Vol.0 (200x) No.0, 000–000

11institutetext: Department of Physics, St. Thomas College, Kozhencherry,
Kerala, India
22institutetext: Inter-University Centre for Astronomy & Astrophysics, Post
Bag 4, Ganeshkhind, Pune, India
33institutetext: Physics Department, Bishop’s University, 2600 College Street,
Sherbrooke, QC, Canada J1M 1Z7

# UV and X-Ray Variability of the Narrow-Line Seyfert 1 Galaxy Ark 564

Savithri H. Ezhikode 11 Gulab C. Dewangan 22 Ranjeev Misra 22 Shruti Tripathi
33 Ninan Sajeeth Philip 11 Ajit K. Kembhavi 22

(Received 2009 month day; accepted 2009 month day)

###### Abstract

We analyse eight XMM-Newton observations of the bright Narrow-Line Seyfert 1
galaxy Arakelian 564 (Ark 564). These observations, separated by $\sim 6$
days, allow us to look for correlations between the simultaneous UV emission
(from the Optical Monitor) with not only the X-ray flux but also with the
different X-ray spectral parameters. The X-ray spectra from all the
observations are found to be adequately fitted by a double Comptonization
model where the soft excess and the hard X-ray power law are represented by
thermal Comptonization in a low temperature plasma and hot corona,
respectively. Apart from the fluxes of each component, the hard X-ray power
law index is found to be variable. These results suggest that the variability
is associated with changes in the geometry of the inner region. The UV
emission is found to be variable and well correlated with the high energy
index while the correlations with the fluxes of each component are found to be
weaker. Using viscous time-scale arguments we rule out the possibility that
the UV variation is due to fluctuating accretion rate in the outer disc. If
the UV variation is driven by X-ray reprocessing, then our results indicate
that the strength of the X-ray reprocessing depends more on the geometry of
the X-ray producing inner region rather than on the X-ray luminosity alone.

###### keywords:

galaxies: active — galaxies: Seyfert — galaxies: individual (Ark 564) —
X-rays: galaxies

##  1 Introduction

Active Galactic Nuclei (AGN) emit over a wide range of the electromagnetic
spectrum and their spectra show strong optical/UV emission lines which are not
present in the spectrum of a normal galaxy. AGN are believed to harbour a
supermassive black hole (SMBH) of mass $\sim 10^{6}-10^{9}M_{\odot}$. The
accretion of matter on to the SMBH is the major source of radiation in AGN,
and they can outshine the stellar emission of the host galaxy (e.g. Peterson
1997; Beckmann & Shrader 2012). According to the standard model of AGN, the
matter accreted from the host galaxy forms an accretion disc surrounding the
central black hole and the spectrum emitted from the disc peaks in the
optical/UV band. Furthermore, there is a hot corona above the disc which
inverse Compton scatters the disc photons resulting in X-ray emission. The
broadband X-ray emission is one of the fundamental characteristics defining
AGN (e.g. Beckmann & Shrader (2012)).

A significant property of AGN is their continuum variability over the entire
electromagnetic spectrum. AGN, in general, show strong X-ray variability and a
subset of AGN called Narrow Line Seyfert 1 (NLS1) shows extreme variability
(e.g. Boller et al. (1996)). Relationship between emission in different bands
provides important insights into the nature of AGN. A number of previous
studies have shown that X-ray and optical/UV variations in AGN are well
correlated (Nandra et al., 1998; Edelson et al., 1996; Smith & Vaughan, 2007;
McHardy et al., 2014). There are two basic models to explain the UV
variability in AGN: the UV variability could be due to accretion rate
variation in the outer disc or it could be due to X-ray reprocessing (e.g.
McHardy et al. (2014)). In the first case, the UV flux variation can provide
information about accretion rate variation while in the second case, the
correlation between the soft/hard X-ray component and the UV emission can
reveal which X-ray spectral component has the greater affect on the outer
disc.

A multiwavelength campaign undertaken by Edelson et al. (1996) observed a
strong correlation between the X-ray, UV and optical variability in the
Seyfert 1 (Sy 1) galaxy NGC 4151, with no detectable lags. The UV observations
were taken with a sampling interval of $\sim$0.05 d while the X-ray
observations were taken twice per day. The obtained results suggest that UV
emission in the source is produced by the reprocessing of primary X-rays. A
recent study by McHardy et al. (2014) investigates the relationship between
the X-ray, UV and optical variability of the Sy 1 galaxy NGC 5548. They
analysed 554 Swift (XRT $\&$ UVOT) observations of the source, typically taken
every two days, over a period of 750 d. The study strengthens the short time-
scale correlations between X-ray and UV/optical bands and the lag measurements
of this object also lead to the conclusion that UV/optical variability is due
to the reprocessing of X-rays. Shemmer et al. (2003) studied the
X-ray$-$optical correlation of the NLS1 galaxy NGC 4051 based on the data
available from 2000 May-July observations. The optical data were retrieved
from Wise Observatory and X-ray data from RXTE. They obtained RXTE data from
251 observations with an interval of 6 h. They conclude that the observed
X-ray$-$optical correlation in the source can be explained as a combined
effect of X-ray reprocessing and the propagation of perturbations from the
outer disc to the X-ray emitting region.

In this work, we study the source Arakelian 564 (Ark 564) which is an X-ray
bright NLS 1 (Brandt et al., 1994; Vaughan et al., 1999) found in the nearby
universe with redshift z = 0.0247. It is a well studied source which is known
to accrete at super-Eddington rate (Mullaney et al., 2009). Earlier studies
have shown that the high energy spectrum of Ark 564 is characterised by a
steep power law (Vignali et al., 2004; Matsumoto et al., 2004). Vignali et al.
(2004) detected absorption corresponding to the O VII K-edge ($\sim$ 0.73 keV)
in two different XMM-Newton observations (2000 & 2001) of the source. They
also obtained evidence for significant X-ray variability of the object both at
low and high energies. The spectral variability analysis of Ark 564 was
carried out by Brinkmann et al. (2007) with the longest exposure observation
available from XMM-Newton. They found that both the soft and hard X-ray flux
are highly variable on short time-scales. The cross-correlation analysis of
light curves showed some delay in the observation of hard band photons with
respect to soft band photons. The high energy photon index $\Gamma$ was also
variable and it was found to be leading the variations in soft and hard energy
bands. A previous study using ASCA observations by Bian & Zhao (2003)
determined the relations between hard X-ray variability, photon index $\Gamma$
and Eddington ratio $\dot{m}$ of a sample of AGN, including Ark 564. They
found that the X-ray variability and central black hole mass $M_{BH}$ are
strongly anti-correlated while a weak correlation exists for the variability
and $\dot{m}$. This means that small value of $M_{BH}$ is responsible for the
variability of NLS1s. The study also discovered a strong correlation between
$\Gamma$ and $\dot{m}$.

Smith & Vaughan (2007) examined the X-ray and optical variability of Ark 564
and seven other Sy 1 galaxies over a period of $\sim$1 d. The source was
variable in X-rays, but not in the optical band. But another study by Shemmer
et al. (2001) investigated the optical$-$UV$-$X-ray connection of Ark 564 over
a longer period of time of $\sim$50 d. It was a two year long multiwavelength
monitoring program in which the X-ray observations were covered with RXTE and
ASCA while the UV observations with HST. They observed a significant
correlation of the continua, where the X-ray continuum was followed by UV with
a time lag of 0.4 d which in turn was followed by the optical band by $\sim$2
d. The soft X-ray flux was also found to be well correlated with the hard band
flux with zero time lag. Rapid X-ray variations of $\lesssim$1 d was observed
in the source, but the mean flux was constant on time-scales $>$30 d.

A study has been carried out by Dewangan et al. (2007) of the soft excess
emission from the NLS1 galaxies Ark 564 and Mrk 1044. They argued that the
soft excess emission from Ark 564 can be explained by considering a two-
component corona in the source. According to their model, the geometry of the
corona is such that it consists two different physical regions, one being
optically thick and cool while the other a high-temperature, optically thin
region. The hot corona extends above the low-temperature corona while the
latter is coupled to the inner part of the accretion disc. The optical/UV
photons emitted from the accretion disc are Comptonized by the optically thick
corona leading to the soft X-ray emission. The geometry suggests that a
fraction of these scattered photons again get inverse Compton scattered by the
optically thin corona giving rise to the hard X-ray spectrum.

In this work, we examine the correlation between the X-ray and UV emission
from Ark 564, by using eight XMM-Newton observations in 2011. The data from
these observations have already been analysed to study the observed time lag
between the soft and hard X-ray emissions (Legg et al., 2012) and the
frequency-dependent Fe K lags (Kara et al., 2013). Legg et al. (2012) detected
a delayed ($\sim 1000s$) hard X-ray emission in the 4$-$7.5 keV with respect
to a flaring in the soft X-ray band (0.4$-$1 keV). In view of these results
Giustini et al. (2015) reported on the X-ray spectral properties using XMM-
Newton and Suzaku observations of Ark 564 by analysing the time-averaged,
flux-selected and time-resolved spectra. They interpreted the delayed hard
excess component as the reprocessing of soft photon flares Compton up-
scattered in a medium situated at 10$-$100 of gravitational radii. However
these studies have been limited to the variations in the X-ray band only.
Moreover, most of the previous UV/X-ray correlation studies of the source have
concentrated on the variation of total X-ray counts or flux with that of the
UV band. The high quality X-ray spectral data available by XMM-Newton provide
us the opportunity to study the variation of different X-ray spectral
components with the UV flux. In the present work, we fit the spectra utilising
some physical models to obtain the spectral parameters and then study the
correlation between these parameters and the UV flux.

The paper is organized as follows. In Section 2, we describe the observations
and data reduction. We discuss the variability of the source and spectral
analysis in Section 3 and Section 4, respectively. Then we present the
correlations in Section 5. Finally, we summarize and discuss our results in
Section 6.

##  2 Observations and Data Reduction

We used eight observations of Ark 564 taken by XMM-Newton between May and July
2011. The details of these pointed observations are given in Table 1. XMM-
Newton has simultaneous X-ray and UV exposures for all these observations.
X-ray data from the European Photon Imaging Camera (EPIC) pn and MOS and UV
data from the Optical Monitor (OM) were retrieved from the HEASARC archive.
The EPIC-pn camera (Strüder et al., 2001) is positioned such that the incoming
radiation from the source enters the primary focus unobstructed, but the EPIC-
MOS cameras (Turner et al., 2001) can receive only half of the radiation. The
EPIC-pn has large effective area at high energies as well as high quantum
efficiency. The pn CCDs are also less susceptible to pile-up during the
observations of bright sources. We therefore focus our analysis on the pn
data, where the camera was operated in small window mode using a thin filter.
OM observations were performed with the UVW2 filter in the imaging mode. The
data reduction is done with SAS version 14.0 using the updated calibration
files available in July 2015.

