#  $\delta$ Scuti pulsations in the bright Pleiades eclipsing binary HD 23642

John Southworth 1, S. J. Murphy2, K. Pavlovski3
1 Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK
2 Centre for Astrophysics, University of Southern Queensland, Toowoomba, QLD
4350, Australia
3 Department of Physics, Faculty of Science, University of Zagreb, Bijenicka
cesta 32, 10000 Zagreb, Croatia

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We announce the discovery of pulsations in HD 23642, the only bright eclipsing
system in the Pleiades, based on light curves from the Transiting Exoplanet
Survey Satellite (TESS). We measure 46 pulsation frequencies and attribute
them to $\delta$ Scuti pulsations in the secondary component. We find four
$\ell=1$ doublets, three of which have frequency splittings consistent with
the rotation rate of the star. The dipole mode amplitude ratios are consistent
with a high stellar inclination angle and the stellar rotation period agrees
with the orbital period. Together, these suggest that the spin axis of the
secondary is aligned with the orbital axis. We also determine precise
effective temperatures and a spectroscopic light ratio, and use the latter to
determine the physical properties of the system alongside the TESS data and
published radial velocities. We measure a distance to the system in agreement
with the Gaia parallax, and an age of $170\pm 20$ Myr based on a comparison to
theoretical stellar evolutionary models.

###### keywords:

stars: fundamental parameters — stars: binaries: eclipsing — stars:
oscillations

††pubyear: 2022††pagerange: $\delta$ Scuti pulsations in the bright Pleiades
eclipsing binary HD 23642–2

##  1 Introduction

The Pleiades is one of the closest and most extensively studied star clusters,
containing $\sim$1300 stars at a distance of $136$ pc (Melis et al., 2014;
Heyl et al., 2022). It is a benchmark for studying theoretical modelling (e.g.
Vandenberg & Bridges, 1984) the kinematics of star clusters (Lodieu et al.,
2019), binarity (Torres et al., 2021), pulsations (Murphy et al., 2022),
rotation (Rebull et al., 2016), and the connection between rotation, lithium
depletion and stellar inflation (Somers & Pinsonneault, 2015; Bouvier et al.,
2018).

Eclipsing binary stars (EBs) are another crucial research area for improving
our understanding of stellar evolution (Torres et al., 2010). The masses,
radii and effective temperatures ($T_{\rm eff}$s) of stars can be determined
to precisions approaching 0.2% (Maxted et al., 2020; Miller et al., 2020),
allowing their use as checks and calibrators of theoretical models (e.g.
Claret & Torres, 2018; Tkachenko et al., 2020). In particular, EBs in open
clusters are enticing targets for multiple scientific goals (e.g. Southworth
et al., 2004a; Brogaard et al., 2011; Torres et al., 2018).

A third phenomenon well suited to extending our understanding of stellar
physics is that of pulsations. $\delta$ Scuti stars pulsate in pressure modes
of low radial order ($n\sim 1\dots 10$) that are mostly sensitive to the
stellar envelope (Aerts et al., 2010; Kurtz, 2022). These modes probe the
stellar density, making them good indicators of age (Aerts, 2015), especially
when mass and/or metallicity are already constrained. Recently, the discovery
that $\delta$ Scuti stars pulsate in regular patterns (Bedding et al., 2020)
has allowed pulsation modes to be identified in many stars (e.g. Kerr et al.,
2022; Currie et al., 2022), including five members of the Pleiades (Murphy et
al., 2022), facilitating detailed asteroseismic modelling. In some cases, this
confers an age precision better than 10% (Murphy et al., 2021).

HD 23642 (V1229 Tau, HII 1431) was found to be a double-lined spectroscopic
binary by Pearce (1957) and Abt (1958). Its eclipsing nature was announced
independently by Miles (1999) and Torres (2003) using data from the Hipparcos
satellite. Detailed analyses of the system using ground-based data have been
presented by Griffin (1995); Torres (2003); Munari et al. (2004); Southworth
et al. (2005) and Groenewegen et al. (2007). HD 23642 was observed in long
cadence mode by the K2 satellite (Howell et al., 2014) in Campaign 4111HD
23642 was requested as a target by nine Guest Observer proposals including
GO4028 (PI Southworth) and GO4035 (PI Murphy).. An analysis of these data was
presented by David et al. (2016).

