questionnaire is given to the participant to gather confidence and to know the information acquired during exploration, which enabled him to make his/her decision (selecting a solution).

Design problems

As already mentioned, we use four bi-objective design problems for our experiment. The first two are used during the training part and a description is available in [START_REF] Deb | Innovization: Innovating design principles through optimization[END_REF]. The first problem for the test is the Vessel problem described as follow: It is a design problem of a cylindrical thin walled pressure vessel with hemispherical ends. There are three design variables (R, T, L) and two performances (W, V). The objectives are minimizing W and maximizing V with controlling R, T and L while satisfying constraints C1 to C7. Vessel problem nomenclature and constants are: W is Weight of the pressure vessel in lbs, V is Volume of the pressure vessel, R is Radius, T is Thickness of the pressure wall, L is Length of the cylinder, P is Pressure inside the cylinder, UTS is Ultimate tensile strength of the vessel material and equals 35 klb, d is density of the vessel material and equals 0.283 and Circ is Circumferential stress (see Figure 3). The second problem for the test is the Disk problem described as follow: In this problem, a multiple clutch brake needs to be designed. Two conflicting objectives are considered: minimization of mass (M) of the brake system in kg and minimization of stopping time (S) in seconds. There are five decision variables Ri, Ro, t, F, and Z (see Figure 3), where Ri is the inner radius in mm, Ro is the outer radius in mm, t is the thickness of discs in mm, F is the actuating force in N and Z is the number of discs (or friction surfaces). All five variables are considered discrete and their allowable values are given below: 60 < Ri < 80; 90 < Ro < 110; 1 < t < 3; 600 < F < 1000; 2 < Z < 10. All performance formulas, constraints and bounds of the two problems are available on the web platform.

Measurements

In our work, variables were either measured during the test with an eye-tracking system the Tobii X2 or collected through the questionnaires. We use three types of measurements: those controlled during the test and those collected with questionnaires. The controlled measurements are used to verify that there are no differences between the three groups of participants (one group per graph) concerning: the knowledge level from the three Multiple-Choice-Questionnaires (MCQs) and the "performance" of the final solution selected (is it a Pareto solution?). Then, we have three measurements realized during the test: 

The number of discoveries / insights realized by the participant knowing that for each problem, there are seven discoveries in order to make a fully informed decision (see Erreur ! Source du renvoi introuvable.). The collect of the number of discoveries was performed by analysis of the video screen in conjunction with eye-tracking record.



We also measure the time passed before the participant realizes the first discovery (in seconds). This time is not dependent on elapsed time to understand the problem because the timer is triggered only when the participant has read the description of the problem and s/he makes the first design points sampling. This second measurement is for us a first clue to the speed of discovery with the three graphs.



The third measurement is the average time to complete a discovery (in seconds) i.e. the total time to perform all the discoveries divided by the number of discoveries realized. This measurement is for us a second clue to the speed of discovery with the three graphs. The solutions tend to t = 1

The solutions tend to F = 1000 Positive correlation between M & Z Finally we use a post-problem questionnaire to know if participants can "justify" their decision based on the information acquired during exploration (correlation, trend, transition point, etc.). Note that for the analysis of this indicator, we only check if the participants justify their decision or not (nominal qualitative variable). We cannot analyse the amount of information that participants use to justify their response because the amount depends on the participant. Similarly, the white forms are not considered (because lack of response does not mean that participants do not know how to justify their decision).

RESULTS

We have a sample of 42 subjects and it is a between approach (3 groups). So we have three samples of N = 14 subjects in each group / graph. For the analysis of data, we apply different statistical tests and we consider a significance level α=10%.

Differences and similarities between the three groups of participants

First of all, we analyse the data obtained with MCQs (Multiple-Choice-Questionnaires ) to verify if we can distinguish designer profiles within testers and / or if there is a difference between groups (i.e. three groups -three graphs). To detect designer profiles, we use the Friedman test (and the Wilcoxon Signed-Rank test if a post-hoc analysis is required) because MCQs answers give us qualitative ordinal variables and the groups are paired (i.e. within-approach). To detect difference between groups, we use Kruskal-Wallis test because in this case the analysis is a between-approach. Abi Akle A., Minel S., Yannou B., 2017. Design space visualization for efficiency in knowledge discovery leading to an informed decision, In 21st International Conference on Engineering Design (ICED), August 21-25, Vancouver, Canada. The results for the three graphs, presented in Table 3, allow us to conclude that all participants have a novice profile at the start of the test and they all gained knowledge (in design space exploration method): MCQ1<MCQ2<MCQ3. The average scores of MCQs are depicted in Figure 4. The results presented in Erreur ! Source du renvoi introuvable. show us that there is no significant difference between the three graphs for the three MCQs answers. Thus, there is no difference of knowledge between the groups. Then, we analyse the solutions selected by participants. For this indicator we analyse the data from the two problems together. Thus, our study has two variables: the group (SPM, PCP and SSP) and the "Pareto" (yes or no). We obtain (from the descriptive statistics analysis) that over 20% of our results (half) had an expected frequency less than 5 (i.e. 95% for the non-dominated) that is why we realize the Chi2 test with Yates correction to address our statistical assumptions: Χ² = 1.647, df = 2 and p = 0.44. We should infer that there is no difference between the three groups.

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The first part of the analysis allows us to conclude that there is no difference in knowledge between the three groups and they have almost all selected a Pareto solution. We can thus free ourselves to divide the groups for the following analyses.

Measurements during test: discovery variables

Unfortunately, we have different sample sizes because all subjects were unable to perform the tests in their entirety. This is due to bugs in the platform. All discovery variables measured are quantitative. Therefore we apply the ANOVA-between statistical test and pairwise t-tests post-hoc analysis if necessary (i.e. if ANOVA is significant). We considered the following statistical hypotheses: H0: there is no difference between the graphs for the discovery phase and H1: there is a difference.

For the Vessel problem, we obtain for the SSP (n =12) an average of 2.66 discoveries, 209.6 seconds elapsed before the first discovery and 198.4 seconds mean time per discovery, for PCP (n =12) 3.25 discoveries, 115.8 seconds elapsed before the first discovery and 131.1 seconds mean time per discovery and for SPM (n =13) 6.3 discoveries, 14.8 seconds elapsed before the first discovery and 65.1 seconds mean time per discovery. ANOVAs give significant results: F(2,34)= 24.27 and p<0.0001 for the number of discoveries, F(2,33)= 8.65 and p= 0.000954 for the time before the first discovery and F(2,33)= 6.45 and p= 0.004321 for the meantime per discovery.

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Regarding the Disk problem, we obtain for the SSP (n =13) an average of 2.92 discoveries, 175.2 seconds elapsed before the first discovery and 129.1 seconds mean time per discovery, for PCP (n =14) 3 discoveries, 107.2 seconds elapsed before the first discovery and 93.3 seconds mean time per discovery and for SPM (n =13) 5.4 discoveries, 52.4 seconds elapsed before the first discovery and 75.3 seconds mean time per discovery. ANOVAs give significant results: F(2,37) = 17.01 and p<0.0001 for the number of discoveries, F(2,37)= 9.17 and p= 0.000583 for the time before the first discovery and F(2,37)= 5.71 and p= 0.006899 for the meantime per discovery. So we perform a post-hoc analysis for the two problems (see Table 5). The averages for each variable are depicted in Figure 5. Statistical analysis performed on three discovery variables allows us to conclude that the Scatter Plot Matrix (SPM) is the most relevant graph for the discovery phase. We should also indicate that the Simple Scatter Plot (SSP) is the graph getting the worst results for the discovery phase.

Justification of the decision

For this indicator we analyse the data from the two problems together. Our study has two qualitative variables: the group (SPM, PCP and SSP) and the answer to question (decision justified or not). We apply a test of Chi2: Χ² = 4.747, df = 2 and p = 0.093. The result is significant. We then operate the Khi2 tests in pairs:  SSP vs PCP : Χ2=0.18, df=1 and p=0.671  SSP vs SPM : Χ2=2.99, df=1 and p=0.084  PCP vs SPM : Χ2=4.447, df=1 and p=0.035 There is a significant difference between the SPM and SSP graphs; and SPM and PCP. Decisions are justified with SPM. We conclude that SPM is the graph with which the participants made the most informed decisions. We have presented an optimization strategy to focus on informed decision as competiveness leverage. Hence, designers must iteratively build the design based on experimentation. They have to choose with a large, and often cumbersome, set of alternatives. Our research helps in better understanding the performance impacts of the data presentation on the system. We identify the Simple Scatter Plot (SSP), the Scatter Plot Matrix (SPM) and the Parallel Coordinate Plot (PCP) as graphical optimization.

Thanks to our experiment we aimed to draw out clear recommendations regarding the choice of a graph for the discovery phase in Design Space Exploration to make an informed decision. We have shown how to select an optimal solution in a set of feasible solutions defined by their design and performance value vectors. From the results of our tests, design space exploration is improved while using SPM graph to present data. Designer seems to be more confident and made informed decisions depending on the graphical interface proposed. SPM is most appropriate for the discovery phase because this phase involves understanding the problem by observing interactions between the variables and SPM is the graph that most reveals these types of interactions (such as clusters, correlations, etc.) [START_REF] Keim | Designing pixel-oriented visualization techniques: Theory and applications[END_REF]. Moreover, we believe that it inspires designers because it uses Cartesian representations (i.e. facilitates interpretation) and gives an overview of the dataset and therefore of the problem. Our results are consistent, however there is still a need to test our models and build on this work to improve the accuracy of our results, especially focusing with the impacts of graphical visualization depending on the designer's expertise.