The event lists for EPIC-pn are filtered for single and double (PATTERN=4)
best quality (FLAG=0) events in the energy range 0.2$-$10 keV. To examine the
flaring particle background, light-curves are extracted in the 10$-$12 keV
band for single events. The intervals of flaring background are then removed
from the event list using the threshold rate of 0.1 counts/s obtained from the
light curve. The source spectra were extracted from circular regions of radius
36around the centre of the source. The background spectra were also extracted
from two circular regions of 30radii on the same chip, but devoid of source
photons. We checked whether any of the observations were affected by pile-up
using the SAS task epatplot. But no significant deviation in the pattern
distribution was observed in any of them. Then we rebinned the data with the
tool specgroup. While binning we ensured that each bin has a minimum number of
20 counts. Also the oversampling factor was set to 5 such that there are no
more than 5 bins to cover the energy resolution.

All the Optical Monitor observations were made with UVW2 filter in the Imaging
mode. The SAS task ‘omichain’ is used to reprocess the OM data, which
automatically produces the combined source list of all the filters. But there
is a chance that the detection algorithm (‘omdetect’) may misidentify the
source as an extended one, yielding incorrect values of count rate. In order
to avoid this, omichain is run with the option ‘omdetectdetectextended=no’ for
the detection and photometry to be performed as on a point source. Then the
count rate of the source is obtained form the combined source list of each
observation.

Table 1: List of XMM-Newton Observations of Ark 564.

Observation | Observation ID | Start Date | Duration | PN Exposure Time1 | Number of OM
---|---|---|---|---|---
Number |  |  | s | s | Exposures
1 | 0670130201 | 2011 May 24 | 59520 | 41180 | 39
2 | 0670130301 | 2011 May 30 | 55919 | 36510 | 40
3 | 0670130401 | 2011 June 05 | 63582 | 31250 | 45
4 | 0670130501 | 2011 June 11 | 67312 | 43850 | 45
5 | 0670130601 | 2011 June 17 | 60919 | 35570 | 45
6 | 0670130701 | 2011 June 25 | 64439 | 29470 | 43
7 | 0670130801 | 2011 June 29 | 58216 | 40500 | 40
8 | 0670130901 | 2011 July 01 | 55915 | 38400 | 40

1Net exposure time for the EPIC-pn camera

Table 2: X-ray count rate from EPIC-pn in the 0.3$-$10 keV band (Column 2), UV count rate from OM UVW2 filter (Column 3) and the corresponding UV flux (Column 4)  Observation | EPIC-pn Count Rate | UVW2 Count Rate |  FUV
---|---|---|---
Number |  counts s-1 |  counts s-1 |  10-15 erg cm-2 s-1 Å-1
1 |  55.37 $\pm 0.04$ |  1.29 $\pm 0.01$ |  7.39 $\pm 0.04$
2 |  36.33 $\pm 0.03$ |  1.31 $\pm 0.01$ |  7.49 $\pm 0.04$
3 |  37.11 $\pm 0.03$ |  1.22 $\pm 0.01$ |  6.99 $\pm 0.04$
4 |  43.52 $\pm 0.03$ |  1.29 $\pm 0.01$ |  7.37 $\pm 0.04$
5 |  39.78 $\pm 0.03$ |  1.25 $\pm 0.01$ |  7.13 $\pm 0.04$
6 |  24.20 $\pm 0.03$ |  1.18 $\pm 0.01$ |  6.71 $\pm 0.04$
7 |  38.28 $\pm 0.03$ |  1.28 $\pm 0.01$ |  7.30 $\pm 0.04$
8 |  50.91 $\pm 0.04$ |  1.33 $\pm 0.01$ |  7.57 $\pm 0.04$
Figure 1: The variation in UV flux FUV obtained from UVW2 filter of Optical
Monitor (Upper panel) and X-ray count rate from EPIC-pn in the range 0.3$-$10
keV (Lower panel). The horizontal dot-dashed line in each panel corresponds to
the constant best-fit value obtained using $\chi^{2}$ analysis.The vertical
error-bars correspond to $1\sigma$ errors and in case of the X-ray count rate
these are very small. Figure 2: The variation in X-ray count rate from EPIC-pn
in the range 0.3$-$10 keV. Here, the vertical error-bars ($1\sigma$) are very
small. The horizontal solid line corresponds to the constant best-fit value
obtained using $\chi^{2}$ analysis.

##  3 UV $-$ X-ray Variability

We calculated the UV flux FUV of the source from each observation by
multiplying the count rate, obtained from the combined source list, with the
conversion factor 5.71$\times$ 10-15 erg cm-2 s-1 Å-1 for the UVW2
filter111See XMM-Newton Optical and UV Monitor (OM) Calibration Status
document, Talavera 2011 (http://xmm2.esac.esa.int/docs/documents/CAL-
TN-0019.pdf). Both EPIC-pn and UVW2 count rates and UVW2 flux values are given
in Table 2. We checked the variability of the source by $\chi^{2}$ analysis,
which measures the deviation of the data points from the best-fit constant.
The analysis yielded large $\chi^{2}$ values for FUV and pn count rate
(0.3$-$10 keV), proving that the source is highly variable in X-ray and UV
bands. This is clear from Fig. 1 and Fig. 2.

In order to confirm that the observed variability is not an artifact of the
instrument we checked the variability in FUV of other sources which happened
to be in the same field of view of all UVW2 observations. Among these 23
sources five were found to be varying in flux while other sources didn’t show
any sign of variability. Some of the non-variable sources observed in the same
field of view are NVSS J224244+293856, NVSS J224252+294533, 2MASX
J22425351+2943125 etc.

We have also calculated the fractional rms variability amplitude of both FUV
and pn count rate of Ark 564 and the respective values are $\sim$0.039 and
$\sim$0.245. This shows that the observed variance in X-ray count rate is
large compared to the UV flux variance.

##  4 Spectral Analysis

The spectral analysis of EPIC-pn and UVW2 data was done with the XSPEC package
(Version 12.8.2). The $\chi^{2}$ statistic was applied for the spectral
fitting and the errors calculated for each parameter correspond to the 90%
confidence range, unless otherwise stated.

The template spectral file and response files for the OM data were obtained
from the XMM-Newton
website222http://xmm2.esac.esa.int/external/xmm_sw_cal/calib/om_files.shtml.
We used the template file and the measured count rate to create the spectral
file for the analysis of UVW2 data.

As an example of how we have systematically done the spectral analysis we
report here the analysis of Observation Number 2 (Observation ID: 0670130301).
The same analysis scheme was applied to the other observations. We started the
spectral analysis by fitting the EPIC-pn data, in the energy range 3 to 10
keV, using powerlaw together with TBabs, the Tuebingen-Boulder ISM absorption
model (Wilms et al., 2000). This model with photon index $\Gamma\sim 2.3-2.5$
and the Galactic column density $N_{H}$ fixed to $5.41\times 10^{20}cm^{-2}$
(Kalberla et al., 2005) provides a reasonable fit for all observations. It
gave a $\chi^{2}$ of 239.66 for 162 degrees of freedom (dof) for the second
observation. Some excess emission features were detected in the range 6.4$-$7
keV which might be attributed to the fluorescent Fe emission line. An
improvement in fit, $\Delta\chi^{2}=-24.29$, was observed in the second
observation, when a red-shifted Gaussian profile (zgauss) at $\sim$ 6.6 keV
was included. The line appeared to be broad having a width,
$\sigma$=0.25${}_{-0.11}^{+0.61}$ keV. But this broad Gaussian did not fit
properly the spectrum of any of the other observations. When we fixed $\sigma$
of the line to 0.5 keV, all observations could achieve a good fit statistic.

Table 3: Best-fit parameters for the model
uvred$\times$zedge$\times$zedge$\times$TBabs(ezdiskbb+Simpl$\otimes$Nthcomp+zgauss)
fitted to the UVW2 and EPIC-pn spectral data

Model | Observation Number
---|---
Component | Parameter | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
zedge 1 | Eedge (keV) | 0.72 $\pm 0.01$ | 0.69 $\pm 0.01$ | 0.71 $\pm 0.01$ | 0.71 $\pm 0.01$ | 0.71 $\pm 0.01$ | 0.7 $\pm 0.01$ | 0.71 $\pm 0.01$ | 0.71 $\pm 0.01$
| $\tau^{(1)}_{max}$ | 0.11 $\pm 0.01$ | 0.12 ${}^{+0.01}_{-0.02}$ | 0.12 $\pm 0.02$ | 0.15 $\pm 0.02$ | 0.11 $\pm 0.02$ | 0.18 $\pm 0.03$ | 0.15 $\pm 0.02$ | 0.12 $\pm 0.02$
zedge 2 | Eedge (keV) | 0.5 $\pm 0.01$ | 0.51 $\pm 0.02$ | 0.53 $\pm 0.02$ | 0.54 $\pm 0.01$ | 0.52 $\pm 0.01$ | 0.54 ${}^{+0.01}_{-0.02}$ | 0.536 ${}^{+0.008}_{-0.003}$ | 0.52 $\pm 0.01$
| $\tau^{(1)}_{max}$ | 0.1 $\pm 0.02$ | 0.05 $\pm 0.02$ | 0.05 $\pm 0.03$ | 0.09 ${}_{-0.02}^{+0.03}$ | 0.07 ${}_{-0.02}^{+0.03}$ | 0.11 $\pm 0.04$ | 0.12 $\pm 0.03$ | 0.06 $\pm 0.02$
Simpl | $\Gamma_{Simpl}$ | 2.60 $\pm 0.01$ | 2.56 $\pm 0.02$ | 2.50 $\pm 0.02$ | 2.55 $\pm 0.01$ | 2.53 $\pm 0.02$ | 2.50 $\pm 0.02$ | 2.53 $\pm 0.01$ | 2.60 $\pm 0.01$
| $f_{sc}$ | 0.232 ${}_{-0.005}^{+0.010}$ | 0.20 $\pm 0.01$ | 0.20 $\pm 0.01$ | 0.24 $\pm 0.01$ | 0.22 ${}_{-0.01}^{+0.02}$ | 0.22 $\pm 0.02$ | 0.23 $\pm 0.01$ | 0.23 $\pm 0.01$
| L${}^{(2)}_{Simpl}$/LEdd | 0.302 $\pm 0.004$ | 0.184 $\pm 0.003$ | 0.201 $\pm 0.003$ | 0.241 $\pm 0.005$ | 0.220 $\pm 0.004$ | 0.126 $\pm 0.004$ | 0.207 $\pm 0.004$ | 0.277 $\pm 0.005$
Nthcomp | $\Gamma_{Nthcomp}$ | 1.89 ${}_{-0.04}^{+0.06}$ | 1.88 ${}_{-0.1}^{+0.08}$ | 1.96 $\pm 0.08$ | 1.71 ${}_{-0.13}^{+0.11}$ | 1.91 ${}_{-0.15}^{+0.11}$ | 1.59 ${}_{-0.19}^{+0.17}$ | 1.55 ${}_{-0.15}^{+0.13}$ | 1.88 ${}_{-0.11}^{+0.08}$
| kTe (eV) | 158.07 ${}_{-4.25}^{+4.04}$ | 160.77 ${}_{-5.40}^{+5.41}$ | 164.07 ${}_{-5.32}^{+5.80}$ | 151.15 ${}_{-5.21}^{+5.27}$ | 155.81 ${}_{-7.49}^{+6.41}$ | 150.31 ${}_{-7.19}^{+8.01}$ | 143.50 ${}_{-4.70}^{+5.02}$ | 155.69 ${}_{-5.55}^{+4.80}$
| kTbb (eV) | $<30.03$ | $<29.81$ | $<188.56$ | 33.12 ${}_{-2.52}^{+1.91}$ | 32.46 ${}_{-2.26}^{+2.03}$ | 33.08 ${}_{-2.55}^{+1.86}$ | 34.97 ${}_{-1.93}^{+1.58}$ | $<31.91$
| L${}^{(3)}_{Nthcomp}$/LEdd | 0.284 ${}_{-0.006}^{+0.005}$ | 0.195 ${}_{-0.005}^{+0.004}$ | 0.186 $\pm 0.005$ | 0.236 $\pm 0.004$ | 0.200 $\pm 0.004$ | 0.135 $\pm 0.003$ | 0.216 $\pm 0.004$ | 0.261 $\pm 0.005$
zgauss | ELine (keV) | 6.80 ${}_{-0.24}^{+0.25}$ | 6.61 $\pm 0.15$ | 6.69 ${}_{-0.21}^{+0.20}$ | 6.77 ${}_{-0.14}^{+0.15}$ | 6.66 ${}_{-0.18}^{+0.19}$ | 6.67 $\pm 0.16$ | 6.55 $\pm 0.13$ | 6.41 ${}_{-0.23}^{+0.22}$
| F${}^{(4)}_{Line}$ (10-5) | 2.58 ${}_{-0.68}^{+0.66}$ | 2.90 $\pm 0.62$ | 2.69 $\pm 0.71$ | 3.33 ${}_{-0.62}^{+0.63}$ | 2.88 ${}_{-0.66}^{+0.68}$ | 2.7 $\pm 0.6$ | 3.36 $\pm 0.62$ | 2.86 ${}_{-0.69}^{+0.71}$
$\chi^{2}$/dof | 307.78/251 | 375.10/248 | 358.36/247 | 357.22/250 | 324.17/248 | 277.84/238 | 359.74/249 | 349.05/248
$\chi^{2}_{\nu}$ | 1.24 | 1.51 | 1.45 | 1.43 | 1.31 | 1.17 | 1.44 | 1.41