In this work we present the discovery of $\delta$ Scuti pulsations in this
important EB and determine the physical properties of the system to high
precision for the first time. We note that independent detections are
presented by Chen et al. (2022) without detailed analysis, and by Bedding et
al. (2023). Section 2 in the current work outlines the observations used,
Sections 3 and 4 present our spectroscopic and photometric analyses, Section 5
is dedicated to the pulsation analysis, and our work is concluded in Section
6.

Figure 1:  TESS simple aperture photometry (SAP) light curve of HD 23642. The
sectors are labelled.

##  2 Observations

HD 23642 was observed using the TESS mission (Ricker et al., 2015) in three
consecutive sectors (42–44) covering 76 days (2021/08/20 to 2021/11/06), at a
cadence of 120 s. We downloaded the data from the Mikulski Archive for Space
Telescopes (MAST) archive and extracted the simple aperture photometry (SAP)
from the fits files (Jenkins et al., 2016); the PDCSAP data are practically
identical. We included only the data with a QUALITY flag of zero, totalling 44
438 points, and converted them into differential magnitude. The data errors
were not used as they were much smaller than the scatter of the measurements.
The full TESS data are shown in Fig. 1.

We observed HD 23642 during two observing runs in November 2006 using the
Nordic Optical Telescope (NOT) and its FIbre Echelle Spectrograph (FIES). A
total of 27 échelle spectra were obtained using the medium-resolution fibre in
bundle A, which covered 364–736 nm with a resolving power of $R\approx
47\,000$. Exposure times of 600 s yielded a signal to noise ratio (S/N) of
approximately 200, although some spectra had a lower S/N due to cloudy
conditions. The data were reduced using iraf échelle package routines, with
particular care taken in the normalisation and merging of the échelle orders
(Kolbas et al., 2015).

##  3 Spectroscopic analysis

We first sought to measure the spectroscopic parameters of the two stars, in
particular their $T_{\rm eff}$s and light ratio. To do this we applied the
method of spectral disentangling (Simon & Sturm, 1994) using the Fourier
approach (Hadrava, 1995) and concentrating on the 650-670 nm spectral range
which includes the H$\alpha$ line. This wavelength interval is contaminated by
telluric lines, which we removed before the disentangling process. We used the
FDbinary code (Ilijić et al., 2004) and our standard methods (Pavlovski &
Hensberge, 2010; Pavlovski et al., 2018). We also fixed the velocity
amplitudes of the stars to the values measured by Torres et al. (2021), so
effectively ran in spectral separation mode.

The H$\alpha$ profiles of the two stars were then modelled to determine the
$T_{\rm eff}$s and light ratio. We fixed the surface gravities and rotational
velocities of the stars to the values determined in Section 4, thus avoiding
the degeneracy between $T_{\rm eff}$ and $\log g$ present in the Balmer lines
of hot stars. This approach required us to iterate our analysis with that
described in Section 4 to ensure internal consistency; the iteration converged
within one step. Optimal fitting was performed with the starfit code (Kolbas
et al., 2015), which uses a genetic algorithm to search for the best fit
within a grid of synthetic spectra pre-calculated using the uclsyn code
(Smalley et al., 2001). The fractional light contributions of the two
components were forced to sum to unity (Tamajo et al., 2011) and only the
wings of the H$\alpha$ line were fitted (with metallic lines masked). The
uncertainties were calculated using the MCMC approach described in Pavlovski
et al. (2018).

We found $T_{\rm eff}$s of $10\,200\pm 90$ K and $7670\pm 85$ K for the two
stars, and a light ratio of $\ell_{\rm B}/\ell_{\rm A}{\rm(H}\alpha)=0.313\pm
0.010$. HD 23642 A is known to be chemically peculiar: Abt & Levato (1978)
classified its spectrum as A0Vp(Si) + Am. Our light ratio should not be
affected by this because it was obtained from only the H$\alpha$ line. We
propagated it to the TESS passband using BT-Settl theoretical spectra (Allard
et al., 2012) to obtain $\ell_{\rm B}/\ell_{\rm A}{\rm(TESS)}=0.328\pm 0.011$
where the errorbar includes the contributions from $\ell_{\rm B}/\ell_{\rm
A}{\rm(H}\alpha)$ and both $T_{\rm eff}$s.

Figure 2:  A short section of the TESS light curve, chosen at random, is shown
(blue points) along with the jktebop best fit (red line). The residuals are
displayed offset to the base of the figure and magnified by a factor of 5 to
make the pulsations visible.