Figure 1 .

 1 Figure 1. Design Space Exploration Process

Figure 2 .

 2 Figure 2. Illustration of the SPM, the SSP and the PCP

Figure 3 .

 3 Figure 3. Illustration of the Vessel problem (on the left) and the Disk problem (on the right)

Problem

  Discoveries from a global point of view Discoveries from a local point of view (Pareto solutions) Vessel Positive correlation (trend) between R & V Positive correlation (trend) between R & T Positive correlation (trend) between T & V Positive correlation between W & L Positive correlation between V & L The solutions tend to T = 4 The solutions tend to R = 36 Disk Negative correlation (trend) between S & Z Positive correlation (trend) between Ri & Ro A transition point for Ro = 100 A transition point for M = 1

Figure 4 .

 4 Figure 4. Average scores of MCQs and standard deviation

Figure 5 .

 5 Figure 5. Average and standard deviation for the three discovery variables

  ., Minel S., Yannou B., 2017. Design space visualization for efficiency in knowledge discovery leading to an informed decision, In 21st International Conference on Engineering Design (ICED), August 21-25, Vancouver, Canada. ICED17 6 DISCUSSION AND CONCLUSION

Table 2 .

 2 List of the discoveries for the two problems of the test

Table 3 .

 3 Results of the Friedman and Wilcoxon tests for the intra-graphs analysis

			SSP	PCP	SPM
	Friedman test		csqr=23.89, df=2	csqr=26.14, df=2	csqr=21.09, df=2
			and p<0.0001	and p<0.0001	and p<0.0001
	Wilcoxon	MCQ1 vs. MCQ2	W=-105, Z=-3.28	W=-105, Z=-3.28	W=-107.5, Z=-3.36
	Signed-Rank		and p=0.0005.	and p=0.0005.	and p=0.0004
	Test	MCQ2 vs. MCQ3	W=-76, Z=-2.64	W=-103, Z=-3.22	W=-53,	Z=-1.83
			and p=0.0041.	and p=0.0006	and p=0.0336

Table 4 .

 4 Results of Kruskal-Wallis tests for the inter-graphs analysis

			MCQ1	MCQ2	MCQ3
	Kruskal-Wallis Test	H=0.61 p=0.7371	H=1.55 p=0.4607	H=4.2 p=0.12
	Rank	SPM	23.1	22	16.8
		PCP	19.5	18.4	26.3
		SSP	21.9	24.1	21.5

Table 5 .

 5 Results of the t-test for the three discovery variables for the 2 problems The results of the t-test pairwise comparison indicate that there is a significant difference between SPM and SSP; SPM and PCP for the number of discoveries and between SSP, PCP and SPM for the other two indicators. SPM is significantly different from the other two and gets the best scores.

	Problem T-test pairwise	Number	of	Time before the first	Mean	time	to
		comparison	discoveries		discovery	complete a discovery
	Vessel SSP vs PCP	t(22)=-0.92 p=0.18		t(21)=-1.57 p= 0.066	t(21)= 1.44 p= 0.0823
		SSP vs SPM	t(23)=6.62 p<0.0001 t(22)= 3.69 p= 0.0006 t(22)= 3.19 p= 0.002
		PCP vs SPM	t(22)=5.86 p<0.0001 t(23)= 4.55 p<0.0001	t(23)= 3.52 p= 0.0009
	Disk	SSP vs PCP	t(25)=0.16 p= 0.437	t(25)= 2.39 p= 0.0123 t(25)= 1.94 p= 0.0319
		SSP vs SPM	t(24)= 4.99 p <.0001 t(24)= 4.44 p <.0001	t(24)= 3.24 p = 0.0017
		PCP vs SPM	t(25)= 5.08 p <.0001 t(25)= 1.89 p = 0.0352 t(25)= 1.44 p = 0.081


Ira Frederick

Moxley

A Schrödinger Equation for Solving the Riemann Hypothesis

Keywords: quantum chaos, Riemann hypothesis, eigenspectra

The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender-Brody-Müller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a unique Hermitian Hamiltonian that maps to the zeros of the analytic continuation of the Riemann zeta function, and provide an analytical expression for the eigenvalues of the results. The holomorphicity of the resulting eigenvalues is examined. Moreover, a second quantization of the resulting Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function converges at σ = 1/2.

I. INTRODUCTION

The unification of number theory with quantum mechanics has been the subject of many research investigations [1][2][3][4][5]. It has been proven that an infinitude of prime numbers exist [START_REF] Ribenboim | The little Book of Big Primes[END_REF]. In Ref. [START_REF] Bender | [END_REF], it was shown that the eigenvalues of a Bender-Brody-Müller (BBM) Hamiltonian operator correspond to the nontrivial zeros of the Riemann zeta function. If the Riemann hypothesis is correct [START_REF] Riemann | On the Number of Prime Numbers less than a Given Quantity[END_REF], the zeros of the Riemann zeta function can be considered as the spectrum of an operator R = Î/2 + i Ĥ, where Ĥ is a self-adjoint Hamiltonian operator [5,[START_REF] Faris | [END_REF], and Î is identity. Hilbert proposed the Riemann Hypothesis as the eighth problem on a list of significant mathematics problems [10]. Although the BBM Hamiltonian is pseudo-Hermitian [11], it is consistent with the Berry-Keating conjecture [START_REF] Berry | Supersymmetry and Trace Formulae[END_REF][START_REF] Berry | [END_REF][14], which states that when x and p commute, the Hamiltonian reduces to the classical H = 2xp. Berry, Keating and Connes proposed the classical Hamiltonian in an effort to map the Riemann zeros to the Hamiltonian spectrum. More recently, the classical Berry-Keating Hamiltonians were quantized, and were found to contain a smooth approximation of the Riemann zeros [15,16]. This reformulation was found to be physically equivalent to the Dirac equation in Rindler spacetime [17]. Herein, the eigenvalues of the BBM Hamiltonian are taken to be the imaginary parts of the nontrivial zeroes of the analytical continuation of the Riemann zeta function

ζ(s) = 1 1 -2 1-s • ∞ n=1 (-1) n-1 n s , (1)
where the complex number s = σ + it, and (s) > 0. The idea that the imaginary parts of the zeros of Eq. ( 1) are given by a self-adjoint operator was conjectured by Hilbert and Pólya [18]. Hilbert and Pólya asserted that the nontrivial zeros of Eq. ( 1) can be considered as the spectrum of a self-adjoint operator in a suitable Hilbert space. The Hilbert-Pólya conjecture has also found applications in quantum field theories [19]. The Riemann Hypothesis states that the zeros of Eq. ( 1) on 0 ≤ σ < 1 have real part equal to 1/2 [START_REF] Riemann | On the Number of Prime Numbers less than a Given Quantity[END_REF]20]. In Ref. [21], Hardy proved that infinitely many zeros are located at σ = 1/2. According to the Prime Number Theorem [22,23], no zeros of Eq.

(1) can exist at σ = 1. In this paper we present a Schrödinger equation that maps to the nontrivial zeros of the Riemann zeta function, and evaluate the convergence of the expression by imposing a normalization constraint on the density. A self-adjoint Hamiltonian is derived from the BBM Hamiltonian, and a second quantization of the resulting Schrödinger equation is performed to obtain the equations of motion. We study the nontrivial zeros of the Riemann zeta function by taking the expectation values of the resulting Schrödinger equation to show that the real part of every nontrivial zero of the analytic continuation of the Riemann zeta function converges at σ = 1/2. Finally we obtain a general solution to the Riemann zeta Schrödinger equation by performing a similarity transformation.

II. RIEMANN ZETA SCHR ÖDINGER EQUATION

We consider the eigenvalues of the Hamiltonian Ĥ = 1 1 -e -i p (xp + px)(1 -e -i p), (2) where p = -i ∂ x , = 1, and x = x. In Ref. [START_REF] Bender | [END_REF], it is conjectured that if the Riemann hypothesis is correct, the eigenvalues of Eq. ( 2) are non-degenerate. Next, we let Ψ s (x) be an eigenfunction of Eq. ( 2) with an eigenvalue t = i(2s -1), such that

Ĥ |Ψ s (x) = t |Ψ s (x) , (3)
and x ∈ R + , s ∈ C. Solutions to Eq. ( 3) are given by the analytic continuation of the Hurwitz zeta function

|Ψ s (x) = -ζ(s, x + 1) = -Γ(1 -s) 1 2πi C z s-1 e (x+1)z 1 -e z dz, (4)
on the positive half line x ∈ R + with eigenvalues i(2s -1), s ∈ C, (s) ≤ 1, the contour C is a loop around the negative real axis, and Γ is the Euler gamma function for (s) > 0

Γ(s) = ∞ 0 x s-1 e -x dx. (5)
As -|Ψ s (x = 1) is 1 -ζ(s * ), this implies that s belongs to the discrete set of zeros of the Riemann zeta function when s * = σ -it, and as -|Ψ s (x = -1) is ζ(s), this implies that s belongs to the discrete set of zeros of the Riemann zeta function when s = σ + it. From inserting Eq. ( 3) into Eq. ( 2), we have the relation