(1)Maximum optical depth for absorption at the threshold energy.

(2)Normalized hard X-ray luminosity.

(3)Normalized soft X-ray luminosity.

(4)Line flux or the normalization of the redshifted gaussian line in units of
photons cm-2 s-1

Extrapolation of the model down to 0.3 keV provided a poor fit indicating the
presence of soft excess in the spectrum. We attempted to describe this soft
excess with the thermal Comptonization XSPEC model Nthcomp. In this model, the
temperature $kT_{bb}$ of seed photons from the accretion disc (blackbody or
disc blackbody) parameterizes the low energy cut-off, while the high energy
roll over is given by the electron temperature $kT_{e}$ ($\sim$ 160 eV). We
assume that the photons from this component are the seed photons to the second
thermal component giving rise to the high energy power law emission. Hence, we
re-analysed the data replacing the powerlaw by introducing the convolution
XSPEC model (Simpl) which transfers a fraction of the seed photons in the
input spectrum into a power law (Steiner et al., 2009). The model yielded an
unacceptable fit statistic of $\chi^{2}$/dof=547.48/253. Inspection of the
residuals revealed that the soft part of the spectrum was affected by an
absorption feature around 0.7 keV. A better $\chi^{2}=385.74$ for 251 dof was
obtained for the same observation, when we fitted the region with a red-
shifted absorption edge model, zedge. Another absorption feature was also
found at $\sim$0.5 keV and it was modelled using one more zedge.
Correspondingly the $\chi^{2}$ was improved by $\Delta\chi^{2}=-9.92$ which
clearly shows the significance of including the new absorption feature. We
also tried to model these features using more complex models such as zxipcf
and grid22soft and obtained fit statistics comparable to the phenomenological
two edge model. Moreover, the relevant spectral parameters such as the hard
and soft X-ray fluxes and high energy spectral index are not sensitive to the
absorption model used. Thus we proceed with the simple phenomenological model
of two edges in this work. For this model, we show the unfolded spectrum and
residuals in Fig. 3.

Figure 3: Top panel: The Unfolded EPIC-pn spectral data and the best-fitting
model zedge$\times$zedge$\times$TBabs(Simpl$\otimes$Nthcomp+zgauss) for
Observation Number 2. Bottom panel: The deviation of the observed data from
the model.

In order to study the relationship between X-ray and UV emission, we need to
fit the spectra simultaneously. For this we loaded the UVW2 data along with
the pn data and then added the model ezdiskbb which describes the accretion
disc spectrum consisting of multiple blackbody components (Zimmerman et al.,
2005). The model is defined by two parameters, the inner disc temperature
$kT_{in}$ and the norm Ndisk, where Ndisk is determined by the inner disc
radius and the inclination of the disc. Further the effect of interstellar
extinction on the source spectrum was taken into account using the model
uvred, based on Seaton’s law (Seaton, 1979). This UV reddening model is valid
only in the range 1000$-$3074 Å, and can be used in combination with
photoelectric absorption models. The parameter E(B-V) was determined from the
reddening law R${}_{V}=$ AV/E(B-V) (Fitzpatrick, 1999) and the AV magnitude of
0.198 was obtained from original SFD98 values assuming R${}_{V}=3.1$. So the
value of E(B-V) was fixed to 0.064 for all observations. Then we fitted the
UVW2 data simultaneously with the EPIC-pn data by tying the parameter
$kT_{in}$ to the $kT_{bb}$ of Nthcomp. This left only the normalisation of the
ezdiskbb model as a free parameter to fit the single UV data point. To ensure
that only the outer disc emission is used to fit the UV data point, we fixed
the normalisation of the Comptonization component at a negligible value for
the UV part of the spectrum. The best fit parameters for this model for all
the observations are given in Table 3.

We have also calculated the X-ray flux for each model component in the range
0.3$-$10 keV using the XSPEC convolution model cflux. The flux corresponding
to the model component Simpl was calculated from the unabsorbed X-ray flux and
the Nthcomp flux. It was obtained by subtracting $(1-f_{sc})$ times the
Nthcomp flux from the unabsorbed flux. Using the luminosity distance of
D=98.5Mpc, by assuming the cosmological parameters H0=73 km s-1 Mpc-1,
$\Omega_{m}$=0.27 and $\Omega_{\Lambda}$=0.73, we obtained the luminosities of
each component.

The normalisation of the ezdiskbb model, Ndisc is related to the inner disc
radius $R_{in}$ by

$R_{in}(km)=f^{2}\left(\dfrac{N_{disc}}{\cos i}\right)^{1/2}D_{10kpc},$ (1)

where $D_{10kpc}$ is the luminosity distance to the source in units of 10 kpc.
i is the inclination angle assumed here to be 30∘ and f stands for the colour
correction factor which we assume to be the generally accepted value of 1.7.
The mass accretion rate can be obtained using (Zimmerman et al., 2005)

$\dot{M}=\frac{8\pi\sigma}{3GM}\left(\frac{T_{\ast}}{f}\right)^{4}R_{in}^{3},$
(2)

where $T_{\ast}=T_{in}/0.488$. Several previous studies have estimated the
black hole mass of Ark 564 to be in the range ($1.15-10$)$\times
10^{6}M_{\odot}$ (e.g. Pounds et al. (2001), Wang & Lu (2001), Bian & Zhao
(2003), Botte et al. (2004), Zhou & Wang (2005), Zhang & Wang (2006)). In this
study, we adopt the value 2.61$\times 10^{6}$ M⊙ (Botte et al., 2004) obtained
from stellar velocity dispersions. This allows us to express the luminosities
in terms of the Eddington Luminosity $L_{Edd}$ and the accretion rate in terms
of the Eddington accretion rate, $\dot{M}_{Edd}=\frac{L_{Edd}}{\eta c^{2}}$,
where $\eta$ is the efficiency factor taken as 0.1. The normalised
luminosities for the different observations are listed in Table 3.

Table 4: Reduced $\chi^{2}$ values for constant model fits to spectral parameters derived from eight observations Model | Parameter | $\chi^{2}_{\nu}$
---|---|---
zedge (at $\sim$0.7 keV) | Eedge | 0.76
| $\tau_{max}$ | 1.16
zedge (at $\sim$0.5 keV) | Eedge | 2.69
| $\tau_{max}$ | 1.18
ezdiskbb | Ndisc | 0.12
Simpl | $\Gamma_{Simpl}$ | 6.13
| $f_{sc}$ | 1.53
Nthcomp | $\Gamma_{Nthcomp}$ | 1.58
| kTe | 1.56
| kTbb | 0.23
| NNthcomp | 26.07
zgauss | ELine | 0.42
| FLine | 0.21
Flux | FUV | 61.79
Luminosity | $L_{Nthcomp}$ | 129.15
| $L_{Simpl}$ | 200.41
Table 5: Spearman’s correlation between different parameters. Column 3: Spearman’s rank-order correlation coefficient. Column 4: p-value Parameter 1 | Parameter 2 | r | p
---|---|---|---
FUV | Count Rate X-ray | 0.69 | 0.06
FUV | LNthcomp/LEdd | 0.71 | 0.05
FUV | LSimpl/LEdd | 0.57 | 0.14
FUV | $\Gamma_{Simpl}$ | 0.93 | 0.0009
LNthcomp/LEdd | LSimpl/LEdd | 0.95 | 0.0003
LNthcomp/LEdd | $\Gamma_{Simpl}$ | 0.78 | 0.02
LSimpl/LEdd | $\Gamma_{Simpl}$ | 0.69 | 0.06
Figure 4: The variation of different parameters with UV flux $F_{UV}$. The
left and right panels respectively show the soft and the hard X-ray
luminosities as a function of $F_{UV}$. The middle panel depicts the
dependence of $\Gamma_{Simpl}$ on $F_{UV}$. The luminosities are expressed
relative to the Eddington value. Figure 5: The correlation of different
parameters. The luminosities are denoted in units of Eddington luminosity.
Left $\&$ Middle panels: The variation of hard X-ray luminosity and hard X-ray
photon index with soft X-ray luminosity. Right Panel: The dependence of hard
X-ray photon index on the hard X-ray luminosity.