##  4 Light curve and physical properties

The light curve of HD 23642 shows shallow partial eclipses and clear
reflection and ellipsoidal effects. The stars are well-separated and the
system is suitable for analysis with the jktebop code (Southworth et al.,
2004b; Southworth, 2013), for which we used version 43. We fitted for the
fractional radii of the stars in the form of their sum ($r_{\rm A}+r_{\rm B}$)
and ratio ($k={r_{\rm B}}/{r_{\rm A}}$), the orbital inclination ($i$), the
central surface brightness ratio of the two stars ($J$), the amount of third
light ($L_{3}$), the reference time of mid-eclipse ($T_{0}$), and the orbital
period ($P$). Limb darkening was included using the power-2 law (Hestroffer,
1997) with the scaling coefficient for each star ($c_{1}$ and $c_{2}$) fitted
and the power-law coefficients ($\alpha_{1}$ and $\alpha_{2}$) fixed at values
from Claret & Southworth (2022). We also included one quadratic function per
TESS half-sector to normalise the light curve to zero differential magnitude,
to allow for the possibility of slow drifts in brightness for either
instrumental or astrophysical reasons. The pulsations are of much lower
amplitude than the eclipses so were treated as red noise. A circular orbit was
assumed.

Our initial solutions gave measurements of the fractional radii to a
disappointing precision. This is caused by the ratio of the radii being poorly
determined for shallow partial eclipses, a well-known phenomenon that has been
noted before for this system (Southworth et al., 2005; David et al., 2016). We
therefore imposed the spectroscopic light ratio from Section 3 as a Gaussian
prior in our solution. This significantly improved the precision of the fitted
parameters, and is the only viable approach in a system like this with shallow
partial eclipses and a noise limit set by the presence of pulsations.
Uncertainties in the fitted parameters were determined by Monte Carlo and
residual-permutation simulations (Southworth, 2008), taking the larger of the
two options for each quantity. The values and uncertainties of the parameters
are given in Table 1. We have added an extra uncertainty of $\pm$0.0010 in
quadrature to the uncertainties in $r_{\rm A}$ and $r_{\rm B}$ to account for
the variations in these parameters between different model choices,
specifically about whether or not to fit for $L_{3}$ or limb darkening.

Table 1:  Properties of the HD 23642 system. The velocity amplitudes $K$ are from Torres et al. (2021). Quantity and unit | Star A | Star B
---|---|---
Spectroscopic parameters:
$K$ ( km s-1) | 100.07 $\pm$ | 0.23 | 142.60 $\pm$ | 0.28
$T_{\rm eff}$ (K) | 10200 $\pm$ | 90 | 7670 $\pm$ | 85
Light ratio at H$\alpha$ | $0.313\pm 0.010$
Light ratio in TESS passband | $0.328\pm 0.011$
jktebop analysis:
$r_{\rm A}+r_{\rm B}$ | $0.2664\pm 0.0011$
$k$ | $0.784\pm 0.012$
$i$ (∘) | $78.63\pm 0.09$
$J$ | $0.5561\pm 0.0011$
$\ell_{3}$ | $0.029\pm 0.014$
$c$ | 0.52 fixed | 0.63 fixed
$\alpha$ | 0.45 fixed | 0.42 fixed
$P$ (d) | $2.46113223\pm 0.00000060$
$T_{0}$ (BJDTDB) | $2459509.282357\pm 0.000007$
Fractional radii | 0.1494 $\pm$ | 0.0016 | 0.1171 $\pm$ | 0.0017
Physical properties:
Mass ($\mathcal{M}^{\rm N}_{\odot}$) | 2.273 $\pm$ | 0.011 | 1.595 $\pm$ | 0.008
Radius ($\mathcal{M}^{\rm N}_{\odot}$) | 1.799 $\pm$ | 0.019 | 1.410 $\pm$ | 0.021
$\log g$ (c.g.s.) | 4.285 $\pm$ | 0.009 | 4.342 $\pm$ | 0.13
$V_{\rm synch}$ ( km s-1) | 36.98 $\pm$ | 0.40 | 28.99 $\pm$ | 0.42
Luminosity $\log(L/\hbox{$\mathcal{L}^{\rm N}_{\odot}$})$ | 1.499 $\pm$ | 0.018 | 0.792 $\pm$ | 0.023
Reddening $E_{B-V}$ (mag) | $0.040\pm 0.010$
$K$-band distance (pc) | $134.7\pm 2.0$