1 1 -e -i p (xp + px)(1 -e -i p) |Ψ s (x) = t |Ψ s (x) . (6)
Given that Eq. ( 2) is not Hermitian, it is useful to symmetrize the system. This can be accomplished by letting

|φ s (x) = [1 -exp(-∂ x )] |Ψ s (x) , = ∆ |Ψ s (x) = |Ψ s (x) -|Ψ s (x -1) , (7)
and defining a shift operator

∆ ≡ 1 -exp(-∂ x ). (8)
For s > 0 the only singularity of ζ(s, x) in the range of 0 ≤ x ≤ 1 is located at x = 0, behaving as x -s . More specifically,

ζ(s, x + 1) = ζ(s, x) - 1 x s , (9)
with ζ(s, x) finite for x ≥ 1 [START_REF] Espinosa | [END_REF]. As such, it can be seen from Eq. ( 7) that the eigenfunction

|φ s (x) = 1 x s . (10)
Upon inserting Eq. ( 7) into Eq. ( 6) we obtain

-i[x∂ x + ∂ x x] |φ s (x) = t |φ s (x) . (11)
Let H be a Hilbert space, and from Eq. (11) we have the Hamiltonian operator

Ĥ = -i x∂ x + ∂ x x = -i 2x∂ x + 1 , (12)
for x ∈ R acting in H , such that Ĥf, g = f, Ĥg ∀ f, g ∈ D( Ĥ). (13)
Restricting x ∈ R + , Eq. ( 12) is then written

Ĥ = -2i √ x∂ x √ x, (14)
where s ∈ C, and x ∈ R + . For the Hamiltonian operator as given by Eq. ( 14), the Hilbert space is H = L 2 (1, ∞) [26][27][28]. We then impose on Eq. ( 14) the following minimal requirements, such that its domain is not too artificially restricted.

i Ĥ is a symmetric (Hermitian) linear operator;

ii Ĥ can be applied on all functions of the form

g(x, s) = P (x, s) exp - x 2 2 , ( 15
)
where P is a polynomial of x and s. Here, it should be pointed out that Ĥ = T + V , and from Eq. ( 12), it can be seen that T = -2i x∂ x , V = -i . From (ii), V g(x, s) must belong to the Hilbert space H = L 2 defined over the space x. This is guaranteed as

| -i |≤ , (16)
where is the reduced Planck constant or Dirac constant. The domain D V of the potential energy V consists of all φ ∈ H for which V φ ∈ H . As such, V is self-adjoint. It is not necessary to specify the domain of Eq. ( 14), as it is only necessary to admit that Eq. ( 14) is defined on a certain D Ĥ such that (i) and (ii) are satisfied. If we denote by D 1 the set of all functions in Eq. ( 15), then (ii) implies that D Ĥ ⊇ D 1 . By letting Ĥ1 be the contraction of Ĥ with domain D 1 , i.e., Ĥ is an extension of Ĥ1 , and letting H1 be the closure of Ĥ, it can be seen that H1 is self-adjoint. Since Ĥ is symmetric and Ĥ ⊇ Ĥ1 , i.e., Ĥ is an extension of Ĥ1 , it follows that H = H1 and Ĥ is essentially self-adjoint, where H is the unique self-adjoint extension [29]. Other than eigenfunctions φ s (x) in configuration space as seen in Eq. (10), it is useful to represent eigenfunctions in momentum space Φ s (p). The transformation between configuration space eigenfunctions and momentum space eigenfunctions can be obtained via Plancherel transforms [30], where the one-to-one correspondence φ s (x) Φ s (p) is linear and isometric.

Theorem 1. The eigenstate φ s (x) :

X → C is measurable. That is, φ s (x) = φ σ (x) + iφ t (x) where φ σ (x), φ t : X → (-∞, -1] ∪ [1, ∞) are measurable.
Proof.

φ 2 ≡ -1 -∞ | φ s (x) | 2 dx + ∞ 1 | φ s (x) | 2 dx = -1 -∞ | Φ s (p) | 2 dp + ∞ 1 | Φ s (p) | 2 dp ≡ Φ 2 , ( 17
) from which Φ s (p) = 1 2π 3/2 ∞ -∞ φ s (x) exp(-ipx)dx = e -1 2 iπs (sgn(p) + 1) sin(πs)Γ(1 -s) |p| s-1 2π 3/2 , 0 < (s) < 1, (18)
φ s (x) = 1 2π 3/2 ∞ -∞ Φ s (p) exp(ipx)dp = 1 x s , (19)
and

φ 2 = Φ 2 = (-1) -2σ 2σ -1 + 1 2σ -1 , (20)
where σ > 1/2. Interestingly, it can be seen that lim

σ→1/2 (-1) -2σ 2σ -1 + 1 2σ -1 = iπ = log(x = -1). (21)
Then,

φ s (x) ∈ L 2 (µ) ⇐⇒ X |φ σ (x)| 2 dµ < ∞, (22)
and

X |φ t (x)| 2 dµ < ∞. (23)
Eqs. ( 18) and ( 19) are two vector representations of the same Hilbert space H . From Eq. ( 12), it can be seen that

T = -2i x∂ x , (24)
such that we define a multiplicative operator T0 in momentum space ( T0 Φ s )(p) = T0 (p)Φ s (p), where

T0 (p) = 2xp. (25)
Here, it should be pointed out that as x = i d/dp, Eq. ( 25) reduces to

T0 (p) = 2i , (26)
and Eq. ( 12) is then rewritten in momentum space as Ĥ(p) = i . The domain D 0 of T0 is defined as the set of all functions Φ s (p) ∈ H such that T0 (p)Φ s (p) ∈ H . As such, T0 is definitively self-adjoint. From Eq. ( 15) we have defined the set D 1 of functions in configuration space. From the Plancherel transform [30] of Eq. ( 15), we obtain the set D 1 of functions in momentum space having the form

G(p, s) = P (p, s) exp - p 2 2 , ( 27
)
where P is a polynomial of p and s. Eqs. (18) and (19) are true if φ s (x) ∈ D 1 or Φ s (p) ∈ D 1 and since Φ s (p) ∈ D 1 → 0 as p → ∞ then D 1 ⊆ D 0 . Moreover, for φ ∈ D 1 , T0 coincides with Eq. ( 24) [29]. Using Eq. ( 18) and Ĥ(p) = i , the eigenrelation

Ĥ(p) |Φ s (p) = λ |Φ s (p) (28)
is obtained. In order to find the expectation value for Ĥ we take the complex conjugate of Eq. ( 28), set = 1, multiply by the eigenfunction Φ s (p), and then integrate over p to obtain

∞ -∞ i e -1 2 iπs (sgn(p) + 1) sin(πs)Γ(1 -s) |p| s-1 2π 3/2 * e -1 2 iπs (sgn(p) + 1) sin(πs)Γ(1 -s) |p| s-1 2π 3/2 dp = λ * Φ 2 , ( 29
)
where λ is the eigenvalue. It can be seen that the integral on the LHS vanishes for σ > 1/2. Hence, using Eq. ( 20) on the RHS, Eq. ( 29) is then rewritten as

lim σ→1/2 λ * (-1) -2σ = -λ * , (30)
demonstrating that the eigenvalues in momentum space vanish along the critical line σ = 1/2. As a result, it is our ambition to study the eigenvalues in configuration space.

Definition 1. The Riemann zeta Schrödinger equation is

-∂ s |Ψ s (x) = i ∆-1 xp ∆ + ∆-1 px ∆ |Ψ s (x) , ( 31
)
where ∆ is given by Eq. ( 8), x = x, p = -i ∂ x , = 1, x ∈ R + ≥ 1 owing to the difference operator ∆ |Ψ s (x) , and s ∈ C.

Upon inserting Eq. ( 7) into Eq. ( 31) for x ∈ R + , we obtain the symmetrized Riemann zeta Schrödinger equation, i.e.,

∂ s |φ s (x) = 1/2(∂ σ -i∂ t ) |φ s (x) = - 2 √ x∂ x √ x |φ s (x) . ( 32
) Theorem 2. The eigenvalues of Ĥ = -2i √ x∂ x √ x occur at σ = ±1/2 ∀ t ∈ C for = 1 and |φ s (x) = x -s when D Ĥ → (-∞, -1] ∪ [1, ∞).
Proof. Let |φ s (x) be an eigenfunction of Ĥ with eigenvalue t, i.e., Ĥ |φ s (x) = t |φ s (x) .