##  5 UV & X-ray Correlations

We investigate the correlation between varying X-ray spectral parameters,
luminosities and $F_{UV}$. We identify the variable parameters by fitting a
constant to the best-fit values obtained from different observations using
$\chi^{2}$ analysis. The reduced $\chi^{2}$ values are listed in Table 4, with
the larger reduced $\chi^{2}$ values corresponding to highly variable
parameters. Apart from the luminosities of the X-ray components and the UV
flux, the high energy photon index $\Gamma_{Simpl}$ is also found to be
variable. Hence we restrict our correlation analysis to these parameters.

Since the nature of the relations between the parameters are unknown we need a
non-parametric method to calculate the correlation. So we use Spearman’s rank-
order correlation (Press et al., 1992) to reveal any correlations. The rank
and significance of correlations for different parameters are listed in Table
5. Fig. 4 and Fig. 5 show the plots of different parameters for which we find
significant correlations.

In the X-ray domain, the soft excess luminosity $L_{Nthcomp}$ is well
correlated to the high energy power law luminosity $L_{Simpl}$. In fact it
seems that $\Gamma_{simpl}$ is better correlated with $L_{Nthcomp}$ than it is
with the high energy luminosity $L_{Simpl}$, in the sense that the null
hypothesis probability $p$ is significantly smaller. As we discuss in the next
section these correlations are consistent with the double Comptonization model
used for the spectral fitting.

For the UV$-$X-ray correlations, we note that there is not much evidence for
any correlation between $F_{UV}$ and the X-ray luminosities in the two
Comptonization component $L_{Nthcomp}$ and $L_{Simpl}$ with probabilities of
$p=0.05$ and $0.14$. However a strong correlation is seen for $F_{UV}$ and
$\Gamma_{simpl}$ with $p=0.0009$.

##  6 Summary and Discussion

We have analysed the simultaneous X-ray (EPIC-pn) and UV (OM UVW2) data from
eight XMM-Newton observations of Ark 564 taken in 2011. We used the thermal
Comptonization models favoured by Dewangan et al. (2007) for the spectral
fitting of the X-ray data. The soft X-ray spectrum (0.3$-$3 keV) was modelled
by Nthcomp assuming that the UV photons emitted from the accretion disc are
Comptonized by the optically thick corona leading to the soft excess emission.
The accretion disc emitting the seed photons is described by the multicolour
black body model ezdiskbb. Further, the hard X-ray power law emission (3$-$10
keV) is taken into account by the second Comptonization model Simpl which
incorporates the physics of Compton upscattering of soft photons by hot
coronal electrons. We report that this double Comptonization model fits all
the eight spectra well.

In the X-ray band, we find that the luminosity of the soft Comptonization
component $L_{nthcomp}$ correlates well with the hard component $L_{Simpl}$.
While there is some evidence that the high energy index $\Gamma_{Simpl}$
correlates well with both luminosities, there is a seemingly stronger
correlation between $\Gamma_{Simpl}$ and $L_{nthcomp}$ in the sense that null
hypothesis probability is smaller. There have been several studies which have
shown that the high energy index is correlated with the X-ray flux
(e.g.Dewangan (2002), Vaughan & Edelson (2001), Perola et al. (1986)).
Recently, Sarma et al. 2015 have studied the index versus flux variation for
Mrk 335 and Ark 564 and have reported that while the correlation exists for
both sources, there is significantly more scatter for Ark 564. Our results are
broadly consistent with their finding and perhaps gives an explanation for the
difference between the two sources. Also the correlations obtained here are
consistent with the double Comptonization model used.

There is little evidence for any correlation between the UV flux and the hard
component luminosity with null hypothesis probability of $p=0.14$. There is a
hint of a correlation between the UV flux and the soft X-ray component
luminosity with $p=0.05$. However, the UV flux is strongly correlated to the
photon index $\Gamma_{Simpl}$.

We can interpret the correlation between UV and X-ray emissions in two
different ways. One interpretation is that the variation in UV emission could
be due to the accretion rate fluctuation. So as the UV flux varies, it
provides a way to measure the accretion rate $\dot{m}$ as a fraction of the
Eddington rate and it is found to vary from $\sim$3.7 to $\sim$4.4. However,
one can estimate from the accretion rate that the measured UV emission should
mostly arise from the outer disc at a distance of $\sim$ 770$R_{g}$, where
$R_{g}$ is the Schwarzschild radius. At this radius the viscous time-scale is
$t_{visc}\sim 9$ years, which is much longer than the 6 day variability seen;
more importantly, the variation in accretion rate could not have propagated to
the inner regions in such short time-scales. We can further estimate that less
than 2% of the UV flux would arise from radii $\sim$30$R_{g}$. If the flux
variation at those radii is very large, it may give rise to the $\sim$1%
variation seen in the UV. However, even at $\sim$ 30$R_{g}$ the viscous time-
scale is too long at $\sim$ 61 days. Moreover, the accretion rate inferred
from this interpretation is significantly higher than the Eddington rate and
hence unlikely. Thus, it seems that the UV emission, or at least its
variability cannot arise due to accretion rate fluctuations.

The second possibility is that the UV flux variation is due to the
reprocessing of X-rays. In such a case our results indicate that the soft
X-ray emitting region and $\Gamma_{Simpl}$ are more important in determining
the X-ray irradiation than the X-ray luminosity itself. We note that recently
Pal et al. (2016) have studied the UV-X-ray correlation on much shorter 20
ksec time-scales and have come to a similar conclusion that the geometry of
the inner X-ray producing region may be playing an important role in
determining the UV emission.

Though we have not used the blurred reflection model (e.g., Fabian et al.
2002) to describe the soft X-ray excess and the broad iron line observed from
Ark 564, the model can be tested against the observed correlations. In the
blurred reflection model, the soft excess and the broad iron line are
physically the same spectral component, and hence these two features must be
strongly correlated. The presence of the broad iron line strongly suggests
some contribution of the blurred reflection to the soft X-ray excess. Indeed,
the observation of reverberation soft lags of $\sim 100\leavevmode\nobreak\ s$
in Ark 564 by Kara et al. (2013) clearly demonstrate the presence of blurred
reflection in the soft ($0.3-1{\rm\leavevmode\nobreak\ keV}$) band. However,
the findings of soft leads in Ark 564 (Dewangan, 2002; Kara et al., 2013)
suggest contribution of additional spectral component in the soft band. In our
analysis, the broad iron line does not appear to follow the strongly variable
soft X-ray excess emission. Illumination of the hard X-ray power law component
should not only result in the blurred reflection (the soft excess, broad iron
line and the hump in the $\sim 20-40{\rm\leavevmode\nobreak\ keV}$) but also
in the reprocessed emission in the UV band. The correlation between $F_{UV}$
and $L_{Simpl}$ may result from the reprocessing of the coronal X-ray emission
in the disc. However, the similar variability amplitudes of the soft excess
and the hard X-ray emission is difficult to explain in the reflection model in
which a compact corona along the symmetrical axis illuminates the disc. In
such a model, due to the bending of light, the reflected emission that
includes the soft excess and the iron line is much less variable than the
illuminating power law (Miniutti & Fabian, 2004). Thus, the observed strong
soft X-ray excess entirely is unlikely to be the reflected emission.

Our results are based on eight observations separated by $\sim$6 days and
clearly there is a need for a larger number of such observations to verify
these interpretations. Further long term simultaneous monitoring of UV and
X-ray emissions over different time-scales can give us more insight into the
variability of the source. Also the results can be compared with correlations
obtained for other AGN. This may be possible with ASTROSAT which has the Ultra
Violet Imaging Telescope (UVIT) for monitoring the UV emission and the Soft
X-ray imaging Telescope (SXT) and the Large Area X-ray Proportional Counters
(LAXPC) for X-ray studies.

## Acknowledgments

This study is based on the observations obtained with XMM-Newton satellite, an
ESA science mission with instruments and contributions directly funded by ESA
Member States and the USA (NASA). This work has made use of the NASA/IPAC
Extragalactic Data base which is operated by the Jet Propulsion Laboratory,
California Institute of technology and data obtained through the High Energy
Astrophysics Science Archive Research Center Online Service, provided by the
NASA/GSFC. The first author would like to thank the Department of Science and
Technology for the support grant (No.SR/S2/HEP-07/2012) for this work.

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# Zeta zero dependence and the critical line

Gordon Chavez

###### Abstract.

On the critical line the limiting conditional distribution of the zeta
function’s log-modulus around zeta zeros exists and predicts the well-known
pair correlation between Riemann zeros. However, this limiting value
distribution does not exist at most distances above or below any zeta zeros
off the critical line. These results readily generalize to L-functions.

AMS 2020 Subject Classification: 11M06, 11M26

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Elliptical distributions
    2. 2.2 Generalized hyperbolic distributions
  3. 3 Application to zeta and L-functions
    1. 3.1 On the critical line
    2. 3.2 Off the critical line
    3. 3.3 Extension to L-functions
  4. 4 Conclusion
  5. A

##  1\. Introduction

Statistical dependence in nontrivial zeros of the Riemann zeta function has
been well-studied empirically and theoretically. Montgomery [29] gave the
first results describing pair-correlation between zeros assuming the Riemann
hypothesis, which predicted a short-range repulsion between consecutive zeros
on the critical line. Odlyzko [30] later calculated large numbers of zeta
zeros and provided empirical corroboration of Montgomery’s predictions. Dyson
[13] famously pointed out that Montgomery’s pair-correlation result was
equivalent to that for the eigenvalues of random hermitian matrices from the
Gaussian Unitary Ensemble (GUE). This connection has since led to a greater
understanding of statistical dependence in zeros of the zeta function [37] and
its generalizations [25]. Hejhal [23] generalized Montgomery’s results to give
the three-point correlation between zeta zeros and Rudnick & Sarnak [34]
generalized further, providing n-point correlations between zeta zeros as well
as the zeros of L-functions [35], reinforcing the connection between the GUE
and short-range statistical dependence in zeros of zeta and L-functions.

Long-range statistical dependence in the zeta zeros was illuminated by related
connections to physics. In particular, Bogomolny & Keating [6] [7] [8]
provided the first predictions for such dependence in the Riemann zeros by
utilizing semi-classical techniques from the field of quantum chaos along with
the Hardy-Littlewood conjecture. Their results describe an effect where
differences between zeros tend to avoid the imaginary parts of the low-lying
zeta zeros themselves. Conrey & Snaith [12] showed similar results from a
conjecture on the ratios of L-functions given by Conrey, Farmer, & Zirnbauer
[11] and Rodgers [33] proved the effect under the Riemann Hypothesis. Ford &
Zaharescu [16] then used uniform versions of Landau’s formula [26] [19] [15]
to give unconditional proof of the effect in zeta and L-function zeros on the
critical line, explaining observations of the effect in L-function zeros by
Perez-Marco [31].

