You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: |
    We construct a new X-ray (2–10 keV) luminosity function of Compton-thin active galactic nuclei (AGNs) in the local universe, using the first MAXI/GSC source catalog surveyed in the 4–10 keV band. The sample consists of 37 non-blazar AGNs at $z=0.002-0.2$, whose identification is highly ($>97\%$) complete. We confirm the trend that the fraction of absorbed AGNs with [$N_{\rm H}$]{} $>
    10^{22}$ cm$^{-2}$ rapidly decreases against luminosity ([$L_{\rm X}$]{}), from 0.73$\pm$0.25 at [$L_{\rm X}$]{} = $10^{42-43.5}$ erg s$^{-1}$ to 0.12$\pm0.09$ at [$L_{\rm X}$]{} = $10^{43.5-45.5}$ erg s$^{-1}$. The obtained luminosity function is well fitted with a smoothly connected double power-law model whose indices are $\gamma_1 = 0.84$ (fixed) and $\gamma_2 =
    2.0\pm0.2$ below and above the break luminosity, $L_{*} =
    10^{43.3\pm0.4}$ ergs s$^{-1}$, respectively. While the result of the MAXI/GSC agrees well with that of HEAO-1 at [$L_{\rm X}$]{} $\gtsim 10^{43.5}$ erg s$^{-1}$, it gives a larger number density at the lower luminosity range. Comparison between our luminosity function in the 2–10 keV band and that in the 14–195 keV band obtained from the Swift/BAT survey indicates that the averaged broad band spectra in the 2–200 keV band should depend on luminosity, approximated by $\Gamma\sim1.7$ for [$L_{\rm X}$]{} $\ltsim 10^{44}$ erg s$^{-1}$ while $\Gamma\sim 2.0$ for [$L_{\rm X}$]{}$\gtsim 10^{44}$ erg s$^{-1}$. This trend is confirmed by the correlation between the luminosities in the 2–10 keV and 14–195 keV bands in our sample. We argue that there is no contradiction in the luminosity functions between above and below 10 keV once this effect is taken into account.
author:
- 'Yoshihiro <span style="font-variant:small-caps;">Ueda</span>, Kazuo <span style="font-variant:small-caps;">Hiroi</span>, Naoki <span style="font-variant:small-caps;">Isobe</span>, Masaaki <span style="font-variant:small-caps;">Hayashida</span>, Satoshi <span style="font-variant:small-caps;">Eguchi</span>, Mutsumi <span style="font-variant:small-caps;">Sugizaki</span>, Nobuyuki <span style="font-variant:small-caps;">Kawai</span>, Hiroshi <span style="font-variant:small-caps;">Tsunemi</span>, Tatehiro <span style="font-variant:small-caps;">Mihara</span>, Masaru <span style="font-variant:small-caps;">Matsuoka</span>, Masaki <span style="font-variant:small-caps;">Ishikawa</span>, Masashi <span style="font-variant:small-caps;">Kimura</span>, Hiroki <span style="font-variant:small-caps;">Kitayama</span>, Mitsuhiro <span style="font-variant:small-caps;">Kohama</span>, Takanori <span style="font-variant:small-caps;">Matsumura</span>, Mikio <span style="font-variant:small-caps;">Morii</span>, Yujin E. <span style="font-variant:small-caps;">Nakagawa</span>, Satoshi <span style="font-variant:small-caps;">Nakahira</span>, Motoki <span style="font-variant:small-caps;">Nakajima</span>, Hitoshi <span style="font-variant:small-caps;">Negoro</span>, Motoko <span style="font-variant:small-caps;">Serino</span>, Megumi <span style="font-variant:small-caps;">Shidatsu</span>, Tetsuya <span style="font-variant:small-caps;">Sootome</span>, Kousuke <span style="font-variant:small-caps;">Sugimori</span>, Fumitoshi <span style="font-variant:small-caps;">Suwa</span>, Takahiro <span style="font-variant:small-caps;">Toizumi</span>, Hiroshi <span style="font-variant:small-caps;">Tomida</span>, Yohko <span style="font-variant:small-caps;">Tsuboi</span>, Shiro <span style="font-variant:small-caps;">Ueno</span>, Ryuichi <span style="font-variant:small-caps;">Usui</span>, Takayuki <span style="font-variant:small-caps;">Yamamoto</span>, Kazutaka <span style="font-variant:small-caps;">Yamaoka</span>, Kyohei <span style="font-variant:small-caps;">Yamazaki</span>, Atsumasa <span style="font-variant:small-caps;">Yoshida</span>, and the MAXI team'
title: ' Revisit of Local X-ray Luminosity Function of Active Galactic Nuclei with the MAXI Extragalactic Survey '
---

INTRODUCTION {#sec:INTRODUCTION}
============

The tight correlation between the mass of a supermassive black hole (SMBH) in a galactic center and that of the budge found in the local universe ([@1998AJ....115.2285M]; [@2000ApJ...539L...9F]; [@2000ApJ...539L..13G]; [@2003ApJ...589L..21M]; [@har04]; [@hop07]; [@kor09]; [@gul09]) leads to an idea of the “co-evolution” of SMBHs and galaxies. Thus, understanding the growth of SMBHs is a fundamental issue to elucidate the cosmological history of the universe in the relation with the evolution of galaxies. The key objects to study this are active galactic nuclei (AGNs), the phenomena where the SMBH gains its mass by accreting gas.