(

) 33
In order to find the expectation value for Ĥ we take the complex conjugate of Eq. ( 33), multiply by the eigenfunction, and then integrate over

x ∈ [1, ∞), [-1, -∞) to obtain 2i -1 -∞ √ x∂ x √ xφ s (x) * φ s (x)dx + 2i ∞ 1 √ x∂ x √ xφ s (x) * φ s (x)dx = t * -1 -∞ φ * s (x)φ s (x)dx + t * ∞ 1 φ * s (x)φ s (x)dx = t * φ 2 . ( 34
)
Integrating by parts on the LHS then gives

-2i φ 2 + -1 -∞ φ * s (x)x d dx φ s (x)dx + ∞ 1 φ * s (x)x d dx φ s (x)dx = t * φ 2 . ( 35
)
Carrying out the integration on the LHS we obtain 2i(-1) -2σ (-1)

2σ + 1 (σ + it) 2σ -1 = (t * + 2i) φ 2 . ( 36
)
Upon inserting Eq. ( 20) it can be seen that 2i(-1) -2σ (-1)

2σ + 1 (σ + it) = (t * + 2i) (-1) -2σ + 1 . (37)
Hence,

σ = ± 1 2 ∀ t, t * . ( 38
)
Taking the complex conjugate of Eq. ( 32), setting = 1, multiplying by the eigenfunction, and then integrating

over x ∈ [1, ∞), [-1, -∞), it can be seen that 1 2 -1 -∞ ∂ σ φ s (x) * φ s (x)dx + 1 2 ∞ 1 ∂ σ φ s (x) * φ s (x)dx = - i 2 -1 -∞ ∂ t φ s (x) * φ s (x)dx - i 2 ∞ 1 ∂ t φ s (x) * φ s (x)dx -2 -1 -∞ √ x∂ x √ xφ s (x) * φ s (x)dx -2 ∞ 1 √ x∂ x √ xφ s (x) * φ s (x)dx. ( 39
)
Integrating by parts the last two terms on the RHS we obtain

1 2 -1 -∞ ∂ σ φ s (x) * φ s (x)dx + 1 2 ∞ 1 ∂ σ φ s (x) * φ s (x)dx = - i 2 -1 -∞ ∂ t φ s (x) * φ s (x)dx - i 2 ∞ 1 ∂ t φ s (x) * φ s (x)dx + 2 φ 2 +2 -1 -∞ φ * s (x)x d dx φ s (x)dx + 2 ∞ 1 φ * s (x)x d dx φ s (x)dx. ( 40
)
Evaluating those integrals on LHS, we have

1 2 -1 -∞ ∂ σ φ s (x) * φ s (x)dx + 1 2 ∞ 1 ∂ σ φ s (x) * φ s (x)dx = - 1 2(1 -2σ) 2 + (-1) -2σ (-1 + iπ(1 -2σ)) 2(1 -2σ) 2 , (41)
for σ > 1/2. Evaluating those first two integrals on the RHS of Eq. ( 40), we have

- i 2 -1 -∞ ∂ t φ s (x) * φ s (x)dx - i 2 ∞ 1 ∂ t φ s (x) * φ s (x)dx = 1 2(1 -2σ) 2 - i(-1) 1-2σ (π(2σ -1) -i) 2(1 -2σ) 2 , ( 42
)
for σ > 1/2. Evaluating those last two integrals on the RHS of Eq. (40), we have

2 -1 -∞ φ * s (x)x d dx φ s (x)dx + 2 ∞ 1 φ * s (x)x d dx φ s (x)dx = 2(σ + it) 1 -2σ + 2(-1) 1-2σ (σ + it) 2σ -1 , (43)
for σ > 1/2. Finally, by using Eqs. ( 20) and (41-43) in Eq. ( 40), it can be seen that

(-1) 1-2σ iπ(2σ -1) + (-1) 2σ + 1 = (-1) -2σ -8σ 2 -8iσt + (-1) 2σ+1 (4σ(2σ + 2it -3) -4it + 3) + (-1) -2σ 2iπσ + 12σ + 4it -iπ -3 . (44)
Hence, in the limit as σ → 1/2, Eq. ( 44) is satisfied ∀ t, t * .

A. Convergence Theorem 3. The eigenfunction φ s (x) = x -s of Ĥ = -2i √ x∂ x √ x normalizes at x = ±1.
Proof. In order to study convergent solutions to the Riemann zeta Schrödinger Eq. ( 32), it can be seen that upon inserting Eq. ( 7) into Eq. ( 32), we obtain

s = σ + it = 1 2 - log(x) 2 . ( 45
)
Hence, at x = 1 and letting

s = |s|e iθ ∈ C θ = -i log 1 2|s| + 2πn, (46)
and

|s| = e -iθ 2 , (47)
for n ∈ Z, such that θ = θ + iθ like

θ = arccos 1 2|s| + 2πn, (48)
and

θ = 2πn. ( 49
) At x = -1, Eq. (45) becomes θ = -i log - i(π + i) 2|s| + 2πn, (50)
and

|s| = 1 2 e -iθ -iπe -iθ , (51)
for n ∈ Z, such that θ = θ + iθ like

θ = arccos 1 2|s| + 2πn, (52)
and

θ = arcsin - π 2|s| + 2πn. ( 53
)
In agreement with Eq. ( 21). This condition is required such that the density is normalized, i.e.,

φ 2 = m n bn (s) b † m (s) φ m |φ n = n | bn (s)| 2 = iπ. ( 54
) B. Second Quantization Theorem 4. If all of the nontrivial zeros of ζ(s) = (1 -2 1-s ) -1 ∞ n=1 (-1) n-1 n -s occur at s = 1/2 + it n , then the numbers t n correspond to the eigenvalues (spectrum) of Ĥ = -2i √ x∂ x √
x, and n ∈ C, for the eigenstates |φ s (x) = bn (s)x -exp (-n+in)|σ| .

Proof. In order to perform a second quantization [32], we can express the complex order parameter as a linear combination of basis states

|φ s (x) = n bn (s) |φ n (x) , (55)
where s = |σ| exp(in) ∈ C, |σ| ∈ R and n is the complex phase. As such, using Eq. ( 10) we can rewrite Eq. ( 68) as

|φ n (x) = n bn (s)x -exp (-n+in)|σ| , (56)
From Eq. ( 32) we find

d ds bn (s) = -t n bn (s). ( 57
)
We now find a Hamiltonian that yields Eq. ( 70) as the equation of motion. Hence, we take

φ m (x)| Ĥ |φ n (x) = -2 ∞ 1 φ m (x)| √ x∂ x √ x |φ n (x) dx -2 -1 -∞ φ m (x)| √ x∂ x √ x |φ n (x) dx, (58)
as the expectation value. Upon substituting Eq. ( 69) into Eq. ( 71), we obtain the harmonic oscillator

φ m (x)| Ĥ |φ n (x) = -2 m n ∞ 1 x -exp (-m-im)|σ| √ x∂ x √ x x -exp (-n+in)|σ| dx -2 m n -1 -∞ x -exp (-m-im)|σ| √ x∂ x √ x x -exp (-n+in)|σ| dx = - m n b † m (s) bn (s) e (1+i)m e n -2e in |σ| (-1) 1+|σ| -e (-1-i)m -e (-1+i)n e n+(1+i)m -|σ| e n + e (1+i)m+in |m, n + m n b † m (s) bn (s) e (1+i)m e n -2e in |σ| e n+(1+i)m -|σ| e n + e (1+i)m+in |m, n , (59)
for |σ| e (-1-i)m + e (-1+i)n > 1. Hence at m = n,

φ n (x)| Ĥ |φ n (x) = -iπ n e (1+i)n (-1) 1+ -e (-1-i)n -e (-1+i
)n |σ| e n -2e in |σ| e (2+i)n -e n + e (1+2i)n |σ| |n

+ iπ n - e (1+i)n e n -2e in |σ| e (2+i)n -e n + e (1+2i)n |σ| |n . ( 60
)
where |n = 1, 2, 3, . . . , ∞. In accordance with Eq. (38), at |σ| = 1/2,

φ n (x)| Ĥ |φ n (x) = -iπ ∞ n=1 (-1) 1+ 1 2 -e (-1-i)n -e (-1+i)n e (1+i)n e n -e in 1 2 -e n -e (1+2i)n + e (2+i)n |n + iπ ∞ n=1 e (1+i)n e n -e in 1 2 -e n -e (1+2i)n + e (2+i)n |n = -iπ ∞ n=1 2 e n -e in e i(n-πe -n cos(n)) 1 + e iπe -n cos(n) e 2in -2e (1+i)n + 1 |n = 2πi ∀ n. ( 61
) Theorem 5. If all of the nontrivial zeros of ζ(s) = (1 -2 1-s ) -1 ∞ n=1 (-1) n-1 n -s occur at s = 1/2 + it n , then the numbers t n correspond to the eigenvalues (spectrum) of Ĥ = -2i √ x∂ x √ x, and |n| ∈ Z ⊆ R, for the eigenstates |φ s (x) = x -n .
Proof. In order to perform a second quantization [32], we can express the complex order parameter as a linear combination of basis states

|φ s (x) = n bn (s) |φ n (x) , (62)
where n ∈ Z ⊂ R. As such, using Eq. ( 10) we can rewrite Eq. (68) as

|φ n (x) = n bn (s)x -n , (63)
From Eq. ( 32) we find

d ds bn (s) = -t n bn (s). ( 64
)
We now find a Hamiltonian that yields Eq. ( 70) as the equation of motion. Hence, we take

φ m (x)| Ĥ |φ n (x) = -2 ∞ 1 φ m (x)| √ x∂ x √ x |φ n (x) dx -2 -1 -∞ φ m (x)| √ x∂ x √ x |φ n (x) dx, (65)
as the expectation value. Upon substituting Eq. (69) into Eq. ( 71), we obtain the harmonic oscillator

φ m (x)| Ĥ |φ n (x) = -2 m n ∞ 1 x -m √ x∂ x √ x x -n dx -2 m n -1 -∞ x -m √ x∂ x √ x x -n dx = m n b † m (s) bn (s) (2n -1)(-1) -m-n (-1) m+n + 1 m + n -1 |m, n = m n iπ(2n -1)(-1) -m-n (-1) m+n + 1 m + n -1 |m, n , (66) for
(m + n) > 1. Hence at m = n, φ n (x)| Ĥ |φ n (x) = n iπ (-1) -2n + 1 |n = 2πi ∀ n, (67)
where |n = 1, 2, 3, . . . , ∞.