The formal results above either only consider zeros on the critical line,
assume the Riemann hypothesis, or include contributions from zeros off the
critical line in an error term. A question then arises: would zeta or
L-function zeros off the critical line have the same statistical dependence
structure as that described by the results above? Gonek [22] applied a finite
Euler product approximation of the zeta function [20] [21] to predict that any
zeros off the critical line have a structurally different generating process
from those on the critical line, which suggests the answer is no. In this
paper we apply a probabilistic log-Euler product approach similar to that used
in many studies of the zeta function’s value distribution [9] [32] [1] [2] [3]
along with an analytic continuation to provide further evidence in the
negative.

Results provided in [10] can be used to give the following result for the
asymptotic distribution of
$\log\left|\zeta\left(1/2+i\left(t+\Delta\right)\right)\right|$ as
$\zeta\left(1/2+it\right)\rightarrow 0$:

######  Theorem 1.1.

Suppose $t$ is uniformly distributed in $[T,2T]$ and $\Delta$ is uniformly
distributed in $\left(0,t\right]$. Then as $T\rightarrow\infty$,

$\displaystyle\lim_{x\rightarrow
0}p\left(\log\left|\zeta\left(1/2+i\left(t+\Delta\right)\right)\right|\middle|\zeta(1/2+it)=x\right)$
(1.1)
$\displaystyle\rightarrow\mathcal{N}\left(-\log\left|\zeta(1+i\Delta)\right|+\varepsilon(\Delta),\frac{1}{2}\log\log
t\right)$

where $p(.|.)$ denotes the conditional probability density function and
$\left|\varepsilon\left(\Delta\right)\right|<1-\gamma\approx.422784.$

This result predicts the well-known statistical dependence between zeta zeros
described above. In particular, it predicts an “average gap” around zeta
zeros, which agrees with the zeros’ known short-range dependence, and, since
$\left|\zeta\left(1+i\Delta\right)\right|$ has local minima at values of
$\Delta$ approximately equal to the Riemann zeros’ imaginary parts, it
predicts the longer range, zero-difference repulsion effect (see Figure 1).

Figure 1. A graph (Red) of the conditional probability  $\lim_{x\rightarrow
0}\mathbb{P}\left\\{\log\left|\zeta\left(1/2+i\left(t+\Delta\right)\right)\right|\leq-3\sigma\middle|\zeta\left(1/2+it\right)=x\right\\}$
approximated by integrating (1.1) with $\sigma=\frac{1}{2}\log\log t$, where
$t$ is the imaginary part of the 100,000th Riemann zero. Vertical lines (Blue)
are positioned at values of $\Delta$ such that $\zeta(1/2+i\Delta)=0$. Since
large negative values of $\log\left|\zeta(s)\right|$ are a necessary condition
for $s$ to be the position of a zeta zero, this result demonstrates the
statistical dependence between zeta zeros on the critical line described
above.

We show that statistical dependence like that described by (1.1) for zeta
zeros on the critical line cannot exist for zeros that are off the critical
line. In particular, for $\sigma>1/2$, we show that the limiting conditional
distribution of
$\log\left|\zeta\left(\sigma+i\left(t+\Delta\right)\right)\right|$ does not
exist as $\zeta\left(\sigma+it\right)\rightarrow 0$ for most values of
$\Delta$, i.e., the conditional moments diverge and the conditional
probability density vanishes everywhere. These results readily generalize to
L-functions.

##  2\. Preliminaries

We study the statistical behavior of series with the form

(2.1) $X(t)=\sum_{p}a_{p}\cos\left(t\log p+\theta_{p}\right),$

where the $a_{p}>0$ are real numbers indexed by the prime numbers $p$,
$t\in\mathbb{R}$, and $\theta_{p}\in[0,2\pi)$. The independence of the terms
$\cos\left(t\log p\right)$ under uniformly distributed $t\in[T,2T]$ as
$T\rightarrow\infty$ is directly related to the fact that, due to the
fundamental theorem of arithmetic, the characteristic function of (2.1) has
the factorizable form

(2.2) $\varphi\left(\lambda\right)=\prod_{p}J_{0}\left(\lambda
a_{p}\right)=\prod_{p}\varphi_{p}\left(\lambda\right),$

where $\varphi_{p}\left(\lambda\right)$ denotes the characteristic function of
$a_{p}\cos\left(t\log p+\theta_{p}\right)$ and

(2.3)
$J_{0}(z)=\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}}{\left(m!\right)^{2}}\left(\frac{z}{2}\right)^{2m}$

is the 0th order Bessel function of the first kind [27] (Cor 5.2 pg. 38, Cor
6.8 pg. 44) [10] (Thm. 2.1). By the independence of (2.1)’s terms, one can
multiply $X(t+\Delta)$ and $X(t)$ and apply the expected value to show that,
for $t$ uniformly distributed in $[T,2T]$ with $T\rightarrow\infty$, the
autocovariance function of (2.1) is given by

(2.4)
$2R(\Delta)=2E\left\\{X(t+\Delta)X(t)\right\\}=\sum_{p}a_{p}^{2}\cos\left(\Delta\log
p\right).$

###  2.1. Elliptical distributions

We next note that (2.2) is an even function since $J_{0}(x)$ is an even
function. This implies that (2.1)’s probability density function $p(x)$ is
even as well. Relatedly, it is clear from (2.2)-(2.3) that there exists a
function $\phi(.)$ such that

$\varphi(\lambda)=\phi\left(\lambda^{2}\right),$

which, with (2.4), shows that $X(t)$ and $X(t+\Delta)$ have an elliptical
joint distribution, which is a large class of distributions that includes the
multivariate normal distribution. Then from (2.4) and the general results of
Fang, Kotz, & Ng [14] (Theorem 2.18 pg. 45, Eqn. 2.43 pg. 46) the conditional
probability density of $X(t+\Delta)$ given the value of $X(t)$ has the form

(2.5)
$p_{t+\Delta|t}(x)=\frac{1}{\sqrt{h(\Delta)}}g\left(\frac{\left(x-\mu_{t+\Delta|t}\right)^{2}}{h(\Delta)}\right)$

where $g(.)$ is a non-negative function,

(2.6) $h(\Delta)=R(0)\left(1-\left(\frac{R(\Delta)}{R(0)}\right)^{2}\right),$

and the conditional expectation is given by

(2.7)
$\mu_{t+\Delta|t}=E\left\\{X\left(t+\Delta\right)|X(t)\right\\}=\frac{R(\Delta)}{R(0)}X(t).$

Additionally, from (2.4) and [14] (Theorem 2.18 pg. 45), one can show that the
conditional variance has the form

(2.8)
$\textnormal{var}\left\\{X\left(t+\Delta\right)|X(t)\right\\}=h\left(\Delta\right)f\left(X(t)\right),$

where $f(.)$ is a function that depends on the specific distribution of $X(t)$
and $X(t+\Delta)$. In the case of $X(t)$ and $X\left(t+\Delta\right)$ normally
distributed, $f\left(.\right)=1$. In general, however, this is not the case
and (2.8) is dependent on the value of $X(t)$.

###  2.2. Generalized hyperbolic distributions

We next describe the tails of distributions of series with the form (2.1) that
additionally satisfy the convergence property

(2.9) $\sum_{p}a_{p}^{2}<\infty.$

In particular, if (2.1) satisfies (2.9), then (2.1)’s probability density
$p(x)$ satisfies

(2.10) $p(x)=O\left(e^{-r|x|}\right)$

for some $r\geq 1$ as $|x|\rightarrow\infty$. This may be shown from
Laurincikas [27] (Eqn. 6.17 pg. 47). We then note that the generalized
hyperbolic distributions are a large family that includes many distributions
that are symmetric, elliptical, and satisfy the tail constraint (2.10). If
$X(t)$ and $X\left(t+\Delta\right)$ have such a distribution, then one can
apply the general results of Blaesild & Jensen [5] ((1) pg.47, Theorem 1 (b)
pg. 49-50) to show that the conditional density of $X(t+\Delta)$ given $X(t)$
has the form

(2.11) $\displaystyle
p_{t+\Delta|t}(x)=\sqrt{\frac{\alpha^{\prime}}{2\pi}}\left(\frac{1}{\delta_{t+\Delta|t}}\right)^{\lambda-1/2}\frac{K_{\lambda-1}\left(\alpha^{\prime}\left(\delta_{t+\Delta|t}^{2}+\frac{\left(x-\mu_{t+\Delta|t}\right)^{2}}{R(0)h(\Delta)}\right)^{1/2}\right)}{K_{\lambda-1/2}(\alpha^{\prime}\delta_{t+\Delta|t})\left(\delta_{t+\Delta|t}^{2}+\frac{\left(x-\mu_{t+\Delta|t}\right)^{2}}{R(0)h(\Delta)}\right)^{\frac{1-\lambda}{2}}},$

where $K_{\nu}(z)$ is the $\nu$th order modified Bessel function of the second
kind,

(2.12) $\alpha^{\prime}=\alpha\sqrt{R(0)},\hskip
11.38092pt\delta_{t+\Delta|t}=\sqrt{\frac{\delta^{2}}{R(0)}+\frac{X^{2}(t)}{R^{2}(0)}},$

and $\lambda$, $\alpha$, and $\delta$ are given parameters. Additionally from
[5] (pg. 50-51) one can show that the conditional variance has the form (2.8)
with

(2.13)
$f\left(X(t)\right)=\frac{1}{\alpha}\sqrt{\delta^{2}+\frac{X^{2}(t)}{R(0)}}\frac{K_{\lambda+1/2}\left(\alpha\sqrt{\delta^{2}+\frac{X^{2}(t)}{R(0)}}\right)}{K_{\lambda-1/2}\left(\alpha\sqrt{\delta^{2}+\frac{X^{2}(t)}{R(0)}}\right)}.$

We then note the asymptotic result [4] (DLMF 10.17.1, 10.40.2)

(2.14) $K_{\nu}(z)\sim\sqrt{\frac{\pi}{2z}}e^{-z},$

which, with (2.6), (2.8), and (2.13) shows that, for the generalized
hyperbolic family,

(2.15)
$\textnormal{var}\left\\{X(t+\Delta)|X(t)\right\\}\sim\left|X(t)\right|\frac{\sqrt{R(0)}}{\alpha}\left(1-\left(\frac{R(\Delta)}{R(0)}\right)^{2}\right)$

as $\left|X(t)\right|\rightarrow\infty$.