The most basic observational quantities to describe the cosmological evolution of AGNs is the luminosity function (LF), the number density per comoving space as a function of luminosity and redshift. To derive the AGN LF, a statistically well-defined sample (unusually, a flux limited one) with complete identification obtained by unbiased surveys is required. Hard X-ray observations at energies above a few keV provide the most efficient and complete surveys to detect the whole AGNs including obscured ones (so-called “type 2” AGNs), the major class in this population (e.g., see [@gil07]), thanks to its strong penetrating power against absorption by the surrounding material and to little contamination from stars in the host galaxy. In the past several years, combinations of hard X-ray surveys above 2 keV with different survey depths have revealed the evolution of LF of AGNs constituting the major part of the Cosmic X-ray background (CXB), which gives strong constraints on the scenario of the SMBH growth from $z=0$ to $z\sim5$ (e.g., [@ued03]; [@laf05]; [@bar05]; [@sil08]; [@ebr09]; [@yen09]; [@air10]).

To establish the X-ray LF of AGNs in the local universe is of great importance among these efforts, since it gives the reference for any evolution models. While @ued03, @sil08, @ebr09, and @yen09 have found that the luminosity dependent density evolution (LDDE; this term was originally introduced by [@miy00]) best describes of the X-ray AGN LF above 2 keV, @air10 recently suggest that the luminosity and density evolution (LADE) where the “shape” of the LF is constant over the whole redshift range gives a similarly good fit to their data. @ued03 employ the local AGN LF in the 2–10 keV band based on the HEAO-1 all sky survey, by using the sample consisting of 49 AGNs compiled by @shi06. @saz04 [^1] has also determined the local AGN LF in the 3–20 keV band from the RXTE/Slew survey [@rev04], whose integrated volume emissivity corrected for incompleteness is found to be by a factor of $\sim$2 smaller than the HEAO-1 result converted into the same energy band, however.

More recently, hard X-ray surveys above 10 keV performed by the Swift and INTEGRAL satellites also determine the local AGN LF in the 14–195 keV or 15–55 keV band (Swift; [@tue08]; [@bur11]) and in the 20–40 keV or 17–60 keV band (INTEGRAL; [@bec06b]; [@saz07]), respectively. The advantage of these surveys is the least biases against heavily obscured AGNs, although the observed fraction of Compton thick objects with an absorption column density of [$N_{\rm H}$]{} $>10^{24}$ cm$^{-2}$ is found to be as small as $<$5 percent in the total sample ([@tue08]; [@bur11]). It is found the shape of the AGN LF above 10 keV as determined by Swift/BAT looks significantly different from the @shi06 result if the luminosity is simply converted into the other band by assuming a typical AGN spectrum (characterized by a power law with a photon index of $\approx$1.8.). The reasons of this discrepancy have not been understood yet.

Thus, it is timely to revisit the local X-ray AGN LF below 10 keV from a new survey independently, in order to check the consistency with the previous works and solve the apparent contradictions among them. The Monitor of All-sky X-ray Image (MAXI) mission on the International Space Station [@mat09], currently in orbit, provides a valuable opportunity for this. @hir11 produce the first source catalog of the MAXI/Gas Slit Camera (GSC; [@mih11]; [@sug11]) at high galactic latitudes ($|b|>10^\circ$), by compiling the data in the 4–10 keV band accumulated for the first 7 month since the start of its nominal operation. The catalog contains 51 AGNs detected with a significance above 7$\sigma$ consisting of 39 Seyfert galaxies and 12 blazars. In this paper, we constrain the local AGN LF in the 4–10 keV band by using only non-blazar AGNs (i.e., Seyferts) in the @hir11 catalog. We also determine the intrinsic distribution of absorption column density of AGNs (so-called [$N_{\rm H}$]{} function) in the local universe using the same sample, and compare it with the previous results. Section 2 briefly describes the source sample and their X-ray spectral properties in terms of an absorption column density and a photon index determined from various observatories. Section 3 describes the analysis method and obtained results. We discuss the implication in Section 4. The cosmological parameters of [($H_0$, $\Omega_{\rm m}$, $\Omega_{\lambda}$)]{} = (70$h_{70}$ km s$^{-1}$ Mpc$^{-1}$, 0.3, 0.7) are adopted throughout the paper. The “log” symbol represents the base-10 logarithm, while “ln” the natural logarithm.

Sample
======

To investigate the local LF and [$N_{\rm H}$]{} function of AGNs, we collect the 37 non-blazar AGNs from the @hir11 catalog at $z=0.002-0.2$ that constitute a statistically unbiased sample detected in the 4–10 keV band from an area of 34,000 deg$^2$. Here we exclude Cen A, located at $z<0.002$, and ESO 509–066, which has double nuclei and may be contaminated by nearby sources (see Table 1 in [@hir11]). The four “confused” sources are ignored even if they contain contribution from AGNs (like NGC 6814), which do not affect our results. As noted in @hir11, the list of the X-ray brightest AGNs in the all sky has significantly changed from that of the HEAO-1 survey performed 30 years ago; among 39 MAXI/GSC detected AGNs, only 17 objects are listed in both sample by @pic82 and that used by @shi06. The flux limit of the MAXI sample corresponds to $1.5\times10^{-11}$ [erg cm$^{-2}$ s$^{-1}$]{} (1.2 mCrab) in the 4–10 keV band. Figure 1 shows the [log $N$ - log $S$ relation]{} (integral form) of these AGNs in the 4–10 keV band, obtained by using the area curve presented in Figure 9 of @hir11.

(80mm,80mm)[figure1.ps]{}

Table 1 summarizes the AGN list, where the first to sixth columns represent the catalog source No., MAXI source name, counterparts, optical type, 4–10 keV flux, and redshift, respectively. Although it is known that using the spectroscopic redshifts to estimate the distance of very nearby objects is subject to uncertainties due to the galaxy proper motion, we adopt these values for consistency with the analysis of the Swift/BAT AGNs in @tue08. We confirm that even if we instead adopt redshifts corrected for the infall into the Virgo cluster by @mou00 to calculate the luminosity of AGNs at $z<0.01$, our results of both LF and [$N_{\rm H}$]{} function are little affected. Here we only distinguish between two optical classes, “AGN1” (Seyfert 1.0–1.5) or “AGN2” (Seyfert 1.8–2.0), for simplicity. We can regard that this AGN sample is nearly complete (99.3%), because 142 out of the total 143 X-ray sources are identified there. The flux errors due to the statistical uncertainties are better than $14\%$ for all the objects, and hence they are not taken into account in the following analysis.