Theorem 6. If all of the nontrivial zeros of ζ(s) = (1 -2 1-s ) -1 ∞ n=1 (-1) n-1 n -s occur at s = 1/2 + it n , then the numbers t n correspond to the eigenvalues (spectrum) of Ĥ = -2i √ x∂ x √ x, and |n| ∈ Z ⊆ R, for the eigenstates |φ s (x) = bn (s)x -exp (in)|σ| .

Proof. In order to perform a second quantization [32], we can express the complex order parameter as a linear combination of basis states

|φ s (x) = n bn (s) |φ n (x) , (68)
where s = |σ| exp(in) ∈ C, and |σ| ∈ R. As such, using Eq. ( 10) we can rewrite Eq. (68) as

|φ n (x) = n bn (s)x -exp (in)|σ| , (69)
From Eq. ( 32) we find

d ds bn (s) = -t n bn (s). ( 70
)
We now find a Hamiltonian that yields Eq. (70) as the equation of motion. Hence, we take

φ m (x)| Ĥ |φ n (x) = -2 ∞ 1 φ m (x)| √ x∂ x √ x |φ n (x) dx -2 -1 -∞ φ m (x)| √ x∂ x √ x |φ n (x) dx, (71)
as the expectation value.

Upon substituting Eq. (69) into Eq. ( 71), we obtain the harmonic oscillator

φ m (x)| Ĥ |φ n (x) = -2 m n ∞ 1 x -exp (-im)|σ| √ x∂ x √ x x -exp (in)|σ| dx -2 m n -1 -∞ x -exp (-im)|σ| √ x∂ x √ x x -exp (in)|σ| dx = m n b † m (s) bn (s) e im -1 + 2e in |σ| 1 + (-1) -e -im |σ| 1+e i(m+n) |σ| 1 + e i(m+n) -e im |m, n , (72)
for |σ| e -im + e in > 1. Hence at m = n,

φ n (x)| Ĥ |φ n (x) = n 2iπe 2in |σ| (-1) 2|σ| cos(n) cos(2π |σ| cos(n) + n) 2 |σ| cos(n) -1 |n + n 2iπe 2in |σ| cos(2π |σ| cos(n) + n) 2 |σ| cos(n) -1 |n - n iπe in (-1) 2|σ| cos(n) cos(2π |σ| cos(n) + n) 2 |σ| cos(n) -1 |n - n iπe in cos(2π |σ| cos(n) + n) 2 |σ| cos(n) -1 |n + n 2πe 2in |σ| (-1) 2|σ| cos(n) sin(2π |σ| cos(n) + n) 2 |σ| cos(n) -1 |n + n 2πe 2in |σ| sin(2π |σ| cos(n) + n) 2 |σ| cos(n) -1 |n - n πe in (-1) 2|σ| cos(n) sin(2π |σ| cos(n) + n) 2 |σ| cos(n) -1 |n - n πe in sin(2π |σ| cos(n) + n) 2 |σ| cos(n) -1 |n , (73)
where |n = 1, 2, 3, . . . , ∞. In accordance with Eq. ( 38), at |σ| = 1/2,

φ n (x)| Ĥ |φ n (x) = π ∞ n=1 1 + e iπ cos(n) cot n 2 + i (cos(π cos(n)) -i sin(π cos(n))) |n = 0 ∀ (2n -1)π. ( 74
)
Taking bn (s) as an operator, and b † n (s) as the adjoint, we obtain the usual properties:

[ bn (s), bm (s)] = [ b † n (s), b † m (s)] = 0, [ bn (s), b † m (s)] = δ nm . ( 75
)
From the analogous Heisenberg equations of motion,

- d ds n bn (s) = [ bn (s), Ĥ] - = m E m bn (s) b † m (s) bm (s) -b † m (s) bm (s) bn (s) = m E m δ nm bm (s) -b † m (s) bn (s) bm (s) -b † m (s) bm (s) bn (s) = m E m δ nm bm (s) + b † m (s) bm (s) bn (s) -b † m (s) bm (s) bn (s) = n b † m (s) bn (s)t n . ( 76
)
The eigenvalues of Ĥ are then

φ n (x)| Ĥ |φ n (x) = n t n |n . (77)
However, since t n = i(σ n -1/2) at the normalization constraint x = 1, we can rewrite Eq. (77) as

φ s (x)| Ĥ |φ s (x) = i n σ n - 1 2 |n . ( 78
)
From Eq. (76) it can be seen that

- d ds bn = i σ n - 1 2 bn , - d ds b † m = -i σ m - 1 2 b † m . (79)
Finally, upon setting t n = i(σ n -1/2) and t m = -i(σ m -1/2), we obtain

- d ds bn = iπ 1 -2e iθ |n| e -iθ |n| + e iθ |n| -1 + e iθ (-1) -e -iθ (|n|+e 2iθ |n|) (1 -2e iθ |n|) e 2iθ |n| + |n| -e iθ bn , - d ds b † m = -iπ 1 -2e iθ |m| e -iθ |m| + e iθ |m| -1 + e iθ (-1) -e -iθ (|m|+e 2iθ |m|) (1 -2e iθ |m|) e 2iθ |m| + |m| -e iθ b † m . (80)
Eq. ( 76) is in agreement with Eq. ( 80), and can be solved using the Wirtinger derivatives.

Remark. Theorem 6 implies the Riemann hypothesis, as the spectrum of a Hermitian operator consists of real numbers as seen in Theorem 2.

Theorem 7. The densely defined operator

Ĥ = -2 √ x∂ x √ x on the Hilbert space H = L 2 [1, ∞) is non-symmetric (non-Hermitian) [31] for the eigenstates φ n (x) = bn x -exp (-n+in)|σ| on x ∈ (-∞, -1] ∪ [1, ∞) when |σ| = 1/2.
Proof. By taking the inner products

( Ĥφ * , φ) = -2iπ ∞ n=1 -1 -∞ x -exp (-n+in)|σ| √ x∂ x √ xx -exp (-n-in)|σ| dx -2iπ ∞ n=1 ∞ 1 x -exp (-n+in)|σ| √ x∂ x √ xx -exp (-n-in)|σ| dx = iπ -2 |σ| + e (1+i)n (-1) -2e -n |σ| cos(n) (-1) 2e -n |σ| cos(n) + 1 e (1+i)n -1 + e 2in |σ| , (81)
for 2 |σ| e -n cos(n) > 1, and

(φ * , Ĥφ) = -2iπ ∞ n=1 -1 -∞ x -exp (-n-in)|σ| √ x∂ x √ xx -exp (-n+in)|σ| dx -2iπ ∞ n=1 ∞ 1 x -exp (-n-in)|σ| √ x∂ x √ xx -exp (-n+in)|σ| dx = iπ e n -2e in |σ| (-1) -2e -n |σ| cos(n) e n -2 |σ| cos(n) + -1 + e 2in σ e 2in σ -e (1+i)n sgn(σ) + σ + 1 , (82)
for 2 |σ| e -n cos(n) > 1. It can be seen that lim

σ→ 1 2 ( Ĥφ * , φ) = iπ -1 + e (1+i)n (-1) -e -n cos(n) (-1) e -n cos(n) + 1 1 2 -1 -e 2in + e (1+i)n , (83)
and lim

σ→ 1 2 (φ * , Ĥφ) = iπ e n -e in (-1) -e -n cos(n) 1 + i 2e -n cos(n) e n -cos(n) , (84)
∀ φ, φ * ∈ D( Ĥ).

Theorem 8. The densely defined operator [31] for the eigenstates

Ĥ = -2 √ x∂ x √ x on the Hilbert space H = L 2 [1, ∞) is symmetric (Hermitian)
φ n (x) = bn x -exp (in)|σ| on x ∈ (-∞, -1] ∪ [1, ∞) when |σ| = 1/2.
Proof. By taking the inner products

( Ĥφ * , φ) = -2iπ ∞ n=1 -1 -∞ x -exp (in)|σ| √ x∂ x √ xx -exp (-in)|σ| dx -2iπ ∞ n=1 ∞ 1 x -exp (in)|σ| √ x∂ x √ xx -exp (-in)|σ| dx = - ∞ n=1 2πe -in -2 |σ| + e in cos 2 (π |σ| cos(n))(tan(π |σ| cos(n)) + i) 2 |σ| cos(n) -1 , (85)
for 2 |σ| (cos(n)) > 1, and

(φ * , Ĥφ) = -2iπ ∞ n=1 -1 -∞ x -exp (-in)|σ| √ x∂ x √ xx -exp (in)|σ| dx -2iπ ∞ n=1 ∞ 1 x -exp (-in)|σ| √ x∂ x √ xx -exp (in)|σ| dx = π -1 + 2e in |σ| sin(2π |σ| cos(n)) + 2i cos 2 (π |σ| cos(n)) 2 |σ| cos(n) -1 , (86)
for 2 |σ| (cos(n)) > 1. It can be seen that lim

σ→ 1 2 ( Ĥφ * , φ) = - 2πe -in -1 + e in cos 2 1 2 π cos(n) tan 1 2 π cos(n) + i cos(n) -1 , (87)
and lim

σ→ 1 2 (φ * , Ĥφ) = π -1 + e in sin(π cos(n)) + 2i cos 2 1 2 π cos(n) cos(n) -1 , (88)
= L 2 [1, ∞) is holomorphic for the eigenstates φ θ (x) = bθ x -|σ| exp(iθ) on x ∈ (-∞, -1] ∪ [1, ∞).
Proof. By taking the inner products