##  3\. Application to zeta and L-functions

An example of a series with the form (2.1) is the real part of the prime zeta
function

(3.1) $P\left(s\right)=\sum_{p}\frac{1}{p^{s}}.$

One can apply the logarithm to $\zeta\left(s\right)$’s Euler product, Taylor
expand, and apply Möbius inversion to give

(3.2)
$P\left(s\right)=\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\log\zeta\left(ns\right),$

where $\mu(.)$ is the Möbius function [18] [17]. The result (3.2) is
convergent everywhere in $\textnormal{Re}(s)>0$ except at the points $ns=1$,
which arise from the simple pole $\zeta(1)$, and at the points $ns$ where
$\zeta\left(ns\right)=0$. Note that since $\zeta(s)$ has no zeros in
$\textnormal{Re}(s)\geq 1$, the singularities of (3.2) in
$1/2\leq\textnormal{Re}(s)<1$ with $\textnormal{Im}(s)\neq 0$ define the
positions of the nontrivial Riemann zeros. The expansion (3.2) gives the
following result as $s$ approaches such a Riemann zero:

(3.3) $\textnormal{Re}P(s)=\log\left|\zeta(s)\right|+O(1).$

###  3.1. On the critical line

To study the statistical behavior of $\log\left|\zeta(s)\right|$ on the
critical line $\textnormal{Re}(s)=1/2$ it is useful to study a truncated
version of the sum (3.1) restricted to the primes $p\leq X$, $P_{X}(1/2+it)$.
This is an important step in Radziwill & Soundararajan’s proof [32] of
Selberg’s [36] central limit theorem, where it is shown that with $t$
uniformly distributed in $[T,2T]$ and, say, $X(T)=T^{1/\left(\log\log\log
T\right)^{2}}$,

(3.4)
$\textnormal{Re}P_{X(T)}\left(1/2+it\right)\overset{p}{\to}\log\left|\zeta\left(1/2+it\right)\right|$

as $T\rightarrow\infty$. One can then show using a finite series version of
(2.4) that the variance is given by

(3.5) $\frac{1}{2}\sum_{p\leq
X(T)}\frac{1}{p}=\frac{1}{2}P_{X(T)}(1)\sim\frac{1}{2}\log\log T$

as $T\rightarrow\infty$. This combination of independent terms and diverging
variance enables a central limit theorem showing that $P_{X(T)}(1/2+it)$ and
hence $\log\left|\zeta(1/2+it)\right|$ are asymptotically normally distributed
with mean zero and variance (3.5). Finite series versions of (2.1)-(2.8) along
with (3.2)-(3.3) then enable a proof of Theorem 1.1 by deriving the
conditional moments of
$\log\left|\zeta\left(1/2+i\left(t+\Delta\right)\right)\right|$ as
$\zeta\left(1/2+it\right)\rightarrow 0$:

###### Proof.

Let $t$ be uniformly distributed in $[T,2T]$ and $\Delta$ be uniformly
distributed in $[0,t]$ with $T\rightarrow\infty$. We apply (3.4) and note from
a finite series version of (2.1)-(2.4) that the autocovariance function is
given by

$\displaystyle
2R(\Delta)=2E\left\\{\textnormal{Re}P_{X(T)}\left(1/2+i\left(t+\Delta\right)\right)\textnormal{Re}P_{X(T)}\left(1/2+it\right)\right\\}$
(3.6)
$\displaystyle=\textnormal{Re}P_{X(T)}\left(1+i\Delta\right)\overset{p}{\to}\textnormal{Re}P\left(1+i\Delta\right),$

where the last step of (3.6) may be shown by noting the independence of
$\cos\left(\Delta\log p\right)$ and $\cos\left(\Delta\log q\right)$ for $p\neq
q$ and that $E\left\\{\cos^{2}\left(\Delta\omega\right)\right\\}=1/2$ for any
$\omega$ to write

$\displaystyle
E\left\\{\left(\textnormal{Re}P_{X(T)}\left(1+i\Delta\right)-\textnormal{Re}P\left(1+i\Delta\right)\right)^{2}\right\\}$
$\displaystyle=E\left\\{\sum_{p>X(T)}\frac{\cos^{2}\left(\Delta\log
p\right)}{p^{2}}+2\sum_{\begin{subarray}{c}p\neq q\\\
p,q>X(T)\end{subarray}}\frac{\cos\left(\Delta\log p\right)\cos\left(\Delta\log
q\right)}{pq}\right\\}$ (3.7)
$\displaystyle=\frac{1}{2}\sum_{p>X(T)}\frac{1}{p^{2}}\rightarrow 0$

as $T\rightarrow\infty$, which shows that
$\textnormal{Re}P_{X(T)}\left(1+i\Delta\right)$ converges in mean square and
hence converges in probability to $\textnormal{Re}P\left(1+i\Delta\right)$.

Since $\textnormal{Re}P_{X(T)}(1/2+it)$ and hence
$\log\left|\zeta(1/2+it)\right|$ are normally distributed we use the
conditional moment formulas (2.6)-(2.8) with (3.4)-(3.6) to write

$\displaystyle
E\left\\{\log\left|\zeta\left(1/2+i\left(t+\Delta\right)\right)\right||\log\left|\zeta\left(1/2+it\right)\right|\right\\}$
(3.8)
$\displaystyle=\frac{\textnormal{Re}P_{X(T)}\left(1+i\Delta\right)}{P_{X(T)}(1)}\log\left|\zeta\left(1/2+it\right)\right|\overset{p}{\to}\frac{\textnormal{Re}P\left(1+i\Delta\right)}{P_{X(T)}(1)}\log\left|\zeta\left(1/2+it\right)\right|$

and

$\displaystyle\textnormal{var}\left\\{\log\left|\zeta\left(1/2+i\left(t+\Delta\right)\right)\right||\log\left|\zeta\left(1/2+it\right)\right|\right\\}$
(3.9)
$\displaystyle=\frac{1}{2}\left(P_{X(T)}(1)-\frac{\left(\textnormal{Re}P_{X(T)}\left(1+i\Delta\right)\right)^{2}}{P_{X(T)}(1)}\right)\overset{p}{\to}\frac{1}{2}\left(P_{X(T)}(1)-\frac{\left(\textnormal{Re}P\left(1+i\Delta\right)\right)^{2}}{P_{X(T)}(1)}\right)$

We then note from (3.2) that $P(1+i\Delta)$ has no singularities, i.e., only
takes finite values with $\Delta>0$, and hence the far right-hand side of
(3.9) vanishes as $T\rightarrow\infty$, which enables the simplification of
(3.9) to give

(3.10)
$\textnormal{var}\left\\{\log\left|\zeta\left(1/2+i\left(t+\Delta\right)\right)\right||\log\left|\zeta\left(1/2+it\right)\right|\right\\}\overset{p}{\to}\frac{1}{2}\log\log
t+o(\log\log t).$

We next use (3.2) and the Laurent expansion of $\zeta(s)$ to note

(3.11)
$\lim_{T\rightarrow\infty}P_{X(T)}(1)=P(1)=\log\zeta(1)+O(1)=\log\left(\lim_{x\rightarrow
1^{+}}\frac{1}{x-1}\right)=\lim_{x\rightarrow
0^{+}}\log\left(\frac{1}{x}\right).$

We additionally note that as $\zeta(1/2+it)\rightarrow 0$

(3.12) $\log\left|\zeta\left(1/2+it\right)\right|=\lim_{x\rightarrow
0^{+}}\log\left(x\right).$

Substituting the two limits (3.11)-(3.12) into (3.8) and cancelling gives

(3.13) $\lim_{x\rightarrow
0}E\left\\{\log\left|\zeta\left(1/2+i\left(t+\Delta\right)\right)\right||\zeta\left(1/2+it\right)=x\right\\}\overset{p}{\to}-\textnormal{Re}P\left(1+i\Delta\right).$

This shows that
$\log\left|\zeta\left(1/2+i\left(t+\Delta\right)\right)\right|$ is
asymptotically normally distributed with limiting variance (3.10) and mean
(3.13) as $\zeta\left(1/2+it\right)\rightarrow 0$.

Lastly we apply (3.2) to write

(3.14)
$P\left(1+i\Delta\right)=\log\zeta\left(1+i\Delta\right)+\sum_{n=2}^{\infty}\frac{\mu(n)}{n}\log\zeta\left(n+in\Delta\right)$

and note that, for $n\geq 2$,
$\left|\zeta(n+in\Delta)-1\right|\leq\left|\zeta(n)-1\right|<1$. Then by the
alternating series expansion of the logarithm,

(3.15)
$\left|\log\zeta(n+in\Delta)\right|=\left|\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\left(\zeta(n+in\Delta)-1\right)^{k}\right|<\left|\zeta(n+in\Delta)-1\right|\leq\zeta(n)-1.$

Hence by (3.15) and since $\mu(n)\in\left\\{-1,0,1\right\\}$ for all $n$, we
have

(3.16)
$\left|\sum_{n=2}^{\infty}\frac{\mu(n)}{n}\log\zeta(n+in\Delta)\right|<\sum_{n=2}^{\infty}\frac{\left|\log\zeta(n+in\Delta)\right|}{n}<\sum_{n=2}^{\infty}\frac{\zeta(n)-1}{n}=1-\gamma.$

Combining (3.10), (3.13), (3.14), and (3.16) completes the proof. ∎

###  3.2. Off the critical line

We may apply statistical properties of the series (3.1) to study $\zeta(s)$
with $\textnormal{Re}(s)>1/2$ because of the latter’s mean-square convergence
to its Euler product in this region [28] (Eqn. 2.8 pg. 65). We hence apply
(2.4) to (3.1)’s real part to give

(3.17)
$2R(\Delta)=2E\left\\{\textnormal{Re}P\left(\sigma+i\left(t+\Delta\right)\right)\textnormal{Re}P\left(\sigma+it\right)\right\\}=\textnormal{Re}P\left(2\sigma+i\Delta\right),$

which shows that the variance of $\textnormal{Re}P\left(\sigma+it\right)$ for
$\sigma>1/2$ is given by

(3.18)
$\frac{1}{2}\sum_{p}\frac{1}{p^{2\sigma}}=\frac{P\left(2\sigma\right)}{2}=O(1).$

Since the variance (3.18) is convergent, in contrast to (3.5), a central limit
theorem does not hold for $\textnormal{Re}P\left(\sigma+it\right)$ with
$\sigma>1/2$. Additionally, in contrast to on the critical line, we can show
the following result describing the nonexistence of
$\log\left|\zeta\left(\sigma+i\left(t+\Delta\right)\right)\right|$’s
conditional distribution as $\zeta\left(\sigma+it\right)\rightarrow 0$:

######  Theorem 3.1.

Suppose $t$ is uniformly distributed in $[T,2T]$ with $T\rightarrow\infty$ and
let $p^{*}_{t+\Delta|t}(x)$ denote the conditional probability density
function of $\log\left|\zeta\left(\sigma+i\left(t+\Delta\right)\right)\right|$
given $\zeta\left(\sigma+it\right)$. Then for $\sigma>1/2$,

$\displaystyle\lim_{x\rightarrow
0}E\left\\{\log\left|\zeta\left(\sigma+i\left(t+\Delta\right)\right)\right||\zeta\left(\sigma+it\right)=x\right\\}$
(3.19) $\displaystyle=\begin{cases}-\infty\hskip
2.84544pt;&\textnormal{Re}P\left(2\sigma+i\Delta\right)>0\\\ +\infty\hskip
2.84544pt;&\textnormal{Re}P\left(2\sigma+i\Delta\right)<0\\\ O(1)\hskip
2.84544pt;&\textnormal{Re}P\left(2\sigma+i\Delta\right)=0\end{cases}$

and as $\zeta(\sigma+it)\rightarrow 0$,

(3.20) $p^{*}_{t+\Delta|t}(x)\rightarrow 0$

for all $|x|<\infty$ and $\Delta$ such that
$0<\left|\textnormal{Re}P\left(2\sigma+i\Delta\right)/P\left(2\sigma\right)\right|<1$.