To compare our result with the previous works easily, we construct the AGN LF in the “intrinsic” 2–10 keV luminosity corrected for the absorption (i.e., before absorption) at the source frame (hereafter represented as [$L_{\rm X}$]{}). Since we have the count rate in the 4–10 keV band from the MAXI/GSC survey, it is necessary to convert it to [$L_{\rm X}$]{}by using the spectral information as well as the redshift for each source. Fortunately, we are able to find results of spectral fits in the 0.2–10 (or 0.5–10) keV band in the literature for 33 (out of 37) AGNs, which were obtained from data of either [[*ASCA*]{}]{}, [[*XMM-Newton*]{}]{}, [[*BeppoSAX*]{}]{}, Swift/XRT, or [[*Suzaku*]{}]{}. The spectral quality is sufficiently good in most cases, and hence we neglect their errors in the following analysis. The best-fit photon index $\Gamma$, absorption column density [$N_{\rm H}$]{} (at the source frame), and calculated luminosity from these parameters ([$L_{\rm X}$]{}) are listed in the 7th to 9th columns of Table 1, respectively, together with the reference for the spectral parameters (10th column). In the conversion from the MAXI/GSC count rate into [$L_{\rm X}$]{}, we consider a reflection component from cold, optically thick matter [@mag95] with a solid angle of $\Omega=2\pi$ as adopted in @ued03, although this does not affect our result of the LF. For the remaining four targets[^2], we perform the same image analysis of the MAXI/GSC data in the 2–4 keV band as that in @hir11 to obtain the hardness ratio between the 2–4 keV and 4–10 keV count rates. We first calculate the corresponding photon index $\Gamma$ without considering any absorption; if we obtain $\Gamma<1.9$, then we derive an absorption column density at the source redshift assuming an intrinsic power law with $\Gamma=1.9$. The results of $\Gamma$ and [$N_{\rm H}$]{} with $1\sigma$ statistical errors estimated in this way are also listed in Table 1 for these 4 targets. Figure 2 shows the redshift ($z$) versus luminosity ([$L_{\rm X}$]{}) plot for our sample. The open and filled circles correspond to those with a column density of log [$N_{\rm H}$]{} $< 22$ and log [$N_{\rm H}$]{} $< 24$, respectively. The optical type-2 AGNs are further marked with the diagonal crosses.

(80mm,80mm)[figure2.ps]{}

Analysis and Results
====================

Analysis Method
---------------

Our goal is to determine both the [$N_{\rm H}$]{} function and absorption-corrected 2-10 keV LF of X-ray AGNs in the local universe. The calculation follows the same procedure as presented in @ued03, to which we refer the reader for details. The same notation convention is adopted in this paper. The [$N_{\rm H}$]{} function, $f(L_{\rm x}, z; N_{\rm H})$ $d$log [$N_{\rm H}$]{}, represents a probability of finding an AGN with an absorption column density between log [$N_{\rm H}$]{} and log [$N_{\rm H}$]{} + $d$log [$N_{\rm H}$]{} at a given luminosity, [$L_{\rm X}$]{}, and redshift, $z$. For convenience, we assign log [$N_{\rm H}$]{}= 20 for AGNs without any significant absorption, and consider only the range of log [$N_{\rm H}$]{} $\leq$ 24, since no Compton thick AGNs are present in the current sample. It is normalized as $$\int_{20}^{24} f (L_{\rm X}, z; N_{\rm H}) d{\rm log} N_{\rm H} = 1.$$ The LF, $\Phi (L_{\rm X}, z)$ in units of Mpc$^{-3}$, is defined so that $
        d \Phi (L_{\rm X}, z)/d{\rm log} L_{\rm X}
$ gives the co-moving space density of all (Compton-thin) AGNs in a luminosity range between log [$L_{\rm X}$]{} and log [$L_{\rm X}$]{} + $d$log [$L_{\rm X}$]{} at a redshift of $z$.

From the list of [$N_{\rm H}$]{} and [$L_{\rm X}$]{} in our sample, the best-fit parameters are searched for by minimizing the likelihood estimator defined as $$\label{eq1}
L = -2 \sum_{i} {\rm ln}
\frac{N(N_{{\rm H}i}, L_{{\rm X}i}, z_i)}
{\int \int \int N( N_{\rm H}, L_{{\rm X}}, z) d{\rm log} N_{\rm H} d{\rm log} L_{\rm X} d z},
\label{eq-likelihood}$$ where the suffix $i$ denotes each object. The term $N(N_{\rm H}, L_{{\rm X}}, z)$ represents the expected number from the survey, $$\begin{aligned}
\nonumber &&N(N_{\rm H}, L_{{\rm X}}, z) = 
f(L_{\rm X}, z; N_{\rm H}) \times \\
&&\;\;\;\;\;
\frac{d \Phi (L_{\rm X}, z)}{d{\rm log} L_{\rm X}}
d_{\rm A}(z)^2 (1+z)^3 c \frac{d\tau}{dz}(z) A (N_{\rm H},L_{\rm X},z), 
\label{eq-likelihood2}\end{aligned}$$ where $d_{\rm A}(z)$ is the angular distance, $d\tau/d z$ the differential look back time, and $A (N_{\rm
H},L_{\rm X},z)$ the survey area, given as a function of flux that is calculated from $N_{\rm H}$, $L_{\rm X}$, and $z$. The minimization process is carried out on the MINUIT software package. The $1\sigma$ error for a single parameter can be estimated from the parameter range that increases the $L$ value by 1. The fit applied to the unbinned data here cannot estimate the normalization of the LF. Hence, we determine it so that the expected source number agrees with the observed one, and estimate its relative uncertainty only from its Poisson error ($1/\sqrt{N}$, where $N=37$).