We determined the physical properties of the system from the results of the
jktebop analysis and the velocity amplitudes of the stars measured by Torres
et al. (2021). This was done using the jktabsdim code (Southworth et al.,
2005) modified to report results on the IAU scale (Prša et al., 2016). The
full set of measured system properties are given in Table 1. Our masses are in
good agreement with those found by other authors, with minor differences due
to the newer velocity amplitudes adopted. The rotational velocities of the
stars determined by Southworth et al. (2005), $37\pm 2$ and $33\pm 3$ km s-1,
are in agreement with the synchronous values, suggesting that both stars are
rotating synchronously.

The measured radii, however, are significantly different from previous values
(Munari et al., 2004; Southworth et al., 2005; Groenewegen et al., 2007; David
et al., 2016) in the sense that $R_{\rm A}$ is larger and $R_{\rm B}$ is
smaller. This solves an existing problem with HD 23642 B, which was previously
found to be significantly larger than predicted by theoretical evolutionary
models for the Pleiades’ age and metallicity. We attribute this change to
differences in the spectroscopic light ratios adopted in those studies, which
are slightly larger than the one measured in the current work. Both components
of HD 23642 are known to be chemically peculiar so light ratios determined
from metal lines are unreliable. Our new light ratio should be preferred
because it is based on the H$\alpha$ line so is not affected by the chemical
peculiarity, and is also close to the wavelengths transmitted by the TESS
passband. A visual comparison of the masses, radii and $T_{\rm eff}$s of the
primary star to predictions from the parsec evolutionary models (Bressan et
al., 2012) for solar metallicity shows good agreement for an age of $170\pm
20$ Myr. We note that this is at the upper limit of accepted ages of the
Pleiades cluster (Gossage et al., 2018; Murphy et al., 2022). The secondary
component is 1.5$\sigma$ smaller than expected: further investigation is
needed to understand this.

To determine the distance ($d$) and interstellar reddening ($E_{B-V}$) to the
system we used the $BV$ apparent magnitudes from the Tycho satellite (Høg et
al., 2000), the $JHK_{s}$ apparent magnitudes from 2MASS (Skrutskie et al.,
2006) converted into the Johnson system using transformations from Carpenter
(2001), and bolometric corrections from Girardi et al. (2002). We determined
the value of $E_{B-V}$ that results in consistent distances across the $BVJHK$
bands, finding $E_{B-V}=0.040\pm 0.010$ and $d=134.7\pm 2.0$ pc. This distance
is slightly shorter than the $138.3\pm 0.1$ pc determined by simple inversion
of the parallax from Gaia EDR3 (Gaia Collaboration, 2021). This $E_{B-V}$ is
specific to HD 23642, is consistent with previous studies (Taylor, 2008), and
is not affected by reddening variations between Pleiades members.

Figure 3:  Fourier transform of the light curve residuals, after subtracting
the eclipse model, revealing several pulsations at the 0.1 mmag level. Mode
identifications (Section 5) are overlaid, showing radial modes as well as a
series of rotationally split dipole modes.

##  5 Analysis of the pulsations

The residual light curve after subtraction of the jktebop model shows several
pulsation modes at the 0.1 mmag level (Fig. 3). We used the period04 code
(Lenz & Breger, 2004) to extract 46 pulsations at frequencies $f>20$ d-1 down
to 0.015 mmag amplitude, and used non-linear least squares to optimise the
frequencies. The table of frequencies is given in the Appendix. From the
frequencies and $T_{\rm eff}$s of the stars we deduce that they represent
$\delta$ Scuti pulsations in the secondary component, as the primary is hotter
than the $\delta$ Scuti instability strip.

We used the echelle package (Hey & Ball, 2020) to manually find values of the
asteroseismic large spacing, $\Delta\nu$, that give vertical patterns in an
échelle diagram. At $\Delta\nu=6.97$ d-1, there are two parallel ridges on the
left-hand side of the échelle (Fig. 4), at an $x$-location suggestive of
$\ell=1$ modes (Bedding et al., 2020; Murphy et al., 2021). With the same
$\Delta\nu$, part of a radial mode series can also be identified, with period
ratios consistent with low-order radial modes (Netzel et al., 2022). It is
noteworthy that the independently-determined $\Delta\nu$ is consistent with
the value of five other $\delta$ Scuti stars in the Pleiades (6.82–6.99 d-1;
Murphy et al., 2022), and also suggests that the system is relatively young
($\lesssim$200 Myr).