###### Proof.

We apply (2.7) with (3.17) to give the following result for the conditional
expectation of $\textnormal{Re}P\left(\sigma+i\left(t+\Delta\right)\right)$
given $\textnormal{Re}P\left(\sigma+it\right)$:

(3.21)
$E\left\\{\textnormal{Re}P\left(\sigma+i\left(t+\Delta\right)\right)|\textnormal{Re}P\left(\sigma+it\right)\right\\}=\frac{\textnormal{Re}P\left(2\sigma+i\Delta\right)}{P\left(2\sigma\right)}\textnormal{Re}P\left(\sigma+it\right).$

We then note from (3.2)-(3.3) that as $\zeta(\sigma+it)\rightarrow 0$

(3.22) $\textnormal{Re}P\left(\sigma+it\right)=\lim_{x\rightarrow
0^{+}}\log(x)=-\infty,$

which, with (3.21) and (3.18), shows that

(3.23) $\lim_{x\rightarrow
0}E\left\\{\textnormal{Re}P\left(\sigma+i\left(t+\Delta\right)\right)|\zeta\left(\sigma+it\right)=x\right\\}\\\
=\begin{cases}-\infty\hskip
2.84544pt;&\textnormal{Re}P\left(2\sigma+i\Delta\right)>0\\\ +\infty\hskip
2.84544pt;&\textnormal{Re}P\left(2\sigma+i\Delta\right)<0\\\ 0\hskip
2.84544pt;&\textnormal{Re}P\left(2\sigma+i\Delta\right)=0\end{cases}$

We next apply (3.2) to write

(3.24)
$\textnormal{Re}P\left(\sigma+i\tau\right)=\log\left|\zeta\left(\sigma+i\tau\right)\right|+\sum_{n=2}^{\infty}\frac{\mu(n)}{n}\log\left|\zeta\left(n\sigma+ni\tau\right)\right|$

and note that, for $\sigma>1/2$, the terms of the series on (3.24)’s far
right-hand side are $O\left(\frac{1}{n2^{n\sigma}}\right)$ as
$n\rightarrow\infty$ and hence the series converges, giving

(3.25)
$\textnormal{Re}P\left(\sigma+i\tau\right)=\log\left|\zeta\left(\sigma+i\tau\right)\right|+O(1).$

Therefore, for an arbitrary event $\mathcal{X}$,

(3.26)
$E\left\\{\textnormal{Re}P\left(\sigma+i\tau\right)|\mathcal{X}\right\\}=E\left\\{\log\left|\zeta\left(\sigma+i\tau\right)\right||\mathcal{X}\right\\}+O(1).$

The results (3.23) and (3.26) complete the proof of (3.19).

We lastly let $p_{t+\Delta|t}(x)$ denote the conditional probability density
function of $\textnormal{Re}P\left(\sigma+i\left(t+\Delta\right)\right)$ given
$\textnormal{Re}P\left(\sigma+it\right)$ and note that it has the elliptical
form (2.5) with $\mu_{t+\Delta|t}$ given by (3.21) and

(3.27)
$h(\Delta)=\frac{P\left(2\sigma\right)}{2}\left(1-\left(\frac{\textnormal{Re}P\left(2\sigma+i\Delta\right)}{P\left(2\sigma\right)}\right)^{2}\right).$

Due to the requirement that $\int_{-\infty}^{\infty}p_{t+\Delta|t}(x)dx$
converges, $g(u)=o(u^{-1/2})$. Hence, from (2.5) and (3.21), for nonzero
(3.27) we have

(3.28)
$p_{t+\Delta|t}(x)=o\left(\left|x-\frac{\textnormal{Re}P\left(2\sigma+i\Delta\right)}{P\left(2\sigma\right)}\textnormal{Re}P\left(\sigma+it\right)\right|^{-1}\right).$

(3.22)-(3.23) and (3.27)-(3.28) show that, for all $|x|<\infty$ and $\Delta$
such that
$0<\left|\textnormal{Re}P\left(2\sigma+i\Delta\right)/P\left(2\sigma\right)\right|<1$,
$p_{t+\Delta|t}(x)\rightarrow 0$ as $\zeta\left(\sigma+it\right)\rightarrow
0$. Then from (3.24) the result (3.20) follows. ∎

Theorem 3.1 shows that, for $\sigma>1/2$,
$\log\left|\zeta\left(\sigma+i\left(t+\Delta\right)\right)\right|$’s
conditional moments diverge and its conditional density vanishes as
$\zeta(\sigma+it)\rightarrow 0$ for all $\Delta$ except where
$\textnormal{Re}P\left(2\sigma+i\Delta\right)=0$, i.e., where
$\log\left|\zeta\left(\sigma+i\left(t+\Delta\right)\right)\right|$ and
$\textnormal{Re}P\left(\sigma+it\right)$ are uncorrelated111Appendix A. At
these distances
$\log\left|\zeta\left(\sigma+i\left(t+\Delta\right)\right)\right|$’s limiting
conditional expectation exists, but the status of its higher order moments and
density is unclear and depends on the specific, non-normal, unconditional
distribution of $\textnormal{Re}P\left(\sigma+it\right)$.

We note from (3.18) that (2.9)-(2.10) is satisfied by
$\textnormal{Re}P\left(\sigma+it\right)$ with $\sigma>1/2$. Thus we next show
that if $\textnormal{Re}P\left(\sigma+it\right)$’s distribution is in the
generalized hyperbolic family, then
$\log\left|\zeta\left(\sigma+i\left(t+\Delta\right)\right)\right|$’s
conditional density also vanishes as $\zeta(\sigma+it)\rightarrow 0$ at
distances where $\textnormal{Re}P\left(2\sigma+i\Delta\right)=0$.

######  Corollary 3.1.

If $\textnormal{Re}P\left(\sigma+it\right)$ with $\sigma>1/2$ has a
generalized hyperbolic distribution under $t$ uniformly distributed in
$[T,2T]$ with $T\rightarrow\infty$, then with
$\textnormal{Re}P\left(2\sigma+i\Delta\right)=0$ and
$\zeta(\sigma+it)\rightarrow 0$,

(3.29) $p^{*}_{t+\Delta|t}(x)\rightarrow 0$

for all $|x|<\infty$.

###### Proof.

If generalized hyperbolic, then
$\textnormal{Re}P\left(\sigma+i\left(t+\Delta\right)\right)$’s conditional
density has the form (2.11) and, by (3.21), $\mu_{t+\Delta|t}=0$ for
$\textnormal{Re}P\left(2\sigma+i\Delta\right)=0$. Additionally from (2.12) it
is clear that as
$\left|\textnormal{Re}P\left(\sigma+it\right)\right|\rightarrow\infty$ ,

(3.30)
$\delta_{t+\Delta|t}\sim\frac{\left|\textnormal{Re}P\left(\sigma+it\right)\right|}{R(0)}.$

Applying $\mu_{t+\Delta|t}=0$ and (3.30) along with (2.14) in (2.11) and
simplifying shows that for all $|x|<\infty$

(3.31)
$p_{t+\Delta|t}(x)=O\left(\left|\textnormal{Re}P\left(\sigma+it\right)\right|^{-1/2}\right),$

which, with (3.22) and (3.24), proves (3.29). ∎

We may also easily show that, under a generalized hyperbolic distribution
assumption for $\textnormal{Re}P\left(\sigma+it\right)$, the conditional
variance of $\log\left|\zeta\left(\sigma+i\left(t+\Delta\right)\right)\right|$
diverges as $\zeta(\sigma+it)\rightarrow 0$ at distances $\Delta$ where
$\textnormal{Re}P\left(2\sigma+i\Delta\right)=0$ and indeed at any distances
where the autocorrelation is not $\pm 1$.

######  Proposition 3.1.

If $\textnormal{Re}P\left(\sigma+it\right)$ with $\sigma>1/2$ has a
generalized hyperbolic distribution under $t$ uniformly distributed in
$[T,2T]$ with $T\rightarrow\infty$, then

(3.32) $\lim_{x\rightarrow
0}\textnormal{var}\left\\{\log\left|\zeta\left(\sigma+i\left(t+\Delta\right)\right)\right||\zeta(\sigma+it)=x\right\\}=\infty$

for all $\Delta\in\mathbb{R}$ such that
$\left|\textnormal{Re}P\left(2\sigma+i\Delta\right)/P(2\sigma)\right|<1$.