[$N_{\rm H}$]{} Function
------------------------

To avoid coupling between the [$N_{\rm H}$]{} function and LF, we determine them step by step in the same way as @ued03, considering the small sample size. First, we constrain the [$N_{\rm H}$]{} function by adopting the “delta-function” approximation for the LF only based on the sample list. It reduces the formula (2) to a simpler form (see equation (6) in [@ued03]) where the intrinsic [$N_{\rm H}$]{} distribution can be evaluated directly from its observed [$N_{\rm H}$]{} histogram by taking into account the [$N_{\rm H}$]{} dependence of the survey area. For the four sources whose absorptions are estimated from the hardness ratios of the MAXI data and hence have non-negligible statistical errors, we take into account the uncertainties in [$N_{\rm H}$]{} by introducing the “[$N_{\rm H}$]{} response matrix function” as done in @ued03 (see their Section 4.1).

As for the shape of the [$N_{\rm H}$]{} function, we adopt a modified version of that used in @ued03. The difference from @ued03 is that (1) we allow such a case that the [$N_{\rm H}$]{} function at log [$N_{\rm H}$]{} $<22$ is smaller than that at log [$N_{\rm H}$]{} $>22$, (2) we assign 4 discrete bins with the same width between log [$N_{\rm H}$]{}= 20–24 for simplicity, considering the practical difficulties to determine [$N_{\rm H}$]{} with an accuracy better than $d$log [$N_{\rm H}$]{} $< 1$ for objects without good X-ray spectral data. We define the absorption fraction $\psi (L_{{\rm X}})$ as that of AGNs with log [$N_{\rm H}$]{} = 22–24 among those with log [$N_{\rm H}$]{} = 20–24, which is given as a function of luminosity. Its possible redshift dependence is ignored, because our sample consists of only local AGNs. The form of the [$N_{\rm H}$]{} function is expressed differently for two $\psi (L_{{\rm X}})$ ranges; $$\begin{aligned}
\label{eq-former}
\nonumber &&({\rm for}\;\; \psi (L_{\rm X}) < \frac{1+\epsilon}{3+\epsilon})\;\;\;\;\; \\
&&f(L_{{\rm X}}, z; N_{{\rm H}}) = 
\small
\begin{tabular}{ll}
$1-\frac{2+\epsilon}{1+\epsilon}\psi (L_{{\rm X}})$ & $(20 \leq {\rm log} N_{\rm H} < 21)$\\
$\frac{1}{1+\epsilon} \psi (L_{{\rm X}})$ & $(21 \leq {\rm log} N_{\rm H} < 22)$\\
$\frac{1}{1+\epsilon} \psi (L_{{\rm X}})$ & $(22 \leq {\rm log} N_{\rm H} < 23)$\\
$\frac{\epsilon}{1+\epsilon} \psi (L_{{\rm X}})$ & $(23 \leq {\rm log} N_{\rm H} < 24).$\\
\end{tabular}
\normalsize\end{aligned}$$ and $$\begin{aligned}
\label{eq-latter}
\nonumber &&({\rm for}\;\; \psi (L_{\rm X}) \geq \frac{1+\epsilon}{3+\epsilon})\;\;\;\;\; \\
&&f(L_{{\rm X}}, z; N_{{\rm H}}) =
\small
\begin{tabular}{ll}
$\frac{2}{3}-\frac{3+2\epsilon}{3+3\epsilon}\psi (L_{{\rm X}})$ & $(20 \leq {\rm log} N_{\rm H} < 21)$\\
$\frac{1}{3}-\frac{\epsilon}{3+3\epsilon}\psi (L_{{\rm X}})$ & $(21 \leq {\rm log} N_{\rm H} < 22)$\\
$\frac{1}{1+\epsilon} \psi (L_{{\rm X}})$ & $(22 \leq {\rm log} N_{\rm H} < 23)$\\
$\frac{\epsilon}{1+\epsilon} \psi (L_{{\rm X}})$ & $(23 \leq {\rm log} N_{\rm H} < 24).$\\
\end{tabular}
\normalsize\end{aligned}$$ Here $\epsilon$ defines the ratio of the [$N_{\rm H}$]{} function in log [$N_{\rm H}$]{} = 23–24 to that in log [$N_{\rm H}$]{} = 22–23. It is fixed at 1.3 (instead of 1.7 as adopted in [@ued03]), according to the observed [$N_{\rm H}$]{} distribution in the Swift/BAT 9-month survey (23/18, [@tue08]), which agrees with the more recent result by @bur11 . In the former case (equation \[eq-former\]), the [$N_{\rm H}$]{} function is flat above log [$N_{\rm H}$]{} = 21, while in the latter case (equation \[eq-latter\]), the value in log [$N_{\rm H}$]{} = 21–22 is taken to be the mean of those at log [$N_{\rm H}$]{} = 20–21 and log [$N_{\rm H}$]{} = 22–23. The maximum absorption fraction is $\psi_{max} =
\frac{1+\epsilon}{3+\epsilon}$, corresponding to the case of $f(N_{{\rm
H}})$ = 0 at log [$N_{\rm H}$]{} = 20–21.

Figure 2 clearly shows that X-ray absorbed AGNs are mostly found in the lower luminosity range (log [$L_{\rm X}$]{} $< 44$). This confirms the trend found in many previous works (e.g., [@ued03]; [@has08]) that the absorption fraction reduces with an increase of the AGN luminosity. Thus, following @ued03, we model the absorption fraction by a linear function of log [$L_{\rm X}$]{} within the maximum (see above) and minimum values, which is taken to be 0.1. It is represented as $$\psi(L_{{\rm X}}) = {\rm min} [\psi_{\rm max}, {\rm max} [\psi_{44} - \beta ({\rm log} L_{\rm X} - 44), 0.1] ],$$ where $\beta$ and $\psi_{44}$ are the free parameters to be determined through the likelihood fit.