( Ĥφ * , φ) = -2iπ ∞ θ=1 -1 -∞ x -|σ| exp(iθ) √ x∂ x √ xx -|σ| exp(-iθ) dx -2iπ ∞ θ=1 ∞ 1 x -|σ| exp(iθ) √ x∂ x √ xx -|σ| exp(-iθ) dx = -2iπ ∞ θ=1 1 2 -exp(-iθ)|σ| -1 -∞ x -2|σ| cos(θ) dx -2iπ ∞ θ=1 1 2 -exp(-iθ)|σ| ∞ 1 x -2|σ| cos(θ) dx = -2iπ ∞ θ=1 1 2 -exp(-iθ)|σ| (-1) -2|σ| cos(θ) 2|σ| cos(θ) -1 -2iπ ∞ θ=1 1 2 -exp(-iθ)|σ| 1 2|σ| cos(θ) -1 = -2iπ ∞ θ=1 1 2 -exp(-iθ)|σ| (-1) -2|σ| cos(θ) + 1 2|σ| cos(θ) -1 , (90)
and

(φ * , Ĥφ) = -2iπ ∞ θ=1 -1 -∞ x -|σ| exp(-iθ) √ x∂ x √ xx -|σ| exp(iθ) dx -2iπ ∞ θ=1 ∞ 1 x -|σ| exp(-iθ) √ x∂ x √ xx -|σ| exp(iθ) dx = -2iπ ∞ θ=1 1 2 -exp(iθ)|σ| -1 -∞ x -2|σ| cos(θ) dx -2iπ ∞ θ=1 1 2 -exp(iθ)|σ| ∞ 1 x -2|σ| cos(θ) dx = -2iπ ∞ θ=1 1 2 -exp(iθ)|σ| (-1) -2|σ| cos(θ) 2|σ| cos(θ) -1 -2iπ ∞ θ=1 1 2 -exp(iθ)|σ| 1 2|σ| cos(θ) -1 = -2iπ ∞ θ=1 1 2 -exp(iθ)|σ| (-1) -2|σ| cos(θ) + 1 2|σ| cos(θ) -1 , (91)
it can be seen that

C ( Ĥφ * , φ)dθ = C (φ * , Ĥφ)dθ = 0, (92)
∀ φ, φ * ∈ D( Ĥ). Here, it is useful to implement the following identities:

cos(θ) = cos(θ ) cosh(θ ) -i sin(θ ) sinh(θ ), (93)
and

exp(±iθ) = exp(-θ ) cos(θ ) ± i exp(-θ ) sin(θ ). ( 94
)
where θ = θ + iθ . The path C is then parametrized as follows:

θ(q) = |θ| cos(q) + i|θ| sin(q), (95)
with 0 ≤ q ≤ 2π, whilst f (θ) is given by Eq. ( 50), and the real and imaginary parts of Eq. ( 50) are given by Eqs.

(52) and (53), and

|θ| = α 2 + β 2 , (96)
where

α = 2|σ| cos(θ ) cosh(θ ) -1, (97)
and

β = -2|σ| sin(θ ) sinh(θ ). ( 98
)
By performing the parametrization of the path C, the real and imaginary parts of (φ * , Ĥφ) are then

(φ * , Ĥφ) = π |θ| 2 -(1 + α) exp(πβ) sin π(1 + α) -β exp(πβ) cos π(1 + α) + exp(πβ) sin π(1 + α) -β + 2|σ|(α + 1) exp(-θ ) cos(θ ) exp(πβ) sin π(1 + α) + 2β|σ| exp(-θ ) cos(θ ) exp(πβ) cos π(1 + α) -2|σ| exp(-θ ) cos(θ ) exp(πβ) sin π(1 + α) + 2|σ|β exp(-θ ) cos(θ ) -2|σ|(1 + α) exp(-θ ) sin(θ ) exp(πβ) cos π(1 + α) + 2|σ|β exp(-θ ) sin(θ ) exp(πβ) sin π(1 + α) + 2|σ| exp(-θ ) sin(θ ) exp(πβ) cos π(1 + α) -2|σ|(α + 1) exp(-θ ) sin(θ ) + 2|σ| exp(-θ ) sin(θ ) , (99) and
Hence, from the relation

C (φ * , Ĥφ)dθ = C (φ * , Ĥφ) + i (φ * , Ĥφ) (dθ + idθ ) = C (φ * , Ĥφ)dθ - C (φ * , Ĥφ)dθ + i C (φ * , Ĥφ)dθ + i C (φ * , Ĥφ)dθ = 2π 0 (φ * , Ĥφ) dθ dq dq - 2π 0 (φ * , Ĥφ) dθ dq dq + i 2π 0 (φ * , Ĥφ) dθ dq dq + i 2π 0 (φ * , Ĥφ) dθ dq dq (103)
at |θ| = 1, we obtain

2π 0 (φ * , Ĥφ) dθ dq dq = 1 2 π 4 2 exp(-θ ) cos(θ ) |θ| -1 , (104) 2π 0
(φ * , Ĥφ) dθ dq dq = 1 2 π 4 2 exp(-θ ) cos(θ ) |θ| -1 , (105) 2π 0
(φ * , Ĥφ) dθ dq dq = -π 4 exp(-θ ) sin(θ ) |θ| , (106) 2π 0
(φ * , Ĥφ) dθ dq dq = π 4 exp(-θ ) sin(θ ) |θ| , (107) such that C
(φ * , Ĥφ)dθ = C ( Ĥφ * , φ)dθ = 0. ( 108
)
III. HOLOMORPHICITY Cauchy's theorem states that if t σ (θ) is an analytic function, and t σ (θ) is continuous at each point within and on a closed contour C, then

C t σ (θ)dθ = 0. ( 109
)
Theorem 10. Let θ = θ + iθ and t σ (θ) = u σ (θ , θ ) + iv σ (θ , θ ) on some region G containing the point θ 0 . Then the eigenvalues t σ are holomorphic.

Proof. Integrating θ along C, Eq. ( 73) is then written

t σ = i C |σ| cos(θ) cos(2σπ cos(θ)) 1 -2|σ| cos(θ) - |σ| sin(θ) sin(2σπ cos(θ)) 2|σ| cos(θ) -1 dθ - C |σ| cos(θ) sin(2σπ cos(θ)) 2|σ| cos(θ) -1 + |σ| cos(2σπ cos(θ)) sin(θ) 1 -2|σ| cos(θ) dθ + i 2 C cos(2σπ cos(θ)) 2|σ| cos(θ) -1 dθ - 1 2 C sin(2σπ cos(θ)) 1 -2|σ| cos(θ) dθ. (110)
The path C is then parametrized as follows:

θ(q) = |θ| cos(q) + i|θ| sin(q), (111)
with 0 ≤ q ≤ 2π, whilst f (θ) is given by Eq. ( 50), and the real and imaginary parts of Eq. ( 50) are given by Eqs.

(52) and (53), and

|θ| = α 2 + β 2 , (112)
where

α = 2|σ| cos(θ ) cosh(θ ) -1, (113)
and

β = -2|σ| sin(θ ) sinh(θ ). (114)
Here, it is useful to implement the following identities:

cos(θ + iθ ) = cos(θ ) cosh(θ ) -i sin(θ ) sinh(θ ), (115)
cos(2σπ cos(θ + iθ )) = cos 2πσ cos(θ ) cosh(θ ) cosh 2πσ sin(θ ) sinh(θ )

+ i sinh 2πσ sin(θ ) sinh(θ ) sin 2πσ cos(θ ) cosh(θ ) , (116) and sin
(θ + iθ ) = sin(θ ) cosh(θ ) + i cos(θ ) sinh(θ ), (117)
sin(2σπ cos(θ + iθ )) = sin 2πσ cos(θ ) cosh(θ ) cosh 2πσ sin(θ ) sinh(θ )

-i sinh 2πσ sin(θ ) sinh(θ ) cos 2πσ cos(θ ) cosh(θ ) . (118)
By performing the parametrization of the path C as given by Eq. ( 111), the real and imaginary parts of t σ = u σ + iv σ are then u σ = π |θ| 2 |θ| 2 sin(πα) exp(πβ) + 2|σ|β sin(θ ) cosh(θ ) exp(πβ) sin(πα) + 2|σ|α sin(θ ) cosh(θ ) + 2|σ|β cos(θ ) sinh(θ ) -2|σ|β cos(θ ) sinh(θ ) exp(πβ) cos(πα) -2|σ|α sin(θ ) cosh(θ ) exp(πβ) cos(πα) -2|σ|α cos(θ ) sinh(θ ) exp(πβ) sin(πα) ,

and

v σ = π |θ| 2 -|θ| 2 + |θ| 2 exp(πβ) cos(πα) + 2|σ|β sin(θ ) cosh(θ ) exp(πβ) cos(πα) + 2|σ|β cos(θ ) sinh(θ ) exp(πβ) sin(πα) -2|σ|α sin(θ ) cosh(θ ) exp(πβ) sin(πα) -2|σ|α cos(θ ) sinh(θ ) exp(πβ) cos(πα) -4|σ| sin(θ ) cosh(θ ) exp(πβ) sin(πα) -2|σ|β sin(θ ) cosh(θ ) + 2|σ|α cos(θ ) sinh(θ ) , (120)
where

u σ = 2π|σ| sin(θ ) sinh(θ ) 1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -4π|σ| 2 cos(θ ) cosh(θ ) sin(θ ) sinh(θ ) exp -2πσ sin(θ ) sinh(θ ) cos π -2πσ cos(θ ) cosh(θ ) 1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -4π|σ| 2 cos 2 (θ ) cosh 2 (θ ) exp -2πσ sin(θ ) sinh(θ ) sin π -2πσ cos(θ ) cosh(θ )
1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 2π|σ| cos(θ ) cosh(θ ) exp -2πσ sin(θ ) sinh(θ ) sin π -2πσ cos(θ ) cosh(θ )