###### Proof.

We apply (2.15) with (3.17)-(3.18) to note that

$\displaystyle\textnormal{var}\left\\{\textnormal{Re}P\left(\sigma+i\left(t+\Delta\right)\right)|\textnormal{Re}P\left(\sigma+it\right)\right\\}$
(3.33)
$\displaystyle\sim\left|\textnormal{Re}P\left(\sigma+it\right)\right|\frac{1}{\alpha}\sqrt{\frac{P\left(2\sigma\right)}{2}}\left(1-\left(\frac{\textnormal{Re}P\left(2\sigma+i\Delta\right)}{P\left(2\sigma\right)}\right)^{2}\right)$

as $\left|\textnormal{Re}P\left(\sigma+it\right)\right|\rightarrow\infty$. The
result (3.33) along with (3.18) and (3.22) shows that

(3.34) $\lim_{x\rightarrow
0}\textnormal{var}\left\\{\textnormal{Re}P\left(\sigma+i\left(t+\Delta\right)\right)|\zeta(\sigma+it)=x\right\\}=\infty$

for all $\Delta\in\mathbb{R}$ such that
$\left|\textnormal{Re}P\left(2\sigma+i\Delta\right)/P(2\sigma)\right|<1$. We
next denote (3.25)’s $O(1)$ term as $B\left(\sigma,t\right)$ and note that,
for an arbitrary event $\mathcal{X}$,

$\displaystyle\textnormal{var}\left\\{\textnormal{Re}P\left(\sigma+it\right)|\mathcal{X}\right\\}=\textnormal{var}\left\\{\log\left|\zeta\left(\sigma+it\right)\right||\mathcal{X}\right\\}+\textnormal{var}\left\\{B\left(\sigma,t\right)|\mathcal{X}\right\\}$
(3.35)
$\displaystyle+2\textnormal{cov}\left\\{\log\left|\zeta\left(\sigma+it\right)\right|,B\left(\sigma,t\right)|\mathcal{X}\right\\}.$

We then apply the Cauchy-Schwartz inequality to (3.35)’s last term to write

$\displaystyle\textnormal{var}\left\\{\textnormal{Re}P\left(\sigma+it\right)|\mathcal{X}\right\\}=\textnormal{var}\left\\{\log\left|\zeta\left(\sigma+it\right)\right||\mathcal{X}\right\\}+\textnormal{var}\left\\{B\left(\sigma,t\right)|\mathcal{X}\right\\}$
(3.36)
$\displaystyle+2\alpha\sqrt{\textnormal{var}\left\\{\log\left|\zeta\left(\sigma+it\right)\right||\mathcal{X}\right\\}\textnormal{var}\left\\{B\left(\sigma,t\right)|\mathcal{X}\right\\}},$

where $|\alpha|\leq 1$. Since $B\left(\sigma,t\right)=O(1)$ it is then clear
that

(3.37)
$\textnormal{var}\left\\{\textnormal{Re}P\left(\sigma+it\right)|\mathcal{X}\right\\}=\infty\implies\textnormal{var}\left\\{\log\left|\zeta\left(\sigma+it\right)\right||\mathcal{X}\right\\}=\infty.$

The results (3.34) and (3.37) complete the proof of (3.32). ∎

###  3.3. Extension to L-functions

The above results may be readily generalized to L-functions by considering the
following series

(3.38) $P_{\chi}\left(s\right)=\sum_{p}\frac{\chi(p)}{p^{s}}$

for a given Dirichlet character $\chi$ s.t. $\chi(p)=e^{i\theta_{p}}$. For
$\textnormal{Re}(s)>1/2$, the real part of (3.38) has the form of (2.1) and
hence is described by Section 2’s results. We then note that one may take the
logarithm of $L(s,\chi)$’s Euler product, Taylor expand, and apply Möbius
inversion to give a generalized version of (3.2) relating L-functions
$L(s,\chi)$ and their corresponding series $P_{\chi}(s)$ for
$\textnormal{Re}(s)>0$:

(3.39) $P_{\chi}(s)=\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\log
L\left(ns,\chi^{n}\right).$

This result has very similar properties to (3.2) and can be used to show that
as $L\left(s,\chi\right)\rightarrow 0$,
$\textnormal{Re}P_{\chi}\left(s\right)=\lim_{x\rightarrow 0^{+}}\log
x=-\infty.$

We then consider a Dirichlet character modulo $k$, $\chi_{k}$, where $k$ has
prime factorization

(3.40) $k=\prod_{p}p^{m_{p}}.$

On the critical line one can show that a result with the form (1.1) applies
for $L\left(s,\chi_{k}\right)$ by applying essentially equivalent reasoning as
in Theorem 1.1’s proof with (3.38)-(3.39) and results from Hsu & Wong [24].
The only change to the result is

(3.41)
$\left|\varepsilon\left(\Delta\right)\right|<1-\gamma+\sum_{\begin{subarray}{c}p\\\
m_{p}\neq 0\end{subarray}}\frac{1}{p}.$

Off the critical line one can use the same reasoning as in Theorem 3.1’s proof
to show that, for $\sigma>1/2$ and $t$ uniform in $[T,2T]$ with
$T\rightarrow\infty$:

(3.42) $\lim_{x\rightarrow
0}E\left\\{\log\left|L\left(\sigma+i\left(t+\Delta\right),\chi_{k}\right)\right||L\left(\sigma+it,\chi_{k}\right)=x\right\\}=\pm\infty$

and $\log\left|L\left(\sigma+i\left(t+\Delta\right),\chi_{k}\right)\right|$’s
conditional probability density given the value of
$L\left(\sigma+it,\chi_{k}\right)$, denoted $p^{*}_{\chi_{k},t+\Delta|t}(x)$,
satisfies

(3.43) $p^{*}_{\chi_{k},t+\Delta|t}(x)\rightarrow 0$

as $L\left(\sigma+it,\chi_{k}\right)\rightarrow 0$ for all
$\Delta\in\mathbb{R}$ except $\Delta$ such that

(3.44)
$\textnormal{Re}\left(P\left(2\sigma+i\Delta\right)-\sum_{\begin{subarray}{c}p\\\
{m_{p}\neq 0}\end{subarray}}\frac{1}{p^{2\sigma+i\Delta}}\right)=0,$

where (3.42)’s right-hand side is then $O(1)$. One may then apply similar
reasoning to that in the proofs of Corollary 3.1 and Proposition 3.1 to show
that, under a generalized hyperbolic distribution assumption for
$\textnormal{Re}P_{\chi_{k}}\left(\sigma+it\right)$, the conditional density
$p^{*}_{\chi_{k},t+\Delta|t}(x)$ also vanishes as
$L\left(\sigma+it,\chi_{k}\right)\rightarrow 0$ at distances $\Delta$ where
(3.44) is satisfied and additionally

(3.45) $\lim_{x\rightarrow
0}\textnormal{var}\left\\{\log\left|L\left(\sigma+i\left(t+\Delta\right),\chi_{k}\right)\right||L\left(\sigma+it,\chi_{k}\right)=x\right\\}=\infty$

for all $\Delta\in\mathbb{R}$ such that the autocorrelation is not $\pm 1$,
i.e.

(3.46)
$\left|\frac{\textnormal{Re}\left(P\left(2\sigma+i\Delta\right)-\sum_{\begin{subarray}{c}p\\\
{m_{p}\neq
0}\end{subarray}}\frac{1}{p^{2\sigma+i\Delta}}\right)}{P\left(2\sigma\right)-\sum_{\begin{subarray}{c}p\\\
m_{p}\neq 0\end{subarray}}\frac{1}{p^{2\sigma}}}\right|<1.$

##  4\. Conclusion

The pole at $\zeta(1)$ is critically important to why
$\log\left|\zeta\left(1/2+i\left(t+\Delta\right)\right)\right|$’s conditional
distribution exists as $\zeta\left(1/2+it\right)\rightarrow 0$. Firstly, it
produces a cancellation in the conditional expectation formula, giving a
finite result around zeta zeros. Secondly, the pole-induced divergence of
$\textnormal{var}\left\\{\log\left|\zeta\left(1/2+it\right)\right|\right\\}$
for $t\in[T,2T]$ as $T\rightarrow\infty$ produces the central limit theorem
and normal distribution on the critical line. The normal distribution implies
that $\log\left|\zeta\left(1/2+i\left(t+\Delta\right)\right)\right|$’s
conditional variance and higher order central moments do not depend on the
value of $\zeta\left(1/2+it\right)$.

In contrast, with $\sigma>1/2$, there is no corresponding pole at
$\zeta(2\sigma)$, so there is no cancellation in
$\log\left|\zeta\left(\sigma+i\left(t+\Delta\right)\right)\right|$’s
conditional expectation and no central limit theorem providing a normal
distribution. As a result the conditional moments and distribution of
$\log\left|\zeta\left(\sigma+i\left(t+\Delta\right)\right)\right|$ do not
exist as $\zeta\left(\sigma+it\right)\rightarrow 0$ except possibly at
specific distances $\Delta$ where the autocorrelation vanishes. Further
reseach is needed on these distances, however, we have shown that under
relatively mild assumptions, the limiting conditional distribution does not
exist here as well. All of this reasoning generalizes to L-functions.

## Appendix A

We consider $\sigma>1/2$ and $t$ uniformly distributed in $[T,2T]$ with
$T\rightarrow\infty$ and note (3.2) to evaluate

$\displaystyle
E\left\\{\textnormal{Re}P\left(\sigma+i\left(t+\Delta\right)\right)\textnormal{Re}P\left(\sigma+it\right)\right\\}=E\left\\{\log\left|\zeta\left(\sigma+i\left(t+\Delta\right)\right)\right|\textnormal{Re}P\left(\sigma+it\right)\right\\}$
(A.1)
$\displaystyle+E\left\\{\left(\sum_{n=2}^{\infty}\frac{\mu(n)}{n}\log\left|\zeta\left(n\sigma+in\left(t+\Delta\right)\right)\right|\right)\textnormal{Re}P\left(\sigma+it\right)\right\\}.$

We consider the summation on (A.1)’s right-hand side. We apply the Euler
product and take logarithms to write

(A.2)
$E\left\\{\left(\sum_{n=2}^{\infty}\frac{\mu(n)}{n}\sum_{k=1}^{\infty}\frac{\textnormal{Re}P\left(kn\sigma+ikn\left(t+\Delta\right)\right)}{k}\right)\textnormal{Re}P\left(\sigma+it\right)\right\\}.$

We then note

$\displaystyle
E\left\\{\textnormal{Re}P\left(kn\sigma+ikn\left(t+\Delta\right)\right)\textnormal{Re}\left(\sigma+it\right)\right\\}$
$\displaystyle=E\left\\{\sum_{p}\frac{\cos\left(kn\left(t+\Delta\right)\log
p\right)\cos\left(t\log p\right)}{p^{(kn+1)\sigma}}\right\\}$ (A.3)
$\displaystyle+2E\left\\{\sum_{p\neq
q}\frac{\cos\left(kn\left(t+\Delta\right)\log p\right)\cos\left(t\log
q\right)}{p^{kn\sigma}q^{\sigma}}\right\\}.$

The expected value of every term in (A.3)’s last summation vanishes by the
independence of its two cosine terms for $p\neq q$. Additionally, the
numerators in the first summation’s terms may be rewritten as

(A.4) $\frac{\cos\left(\left(kn-1\right)t\log p+kn\Delta\log
p\right)+\cos\left(\left(kn+1\right)t\log p+kn\Delta\log p\right)}{2}.$

The expected value of the second cosine term in (A.4) always vanishes, while
for $kn>1$, the first term’s expected value vanishes as well. Hence
(A.2)-(A.3) vanishes entirely. Therefore by (A.1)

(A.5)
$E\left\\{\textnormal{Re}P\left(\sigma+i\left(t+\Delta\right)\right)\textnormal{Re}P\left(\sigma+it\right)\right\\}=E\left\\{\log\left|\zeta\left(\sigma+i\left(t+\Delta\right)\right)\right|\textnormal{Re}P\left(\sigma+it\right)\right\\}.$

The result (3.17) shows that (A.5) is given by
$\textnormal{Re}P\left(2\sigma+i\Delta\right)/2$ and hence (A.5) vanishes if
$\textnormal{Re}P\left(2\sigma+i\Delta\right)=0$.

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