Table 2 summarizes the best-fit parameters of the [$N_{\rm H}$]{} function and their $1\sigma$ errors. Figure 3(a) plots the “intrinsic” [$N_{\rm H}$]{}function (corrected for the observation bias) for the total sample (upper), that for low luminosities of log [$L_{\rm X}$]{} $< 43.5$ (middle), and that for log [$L_{\rm X}$]{} $>43.5$ (lower). The dependence of the absorption fraction on the luminosity is obvious. The best-fit model of the [$N_{\rm H}$]{}function calculated at the mean [$L_{\rm X}$]{} value in each region is overplotted. Figure 3(b) shows the “observed” histogram of [$N_{\rm H}$]{} for these 3 luminosity ranges, on which those predicted from the best-fit model are superposed.

(80mm,80mm)[figure3a.ps]{} (80mm,80mm)[figure3b.ps]{}

Luminosity Function
-------------------

Using the [$N_{\rm H}$]{} function obtained above, we finally determine the local AGN LF by maximum-likelihood fit according to the formula (2). We adopt the smoothly connected double power law model, one of the most standard descriptions for X-ray AGN LFs, given as $$\frac{d \Phi (L_{\rm X}, z=0)}{d{\rm log} L_{\rm X}} 
= A [(L_{\rm X}/L_{*})^{\gamma_1} + (L_{\rm X}/L_{*})^{\gamma_2}]^{-1}.$$ To implement the effect of the cosmological evolution, we introduce the evolution factor represented by $(1+z)^{p1}$, $$\frac{d \Phi (L_{\rm X}, z)}{d{\rm log} L_{\rm X}} 
= \frac{d \Phi (L_{\rm X}, 0)}{d{\rm log} L_{\rm X}} (1+z)^{p1}, 
\label{eq-PDE}$$ where we fix $p1=4.2$ based on the result obtained for the LDDE model in @ued03. Note that at $z<0.2$ and log [$L_{\rm X}$]{} $> 42$, their LDDE model is identical with the pure density evolution model as represented above.

Due to the limited sample size, we find it difficult to constrain the three free parameters of the LF, $\gamma_1$, $\gamma_2$, and $\L_{*}$ simultaneously. Hence, we fix the power law slope in the low luminosity range at three different values, $\gamma_1 =1.0$, $\gamma_1 =0.84$ (the best-fit obtained from the Swift/BAT 9-month survey in [@tue08]), and $\gamma_1 = 0.62$ (the best-fit from the LADE model in [@air10]). The results of the likelihood fit for these three cases are summarized in Table 2. Figure 4 plots the best fit local LF determined from the MAXI survey for the case of $\gamma_1=0.84$ (black curve). The data points are calculated by the $N_{\rm data}/N_{\rm
model}$ method [@miy01], to which the $1\sigma$ statistical errors are attached according to the formula by @geh86. We find the local AGN emissivity in the 2–10 keV band integrated over the log [$L_{\rm X}$]{} = 41–47 range is $W_{\rm 2-10} = (1.37\pm0.23) \times 10^{39}$ $h_{70}$ erg s$^{-1}$ Mpc$^{-3}$ for $\gamma_1 = 0.84$. This value is close to that obtained by @shi06, $W_{\rm 2-10} = (9.3\pm1.3)
\times 10^{38}$ $h_{70}$ erg s$^{-1}$ Mpc$^{-3}$, [^3] but significantly larger than that by @saz04, $W_{\rm 2-10} =
(5.2\pm0.5) \times 10^{38}$ $h_{70}$ erg s$^{-1}$ Mpc$^{-3}$, converted from the 3–20 keV band LF by assuming a power law photon index of 1.7.

(80mm,80mm)[figure4.ps]{}

Discussion
==========

We revisit the local X-ray luminosity function of non-blazar AGNs, together with the absorption distribution function, based on the first source catalog of the on-going MAXI/GSC extragalactic survey. In spite of the fact that the current MAXI/GSC source sample is smaller than those from the HEAO-1 (49 AGNs in [@shi06]) and RXTE (76 AGNs in [@saz04]) all sky surveys performed in similar energy bands, it has some advantages to firmly establish the statistical properties of AGNs below 10 keV in the following points. (1) The sample is highly complete (99.3% = 142/143, or $97$% = 37/38 in the worst case). (2) Since we have adopted a relatively conservative threshold in the source selection in our catalog ($7\sigma$), our sample is less subject to the flux uncertainties in the faintest end ($<14\%$) and is considered to be free from Eddington’s bias as verified in [log $N$ - log $S$ relation]{}(Figure 1). This could actually be a problem in the sample in @shi06, who had to correct for such biases by simulation (see their Appendix). (3) The AGN fluxes are determined from the data collected from many scans (15 times per day), and hence can be regarded as the long-term averaged flux, less affected by short term variability than those obtained from a few snap-shot observations. (4) The energy band of 4–10 keV band is more suitable for detecting obscured AGNs (except for Compton thick ones), thus reducing the observation biases for the [$N_{\rm H}$]{} function determination. It is expected that the MAXI/GSC 4–10 keV sample has intermediate characteristics between the surveys in softer X-rays than 4 keV, and hard X-ray surveys in above 10 keV.