Figure 4:  Échelle diagram for HD 23642, marked with mode identifications.
Radial modes are shown as circles, dipole modes are triangles. Modes without
an identifications are shown as crosses, some of which may belong to the other
star; most will be modes of higher degree ($\ell\geq 2$).

Since $\ell=1$ doublets are seen, which are interpretable as $m=\pm 1$ pairs,
the inclination of the pulsating star is not small (Gizon & Solanki, 2003).
However, at some orders, most notably at $n=3$, there is a peak slightly
offset from halfway between the two identified $\ell=1$ modes that could be a
central component, after second-order effects of rotation are accounted for.
To evaluate this possibility would require the calculation of rotating
evolutionary and pulsation models. If confirmed, it implies that the stellar
inclination is also not completely edge-on and, given the measured orbital
inclination, implies that the spin and orbital axes of the pulsating star are
aligned.

The identification of four $\ell=1$ doublets offers the chance of a
consistency check on the mode identification. If the doublets all have
approximately the same splitting, it strengthens the proposed mode
identifications. For the doublets at $n=6$, 3, 2, and 1 we measure splittings
of 0.813, 0.813, 0.812, and 0.586 d-1, respectively. Thus, the $n=6$, 3, and 2
doublets seem secure, but the $n=1$ doublet does not. Since the Ledoux
constant, $C_{\rm n,\ell}$, is close to zero for p modes, we can estimate the
stellar rotation rate, $\Omega$, to first order based on these frequency
splittings (Aerts et al., 2010):

$\displaystyle\nu_{n,\ell,m}=\nu_{0}+m\Omega(1-C_{n,\ell}),$ (1)

where $\nu_{0}$ is the rest-frame frequency of the pulsation mode (we adopt
the convention that prograde modes have positive $m$). The stellar rotation
frequency is therefore 0.41 d-1, corresponding to a rotation period of 2.46 d.
This is equal to the orbital period of the system, and the stars are known to
rotate approximately synchronously (see Section 4), so this supports our
inference of the stellar rotational inclination.

##  6 Summary and conclusions

HD 23642 has it all: youth, eclipses, pulsations, chemical peculiarity, and
membership of the Pleiades. We establish its physical properties to high
precision for the first time, helped particularly by the measurement of a
spectroscopic light ratio immune to the chemical peculiarity. We determine a
distance and interstellar reddening to the system in good agreement with
previous measurements, and infer an age of $170\pm 20$ Myr for the Pleiades by
comparison to parsec evolutionary model predictions.

A total of 46 pulsation frequencies are detected to high significance from the
three consecutive sectors of TESS photometry. We attribute them to $\delta$
Scuti pulsations in the secondary star. We use an échelle diagram to assign
modes to 11 of the pulsations, based on the identification of $\ell=1$
doublets. An asteroseismic large spacing of $\Delta\nu=6.97$ d-1 allows
identification of a series of radial modes with period ratios consistent with
other $\delta$ Scuti stars in the Pleiades. The stellar rotation rate we find
from the mode splittings is in agreement with the spectroscopic $v\sin i$ and
the orbital period of the system. This implies that the spin axis of the
pulsating secondary component is aligned with the orbital axis of the system.
HD 23642 is well suited to detailed analysis with evolutionary and pulsation
models including rotation.

## Data availability

The TESS data used in this work are available in the MAST archive
(https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html). The FIES
spectra are available in reduced form from the NOT archive
(http://www.not.iac.es/archive/).

## Acknowledgements

We thank Dominic Bowman and Hiromoto Shibahashi for useful discussions. The
TESS data presented in this paper were obtained from the Mikulski Archive for
Space Telescopes (MAST) at the Space Telescope Science Institute (STScI).
STScI is operated by the Association of Universities for Research in
Astronomy, Inc. Support to MAST for these data is provided by the NASA Office
of Space Science. Funding for the TESS mission is provided by the NASA
Explorer Program. SJM was supported by the Australian Research Council through
Future Fellowship FT210100485.