1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 4π|σ| 2 sin(θ ) sinh(θ ) cos(θ ) cosh(θ ) exp -2πσ sin(θ ) sinh(θ ) cos π -2πσ cos(θ ) cosh(θ )

1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(

Hence, from the relation

C t σ (θ)dθ = C (u σ + iv σ )(dθ + idθ ) = C u σ dθ - C v σ dθ + i C u σ dθ + i C v σ dθ = 2π 0 u σ dθ dq dq - 2π 0 v σ dθ dq dq + i 2π 0 u σ dθ dq dq + i 2π 0 v σ dθ dq dq (123)
at |θ| = 1, we obtain

2π 0 u σ dθ dq dq = π 4 |σ| cos(θ ) sinh(θ ), ( 124
) 2π 0 v σ dθ dq dq = π 3 |σ| π cos(θ ) sinh(θ ) -4 sin(θ ) cosh(θ ) , ( 125
) 2π 0 u σ dθ dq dq = 1 2 π 3 ((4 + π) sin(θ ) |σ| cosh(θ ) + 2), ( 126
) 2π 0 v σ dθ dq dq = - 1 2 π 2 4 + π 2 sin(θ ) |σ| cosh(θ ) + 1 2 π 4 sin(θ ) |σ| cosh(θ ) + 2π 2 sin(θ ) |σ| cosh(θ ) -π 3 (127) such that C t σ (θ)dθ = iπ 3 2 sin -1 -π 2|s| +2πσ sin -1 -π 2|s| cos -1 1 2|s| +2πσ cos -1 1 2|s|
π + (4 -8i) sin(θ ) |σ| cosh(θ )dθ dθ = 2 8 + 4i + iπ π 3 |σ| sin(πσ) sinh(πσ) cosh πσ -sin -1 π 2|s|

• cos i log

√ 4ss * -1 + i |s| + πσ -i log(2) = 0 ∀ σ. (128)
Here, it should be pointed out, that in accordance with Eq. (38), when |σ| = 1/2, and since θ = 0, Eq. ( 121) then reduces to

u σ (θ) = -π sin π cos(θ) -π cos π cos(θ) + 1 cot θ 2 , (129)
and Eq. ( 122) then reduces to

v σ (θ) = -π cos π cos(θ) -cot θ 2 sin π cos(θ) + 1 , (130)
such that t σ = u σ + iv σ in agreement with Eq. (74).

Theorem 11. Imaginary solutions of

∂ s |φ s (x) = -2 √ x∂ x √ x |φ s (x) at x = 1 map to the nontrivial zeros of ζ(s) = (1 -2 1-s ) -1 ∞
n=1 (-1) n-1 n -s . Proof. At x = 1, the normalization constraint Eq. ( 54) is satisfied, σ = 1 2 -it, and Eq. ( 4) can be written + ⅈ t The expectation value of Ĥ can be expressed via the density operator, viz.,

Ĥ (s) = m n b † m (s) bn (s) 1 -2e iθ |n| e -iθ |m| + e iθ |n| -1 |m, n + m n b † m (s) bn (s) e iθ (1 -2e iθ |n|)(-1) -e -iθ
IV. SIMILARITY SOLUTIONS Since Eq. ( 32), the Riemann zeta Schrödinger equation (RZSE) possesses symmetry about the origin x = 0, we then seek a similarity solution [START_REF] Pakdemirli | [END_REF] of the form:

φ s (x) = x α f (η), (140)
where η = s/x β , and the RZSE becomes an ordinary differential equation (ODE) for f . As such, we consider Eq. ( 32), and introduce the transformation ξ = a x, and τ = b s, so that

w(ξ, τ ) = c φ( -a ξ, -b τ ), (141)
where ∈ R, and τ ∈ C.

From performing this change of variable we obtain

∂ ∂s φ = -c ∂w ∂τ ∂τ ∂s = b-c ∂w ∂τ , (142)
and

-2 √ x ∂ ∂x √ xφ = -2 √ x ∂ √ x ∂x φ + √ x ∂φ ∂x = -2 √ x 1 2 √ x φ -2 √ x √ x ∂φ ∂x = -φ -2x ∂φ ∂x , (143)
where

∂φ ∂x = -c ∂w ∂ξ ∂ξ ∂x = a-c ∂w ∂ξ . (144)
By using Eqs. ( 142)-(144) in Eq. ( 32), the RZSE is then written

-c b ∂w ∂τ + w + 2ξ ∂w ∂ξ = 0, (145)
and is invariant under the transformation ∀ if b = 2, i.e.,

Since the RZSE is invariant under the transformation, it is to be expected that the solution will also be invariant under the variable transformation. Taking a = c = log -1 ( ), the partial derivatives transform like

where

∂η ∂s = - s -1 2iπn + log(2) , (154)
and 2) .

∂η ∂x = s - 1 2iπn+log (
(155)

The RZSE then reduces to the ODE s -1 + log(2) + 2iπn ν(η) + -s -1 + 2 log(2)η + 4iπnη ν (η) = 0, ∀ n ∈ Z.

A. General Solution

The homogenous linear differential Eq. ( 156) is separable [35], viz., (161)

V. CONCLUSION

In this study, we have discussed the convergence of the real part of every nontrivial zero of the analytic continuation of the Riemann zeta function. This was accomplished by developing a Riemann zeta Schrödinger equation and comparing it with the Bender-Brody-Müller conjecture in both configuration space and momentum space. A symmetrization procedure was implemented to study the convergence of the system, and the expectation values were calculated from the resulting system to study the nontrivial zeros of the analytic continuation of the Riemann zeta function. Moreover, a second quantization procedure was performed for the Riemann zeta Schrödinger equation to obtain the equations of motion and an analytical expression for the eigenvalues. It was also demonstrated that the eigenvalues are holomorphic. A normalized convergent expression for the analytic continuation of the nontrivial zeros of the Riemann zeta function was obtained, and a convergence test for the expression was performed demonstrating that the real part of every nontrivial zero of the Riemann zeta function converges at σ = 1/2. Finally, a general solution to the Riemann zeta Schrödinger equation was found from performing an invariant similarity transformation.

∀ 1 2( 2 (Theorem 9 .

 129 φ, φ * ∈ D( Ĥ). Finally, lim σ→ Ĥφ * , φ) = lim σ→ 1 φ * , Ĥφ), ∀ 2πn. (89) The densely defined operator Ĥ = -2 √ x∂ x √ x on the Hilbert space H

(φ * , Ĥφ) = π |θ| 2 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 -π 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 -

 211111111111111111111 (1 + α) exp(πβ) cos π(1 + α) + β exp(πβ) sin π(1 + α) + exp(πβ) cos π(1 + α) -α + 2|σ|(1 + α) cos(θ ) exp(-θ ) exp(πβ) cos π(1 + α) -2|σ|β cos(θ ) exp(-θ ) exp(πβ) sin π(1 + α) -2|σ| cos(θ ) exp(-θ ) exp(πβ) cos π(1 + α) + 2|σ|(α + 1) cos(θ ) exp(-θ ) -2|σ| cos(θ ) exp(-θ ) + 2|σ|β sin(θ ) exp(-θ ) + 2|σ|(α + 1) sin(θ ) exp(-θ ) exp(πβ) sin π(1 + α)+ 2|σ|β sin(θ ) exp(-θ ) exp(πβ) cos π(1 + α) -2|σ| sin(θ ) exp(-θ ) exp(πβ) sin π(1 + α) ,(100)where(φ * , Ĥφ) = -2π|σ| cos(θ ) cosh(θ ) exp -2πσ sin(θ ) sinh(θ ) sin 2πσ cos(θ ) cosh(θ )1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 2π|σ| sin(θ ) sinh(θ ) exp -2πσ sin(θ ) sinh(θ ) cos 2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + π exp -2πσ sin(θ ) sinh(θ ) sin 2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 2π|σ| sin(θ ) sinh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 4π|σ| 2 cos 2 (θ ) cosh(θ ) exp(-θ ) exp -2πσ sin(θ ) sinh(θ ) sin 2πσ cos(θ ) cosh(θ )1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -4π|σ| 2 cos(θ ) sin(θ ) sinh(θ ) exp(-θ ) exp -2πσ sin(θ ) sinh(θ ) cos 2πσ cos(θ ) cosh(θ ) 1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -2π|σ| cos(θ ) exp(-θ ) exp -2πσ sin(θ ) sinh(θ ) sin 2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -4π|σ| 2 exp(-θ ) cos(θ ) sin(θ ) sinh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -2π|σ| cos(θ ) exp(-θ ) exp -2πσ sin(θ ) sinh(θ ) sin 2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -4π|σ| 2 exp(-θ ) cos(θ ) sin(θ ) cosh(θ ) exp -2πσ sin(θ ) sinh(θ ) cos 2πσ cos(θ ) cosh(θ )1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -4π|σ| 2 exp(-θ ) sin 2 (θ ) sinh(θ ) exp -2πσ sin(θ ) sinh(θ ) sin 2πσ cos(θ ) cosh(θ )1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 2π|σ| exp(-θ ) sin(θ ) exp -2πσ sin(θ ) sinh(θ ) cos 2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -4π|σ| 2 exp(-θ ) sin(θ ) cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 2π|σ| exp(-θ ) sin(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) , Ĥφ) = -2π|σ| cos(θ ) cosh(θ ) exp -2πσ sin(θ ) sinh(θ ) cos 2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -2π|σ| sin(θ ) sinh(θ ) exp -2πσ sin(θ ) sinh(θ ) sin 2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ )+ π exp -2πσ sin(θ ) sinh(θ ) cos 2πσ cos(θ ) cosh(θ ) 1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -2π|σ| cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 4π|σ| 2 cos 2 (θ ) cosh(θ ) exp(-θ ) exp -2πσ sin(θ ) sinh(θ )cos 2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 4π|σ| 2 sin(θ ) sinh(θ ) exp(-θ ) cos(θ ) exp -2πσ sin(θ ) sinh(θ ) sin 2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -2π|σ| cos(θ ) exp(-θ ) exp -2πσ sin(θ ) sinh(θ ) cos 2πσ cos(θ ) cosh(θ ) 1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 4π|σ| 2 cos 2 (θ ) cosh(θ ) exp(-θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -2π|σ| cos(θ ) exp(-θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 4π|σ| 2 cos(θ ) sin(θ ) cosh(θ ) exp(-θ ) exp -2πσ sin(θ ) sinh(θ ) sin 2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -4π|σ| 2 sin 2 (θ ) sinh(θ ) exp(-θ ) exp -2πσ sin(θ ) sinh(θ ) cos 2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -2π|σ| sin(θ ) exp(-θ ) exp -2πσ sin(θ ) sinh(θ ) sin 2πσ cos(θ ) cosh(θ ) 1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -4π|σ| 2 sin 2 (θ ) sinh(θ ) exp(-θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ).

+- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 2 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 -

 11111111112111111111111 2π|σ| sin(θ ) sinh(θ ) exp -2πσ sin(θ ) sinh(θ ) cos π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -4π|σ| 2 sin 2 (θ ) sinh 2 (θ ) exp -2πσ sin(θ ) sinh(θ ) sin π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -2π|σ| sin(θ ) sinh(θ ) exp -2πσ sin(θ ) sinh(θ ) cos 2π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -2π|σ| cos(θ ) cosh(θ ) exp -2πσ sin(θ ) sinh(θ ) sin 2π -2πσ cos(θ ) cosh(θ ) 1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + π exp -2πσ sin(θ ) sinh(θ ) sin 2π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -4π|σ| 2 sin(θ ) cos(θ ) cosh 2 (θ ) exp -2πσ sin(θ ) sinh(θ ) cos π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 2π|σ| sin(θ ) cosh(θ ) exp -2πσ sin(θ ) sinh(θ ) cos π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 4π|σ| 2 sin 2 (θ ) cosh(θ ) sinh(θ ) exp -2πσ sin(θ ) sinh(θ ) sin π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ )+ 4π|σ| 2 sin(θ ) cos(θ ) sinh 2 (θ ) exp -2πσ sin(θ ) sinh(θ ) cos π -2πσ cos(θ ) cosh(θ ) 1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 4π|σ| 2 cos 2 (θ ) sinh(θ ) cosh(θ ) exp -2πσ sin(θ ) sinh(θ ) sin π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -2π|σ| cos(θ ) sinh(θ ) exp -2πσ sin(θ ) sinh(θ ) sin π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -2π|σ| sin(θ ) sinh(θ ) + 4π|σ| 2 sin(θ ) cosh 2 (θ ) cos(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -2π|σ| sin(θ ) cosh(θ ) -4π|σ| 2 cos(θ ) sin(θ ) sinh 2 (θ )1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) cos 2 (θ ) cosh 2 (θ ) -4π|σ| 2 sin 2 (θ ) sinh 2 (θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 4π|σ| cos(θ ) cosh(θ ) -π 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 4π|σ| 2 cos 2 (θ ) cosh 2 (θ ) exp -2πσ sin(θ ) sinh(θ ) cos π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -2π|σ| cos(θ ) cosh(θ ) exp -2πσ sin(θ ) sinh(θ ) cos π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ )+ 4π|σ| 2 sin 2 (θ ) sinh 2 (θ ) exp -2πσ sin(θ ) sinh(θ ) cos π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -2π|σ| sin(θ ) sinh(θ ) exp -2πσ sin(θ ) sinh(θ ) sin π -2πσ cos(θ ) cosh(θ ) 1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 2π|σ| cos(θ ) cosh(θ ) exp -2πσ sin(θ ) sinh(θ ) cos 2π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -π exp -2πσ sin(θ ) sinh(θ ) cos 2π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ )+ -2π|σ| sin(θ ) sinh(θ ) exp -2πσ sin(θ ) sinh(θ ) sin 2π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -4π|σ| 2 sin 2 (θ ) cosh(θ ) sinh(θ ) exp -2πσ sin(θ ) sinh(θ ) cos π -2πσ cos(θ ) cosh(θ )1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -4π|σ| 2 sin(θ ) cos(θ ) cosh 2 (θ ) exp -2πσ sin(θ ) sinh(θ ) sin π -2πσ cos(θ ) cosh(θ )1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 2π|σ| sin(θ ) cosh(θ ) exp -2πσ sin(θ ) sinh(θ ) sin π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + -4π|σ| 2 cos 2 (θ ) sinh(θ ) exp -2πσ sin(θ ) sinh(θ ) cos π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 2π|σ| cos(θ ) sinh(θ ) exp -2πσ sin(θ ) sinh(θ ) cos π -2πσ cos(θ ) cosh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 4π|σ| 2 cos(θ ) sin(θ ) sinh 2 (θ ) exp -2πσ sin(θ ) sinh(θ ) sin π -2πσ cos(θ ) cosh(θ )1 -4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ) + 4π|σ| 2 sin 2 (θ ) cosh(θ ) sinh(θ ) + 4π|σ| 2 cos 2 (θ ) sinh(θ ) cosh(θ ) -2π|σ| cos(θ ) sinh(θ ) 4|σ| cos(θ ) cosh(θ ) + 2σ 2 cos(2θ ) + cosh(2θ ).

Figure 1 :

 1 Figure 1: Plot of the imaginary components of Eq. (1). Results are compared with Eq. (138) (color online).

  can be seen that since φ solves the RZSE for x and s, then w = -c φ solves the RZSE at x = -a ξ, and s = -b τ . We now construct a group of independent variables such thatξ τ a/b = a x ( b s) a/b = x s a/b = η(x, s),(148)and the similarity variable is thenη(x, s) = xs -a log( ) log(2)+2iπn . b = ν(η),(150)suggesting that we seek a solution of the RZSE with the form φ s (x) = s c log( ) log(2)+2iπn ν(η).

,

  s -1 + log(2) s -1 -4iπnη -η log(4) dη.(157)Integrating on both sides, we obtainln |ν| = c 1 -2iπn + s -1 + log(2) log s -1 -4iπnη -η log(4) 4iπn + log(4) .(158)Exponentiating both sides,|ν| = exp(c 1 ) s -1 -4iπnη -η log(4) exp(c 1 ) = C and dropping the absolute value recovers the lost solution ν(η) = 0, giving the general solution to Eq. (156)ν n (η) = C s -1 -4iπnη -η log(4) ∀ n ∈ Z, ∀ C ∈ R. (160)By setting C = 1, and using Eqs. (149) and (151) in Eq. (160), we obtain the general solution to the RZSE Eq. (32), writtenφ s (x) = s 1 log(2)+2iπn 1 s + s - 1 log(2)+2iπn-x log(4) -4iπnx -2πns+is log(2)+i 4πns-is log(4) , ∀ n ∈ Z.

Figure 2 :

 2 Figure 2: Zoom in of the parametric plot of the eigenvalues in Eq. (139) with coordinates (tn) and (tn) as a function of θ. The lower bound of the summation is n = 0 and the upper bound of the summation is n = 10 (color online).

where the contour C is about R -. From the analytic continuation relations of Eq. ( 1)

Owing to t σ = u σ + iv σ and Eqs. ( 115) and (117), it can be seen that

Finally, the real part of every nontrivial zero of Eq. ( 1) can be obtained from setting the imaginary part of Eq. ( 134) and v σ to zero, i.e.,

Owing to Eq. (38), at σ = 1/2 we obtain

Since v σ = 0, then according to Eq. ( 130), θ = (2n -1)π. Furthermore, from Eq. (129) it can be seen that at θ = (2n -1)π, u σ = 0, and the LHS of Eq. ( 138) is then 0 ∀ n.

Remark. The Riemann Hypothesis states that the real part of all of the nontrivial zeros of the Riemann zeta function are located at σ = 1/2 [START_REF] Riemann | On the Number of Prime Numbers less than a Given Quantity[END_REF].

Remark. Solutions to Eq. ( 3) are symmetric about the origin, i.e., x ∈ [1, ∞), [-1, -∞), and subject to the singularity at φ s (x = 0) = 0 [START_REF] Espinosa | [END_REF].