In fact, we find that the “observed” fraction of absorbed AGNs with log [$N_{\rm H}$]{} $> 22$, 32% (=12/37), is higher than the HEAO-1 (20%=10/49) and RXTE (22%=17/76) results, while it is lower than that obtained from the Swift/BAT survey above 15 keV, 49% (=42/86; [@tue08]). We obtain the intrinsic [$N_{\rm H}$]{} distribution by correcting for the observational biases that AGNs with heavier absorptions are harder to be detected due to the reduction of the count rates. The overall shape combined from both low and high luminosity samples (Figure 3) is well consistent with the [$N_{\rm H}$]{} distribution obtained from the Swift/BAT survey, which show an almost flat distribution above log [$N_{\rm H}$]{} $> 21$ with a weak peak in the log [$N_{\rm H}$]{} = 20–21 bin [@tue08]. We also confirm the strong dependence of the absorption fraction on the X-ray luminosity. We obtain the best-fit formula to describe this relation slightly different from that in @ued03, who included much fainter AGN samples in the analysis. The slope of the absorption fraction with respect to log [$L_{\rm X}$]{} is steep ($\beta=0.23\pm0.02$ instead of $\beta=0.10$ in [@ued03]), and reaches a higher maximum value ($\psi
= 0.82$ instead of $\psi = 0.57$) at low luminosities. Such sharp (even sharper) change of the absorption fraction against the luminosity around log [$L_{\rm X}$]{} $\approx$43.5 is also found in the Swift/BAT sample ([@tue08]; [@bur11]). The statistical uncertainty is quite large at present, however. It is of great importance to investigate redshift evolutions of the [$N_{\rm H}$]{} function and the relation between the absorption fraction and luminosity by using much larger samples, which shall be left for future work.

To compare with the past results obtained from surveys in similar energy bands, we overlay the best-fit LFs obtained by @shi06 and @saz04 with the thin solid (red) and thin dashed (cyan) curves, respectively, in Figure 4. The RXTE LF is converted from the 3–20 keV band into the 2–10 keV band by assuming a photon index of 1.7 in our adopted cosmological parameters ($H_0=70$ km s$^{-1}$ Mpc$^{-1}$ instead of $H_0=75$ km s$^{-1}$ Mpc$^{-1}$ in [@saz04]). The systematic uncertainties due to the choice of photon index are small in this case ($18$% in the luminosity within a range of 1.7–2.0). Here, the normalization of the LF is corrected for the maximum factor of incompleteness ($1/0.7$), assuming that the unidentified targets are all AGNs whose luminosity and redshift distribution is the same as those of the identified sample. As already reported by @saz04, the RXTE result lies significantly lower than the HEAO-1 results by a factor of $\sim 2$. The origin for this discrepancy is unclear, but we do not pursue it further in this paper.

As clearly seen in Figure 4, our MAXI LF is closer to the HEAO-1 LF than the RXTE LF. In particular, it is in good agreement with the HEAO-1 result at high luminosity range above log [$L_{\rm X}$]{} = 43.5. However, the MAXI LF gives a larger number density at lower luminosities by a factor of $\sim$2–3. By assuming a similar slope of the LF in the low luminosity range ($\gamma_1 = 0.84-1.0$), the MAXI LF favors a smaller break luminosity, log $L_{*}$, 42.9–43.9, than the best-fit HEAO-1 value (log $L_{*}$ = $44.0\pm0.4$), though within the statistical errors. The discrepancy can be partially explained if the absorption fraction at these low luminosity range is underestimated than the reality in the previous work. In fact, according to the best-fit [$N_{\rm H}$]{}function, the absorption fraction at log [$L_{\rm X}$]{} = 42.5 is estimated to be $\psi = 0.72\pm0.04$ in our work, while it is $\psi = 0.65$ in @ued03. Due to the coupling with the [$N_{\rm H}$]{} function in constraining the LF parameters through the maximum likelihood fit (see equation (2)), the estimated LF for all AGNs with log [$N_{\rm H}$]{} = 20–24 would become smaller if we assume a lower absorption fraction in the [$N_{\rm H}$]{} function, because it is hard to detect objects with large column densities of log [$N_{\rm H}$]{} $>23$ in the 2–10 keV band survey, and its space density can be only constrained by the extrapolation from the lower column-density range.

Comparison of our AGN LF in the 2–10 keV band with the hard X-ray ($>10$ keV) LF determined with the Swift/BAT and INTEGRAL surveys provides insights on the broad band properties of local AGNs. Since the fraction of Compton-thick AGNs in those hard X-ray surveys are negligibly small, we can directly compare them with our result obtained for Compton-thin AGNs. In Figure 4, we also plot the best-fit form of the LF by @tue08 by converting the luminosity from 14–195 keV to 2–10 keV. In this case, the assumption of the spectrum strongly affects the result. We adopt two photon index, $\Gamma=1.7$ (thick dot-dashed, magenta) and $\Gamma=2.0$ (thick dashed, blue). Obviously, the shape of the LF is not the same between these bands if a single spectrum index is assumed for all AGNs. At low luminosity range of log [$L_{\rm X}$]{} $\ltsim 44$, the normalizations of the two LFs become consistent with each other for $\Gamma \sim 1.7$, while at higher luminosities, the conversion with $\Gamma \sim 2.0$ gives a better agreement.

This result indicates that the averaged shape of broad band X-ray spectra of these AGNs depends on the luminosity, in the sense that more luminous AGNs show a steeper slope in the 2–200 keV range on average. To confirm this picture, we make the correlation plot of luminosity between the 2–10 keV and 14–195 keV bands using our MAXI sample (Figure 5). Here the hard X-ray luminosities are taken from the Swift/BAT 22-month catalog except for two AGNs that are not detected there; 2MASX J18470283–7831494 for which we refer to the Swift/BAT 58-month catalog, and 4C +18.51, which is not yet detected by the 58 month data and has only a flux upper limit of $1.1\times10^{-11}$ [erg cm$^{-2}$ s$^{-1}$]{}in the 14–195 keV band. We plot two lines corresponding to a power law photon index of $\Gamma=1.7$ (solid, magenta) and $\Gamma=2.0$ (dashed, blue). The trend that the AGNs with lower luminosities have flatter slope is indeed seen.

(80mm,80mm)[figure5.ps]{}

We have shown that since the dependence of the averaged X-ray broad band spectra on luminosity makes the direct comparison of LFs constructed in different energy bands (below and above 10 keV) not straightforward, its apparent difference in the LF shape is not “contradiction”. This effect must be taken into account when one constructs a LF in a uniform way by compiling results of X-ray surveys performed in different energy bands. There are two explanations for the reasons of the luminosity dependence of the 2–200 keV spectra. Recent studies of nearby AGNs have suggested that the intrinsic power law components of Seyfert 1 are steeper than those of Seyfert 2 (e.g., [@mal03]; [@bec06a]; [@tue08]). Because of the strong dependence of absorption fraction on the luminosity, we mostly detect only type 1 AGNs in the surveys at the high luminosity range, leading to the trend we see. Second possibility is the effect of a reflection component, which could be more significant at lower luminosities, as implied by the “X-ray Baldwin” effects [@iwa93]. The presence of a reflection hump in the spectra increases the observed flux in the hard X-ray band, peaked at $\sim 20 $ keV, and hence the apparent slope over the 2–200 keV band becomes flatter. To distinguish these two effects, systematic studies of the broad X-ray band spectra of both type 1 and type 2 AGNs at various luminosity ranges are necessary.

Conclusion
==========

We have constructed the local AGN X-ray luminosity function, utilizing our new sample consisting of 37 non-blazar AGNs at $z=0.002-0.2$ detected in the 4–10 keV band from the first MAXI/GSC source catalog by @hir11. The sample is highly complete $>97\%$, and is less subject to uncertainties in the measured fluxes compared with the past all-sky survey missions above 2 keV. The conclusion of our work is summarized as follows.

-   We strongly confirm the trend that there exist more absorbed AGNs at lower luminosities. The fraction of absorbed AGNs with log [$N_{\rm H}$]{} = 22–24 among those with log [$N_{\rm H}$]{} $< 24$ corrected for the observational biases changes from 0.73$\pm$0.25 at log [$L_{\rm X}$]{} = 42–43.5 to 0.12$\pm0.09$ at log [$L_{\rm X}$]{} = 43.5–45.5. The estimated absorption distribution ([$N_{\rm H}$]{} function) is consistent with the Swift/BAT and INTEGRAL results obtained above 10 keV.

-   The shape of the intrinsic luminosity function of Compton thin AGNs can be fit with a smoothly connected double power law. For a fixed slope of $\gamma_1=0.84$ at a lower luminosity range, we obtain a break luminosity of log $L_{*}$ = 43.3$\pm$0.4 and a higher luminosity slope of $\gamma_2 = 2.0\pm0.2$. The $L_{*}$ value is somewhat smaller than the HEAO-1 result. The integrated emissivity over log [$L_{\rm X}$]{} = 41–47 is found to be $(1.37\pm0.23) \times 10^{39}$ erg s$^{-1}$ Mpc$^{-3}$, which is only slightly larger than the previous estimate by HEAO-1. The space density agrees with the HEAO-1 result at log [$L_{\rm X}$]{} $gtsim 43.5$ but is larger at the lower luminosity range. This may be partially explained by the smaller biases against absorption in our survey in the 4–10 keV band, which lead to a better estimate of the [$N_{\rm H}$]{} function.

-   We compare our AGN luminosity function in the 2–10 keV band with those derived above 10 keV, by converting the luminosities by assuming a single power law spectrum. We find that the space densities matches with each other for $\Gamma \sim 1.7$ at log [$L_{\rm X}$]{} $<44$, while they do for $\Gamma \sim 2.0$ at higher luminosities. This suggests the luminosity dependence of the averaged broad X-ray band spectra over the $\sim 2-200$ keV band. The trend is indeed confirmed by the luminosity correlation between the MAXI and Swift/BAT data in our sample.

The work is partially supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Grant-in-Aid No.19047001, 20041008, 20244015, 20540237, 21340043, 21740140, 22740120, 23000004, 23540265, and Global-COE from MEXT “The Next Generation of Physics, Spun from Universality and Emergence” and “Nanoscience and Quantum Physics”.

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[rlllrrccrr]{} No. & MAXI Name & Counterpart & Type & Flux$^a$ & Redshift & $\Gamma$& $N_{\rm H}^b$& Log $L_{\rm X}^c$ &Ref.$^d$\
3& MAXI J0048+320& Mrk 348 & AGN2& 2.42& 0.0150& 1.69& 16& 43.49 & (1)\
9& MAXI J0229+315& NGC 931 & AGN1& 1.77& 0.0167& 1.75& 0.36& 43.27 & (1)\
18& MAXI J0333–364& NGC 1365 & AGN2& 1.97& 0.0055& 2.11& 29& 42.74 & (2)\
24& MAXI J0423–569& 1RXS J042601.6–571202 & AGN1& 1.71& 0.1040& 1.48& 0& 44.85 & (1)\
26& MAXI J0433+053& 3C 120 & AGN1& 2.81& 0.0330& 2.00& 0.05& 44.10 & (3)\
29& MAXI J0510+166& IRAS 05078+1626 & AGN1& 2.77& 0.0179& 2.07$^{+0.55}_{-0.17}$& 0.0$^{+1.9}$& 43.56 & (4)\
31& MAXI J0516–001& Ark 120 & AGN1& 2.24& 0.0323& 1.90& 0& 43.97 & (1)\
40& MAXI J0552–075& NGC 2110 & AGN2& 8.13& 0.0078& 1.54& 2.84& 43.28 & (1)\
41& MAXI J0555+464& MCG +08-11-011 & AGN1& 3.66& 0.0205& 1.64& 0.25& 43.75 & (1)\
49& MAXI J0924–317& 2MASX J09235371–3141305 & AGN2& 5.78& 0.0423& 2.07$^{+0.31}_{-0.17}$& 0.0$^{+0.7}$& 44.64 & (4)\
50& MAXI J0947–309& MCG –05-23-016 & AGN2& 6.58& 0.0085& 1.90& 1.6& 43.28 & (1)\
52& MAXI J1023+197& NGC 3227 & AGN1& 1.85& 0.0039& 1.52& 6.6& 42.06 & (3)\
56& MAXI J1105+725& NGC 3516 & AGN1& 1.84& 0.0088& 1.73& 0.35& 42.73 & (1)\
58& MAXI J1139–378& NGC 3783 & AGN1& 3.21& 0.0097& 1.60& 0.09& 43.04 & (3)\
61& MAXI J1144–184& 2MASX J11454045–1827149 & AGN1& 1.87& 0.0330& 1.92& 0& 43.91 & (1)\
62& MAXI J1210+394& NGC 4151 & AGN1& 6.25& 0.0033& 1.65& 3.4& 42.44 & (3)\
67& MAXI J1240–052& NGC 4593 & AGN1& 1.96& 0.0090& 1.69& 0& 42.76 & (3)\
75& MAXI J1335–342& MCG –06-30-015 & AGN1& 3.42& 0.0077& 1.92& 0.02& 42.90 & (3)\
77& MAXI J1338+045& NGC 5252 & AGN2& 2.93& 0.0230& 1.55& 4.34& 43.80 & (1)\
80& MAXI J1349–302& IC 4329A & AGN1& 7.44& 0.0160& 1.74& 0.36& 43.86 & (3)\
83& MAXI J1413–031& NGC 5506 & AGN2& 5.73& 0.0062& 1.72& 3.23& 42.95 & (3)\
84& MAXI J1418+251& NGC 5548 & AGN1& 3.06& 0.0172& 1.57& 0& 43.51 & (3)\
108& MAXI J1716–629& NGC 6300 & AGN2& 1.65& 0.0037& 1.83& 21.5& 42.19 & (1)\
109& MAXI J1741+185& 4C +18.51 & AGN1& 1.66& 0.1860& 1.90$^{+0.17}_{-0.00}$& 4.0$^{+8.3}_{-4.0}$& 45.47 & (4)\
115& MAXI J1835+328& 3C 382 & AGN1& 2.69& 0.0579& 1.86& 0.06& 44.56 & (5)\
117& MAXI J1837–653& ESO 103-035 & AGN2& 2.26& 0.0133& 1.96& 20.3& 43.45 & (3)\
118& MAXI J1839+798& 3C 390.3 & AGN1& 2.85& 0.0561& 1.64& 0.03& 44.53 & (3)\
119& MAXI J1851–783& 2MASX J18470283–7831494 & AGN1& 1.45& 0.0741& 1.93& 0.01& 44.53 & (3)\
122& MAXI J1920–586& ESO 141-G055 & AGN1& 2.52& 0.0360& 1.72& 0& 44.09 & (3)\
127& MAXI J2009–611& NGC 6860 & AGN1& 1.61& 0.0149& 1.64& 0.10& 43.11 & (6)\
129& MAXI J2041+750& 4C +74.26 & AGN1& 1.83& 0.1040& 1.86& 0.18& 44.93 & (1)\
130& MAXI J2044–107& Mrk 509 & AGN1& 3.32& 0.0344& 1.49& 0& 44.15 & (3)\
132& MAXI J2135–626& 1RXS J213623.1–622400 & AGN1& 1.48& 0.0588& 2.10$^{+0.87}_{-0.20}$& 0.0$^{+3.9}$& 44.35 & (4)\
135& MAXI J2202–319& NGC 7172 & AGN2& 2.02& 0.0087& 1.69& 8.2& 42.85 & (3)\
138& MAXI J2235–259& NGC 7314 & AGN2& 2.46& 0.0048& 1.85& 0.72& 42.33 & (3)\
140& MAXI J2253–177& MR 2251–178 & AGN1& 2.98& 0.0640& 1.41& 0.28& 44.64 & (1)\
142& MAXI J2305–085& Mrk 926 & AGN1& 3.92& 0.0469& 1.61& 0& 44.51 & (3)\
\
\
\
\[tab:ref\_catalog\]

[ccccccc]{} &\
$\epsilon$ & $\Psi_{44}$ & $\beta$ & $A^a$ & $\gamma_1$ &$\gamma_2$& $L_{*}^b$\
1.3 (fixed) & 0.37$\pm$0.03 & 0.23$\pm$0.02 & $(1.49\pm0.25)\times 10^{-5}$ &1.00(fixed) &2.13$^{+0.28}_{-0.21}$ & 43.59$^{+0.33}_{-0.41}$\
&&& $(4.2\pm0.7)\times 10^{-5}$ &0.84(fixed) &2.03$^{+0.21}_{-0.17}$ & 43.30$^{+0.31}_{-0.38}$\
&&& $(8.8\pm1.5)\times 10^{-5}$ &0.62(fixed) &1.95$^{+0.17}_{-0.14}$ & 43.03$^{+0.28}_{-0.33}$\
\
\
\
\[tab2\]

[^1]: The AGN X-ray fluxes (and consequently luminosities) used by @saz04 were underestimated by a factor of 1.4 due to an error in the count rate to flux conversion [@saz08]. In this paper, we correct for this error whenever we refer to the results of @saz04.

[^2]: IRAS 05078+1626, 2MASX J09235371–3141305, 4C +18.51, 1RXS J213623.1–622400

[^3]: It is larger than that presented in Section 6.1 of @shi06, $(5.85\pm1.17)\times 10^{38}$ $h_{70}$ erg s$^{-1}$ Mpc$^{-3}$. This is because while @shi06 calculated the volume emissivity using observed (i.e., absorbed) fluxes of each AGN at luminosity range of log [$L_{\rm X}$]{} $>42$, we here calculate it for intrinsic (de-absorbed) luminosities by integrating the analytical expression of the LF down to log [$L_{\rm X}$]{} $=41$.