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##  Appendix A Measured pulsation frequencies in HD 23642

Table 2:  Extracted frequencies with corresponding amplitudes and provisional mode identifications for the pulsating component of HD 23642. $f$ (d-1) | amplitude (mmag) | $n$ | $\ell$ | $m$
---|---|---|---|---
57.2112 | 0.0175 | 6 | 1 | 1
56.3981 | 0.0161 | 6 | 1 | $-1$
36.2304 | 0.1029 | 3 | 1 | 1
35.4177 | 0.0768 | 3 | 1 | $-1$
29.3618 | 0.0483 | 2 | 1 | 1
28.5496 | 0.0431 | 2 | 1 | $-1$
22.4772 | 0.1365 | 1 | 1 | 1
21.8913 | 0.1713 | 1 | 1 | $-1$
39.3467 | 0.0938 | 4 | 0 | 0
33.1444 | 0.0869 | 3 | 0 | 0
26.9261 | 0.0260 | 2 | 0 | 0
53.5132 | 0.0188 | – | – | –
39.4291 | 0.0302 | – | – | –
39.1434 | 0.0157 | – | – | –
39.0450 | 0.0199 | – | – | –
38.7352 | 0.0514 | – | – | –
38.6162 | 0.0407 | – | – | –
38.5353 | 0.0362 | – | – | –
38.3290 | 0.0296 | – | – | –
37.9217 | 0.0272 | – | – | –
37.8036 | 0.1171 | – | – | –
36.1084 | 0.0165 | – | – | –
35.8956 | 0.0370 | – | – | –
35.2961 | 0.0289 | – | – | –
34.1939 | 0.0286 | – | – | –
33.8544 | 0.0206 | – | – | –
32.3307 | 0.0221 | – | – | –
29.4768 | 0.0239 | – | – | –
29.4484 | 0.0198 | – | – | –
29.1269 | 0.0368 | – | – | –
28.6681 | 0.0302 | – | – | –
28.3104 | 0.0203 | – | – | –
28.1144 | 0.0378 | – | – | –
28.0038 | 0.0230 | – | – | –
27.7391 | 0.0347 | – | – | –
27.3022 | 0.0181 | – | – | –
26.1835 | 0.0203 | – | – | –
25.3700 | 0.0252 | – | – | –
24.9365 | 0.0264 | – | – | –
24.5100 | 0.0159 | – | – | –
24.2900 | 0.0160 | – | – | –
23.7500 | 0.0281 | – | – | –
22.9355 | 0.0181 | – | – | –
22.0604 | 0.0403 | – | – | –
20.3501 | 0.0157 | – | – | –


# Spatial Consistency Evaluation Based on Massive SIMO Measurements

Sida Dai, Martin Kurras  Fraunhofer Heinrich Hertz Institute, Einsteinufer 37,
10587 Berlin, Germany
Email<EMAIL_ADDRESS>
###### Abstract

In this paper, the spatial consistency of wireless massive single-input
multiple-output channels in a cellular small cell scenario is evaluated based
on measurements taken in Berlin city. The evaluation is done by computing the
similarity of covariance matrices over distance. As similarity measure the
correlation matrix distance is used. A classification of the measurement
tracks based on the shape of the curves into four different categories is
done.

The results in this paper indicate that spatial consistency is a highly
deterministic property in the sense that it depends strongly on the individual
environment and not so much on large scale parameters. Therefore, we conclude
that spatial consistency is not sufficiently modeled by the current 3rd
Generation Partnership Project feature.

###### Index Terms:

3GPP, GSCM, MIMO, massive MIMO, Channel, Model, Spatial, Consistency,
Measurements

##  I Introduction

In wireless communications research, simulations play an important role. They
enable early and accurate evaluation or comparison of new techniques, for
example in 3rd Generation Partnership Project (3GPP) standardization [1] or in
most of the recent research papers. An essential part of each wireless
transmission simulation is the underlying channel and the modeling of it [2].
There are various channel models but it is out of the scope of this paper to
provide an overview; the interested reader is referred to [3]. Assumptions
taken for the modeling of the channel directly impact or limit simulation
results. For example, while the assumption of independent fading between users
may hold in a multiple-user multiple-input multiple-output (MIMO)
transmission, this assumption cannot be used to evaluate techniques utilizing
channel correlation. One example of such a scheme is joint spatial division
and multiplexing (JSDM) [4], where users are clustered into groups with
similar channel properties prior to data-transmission.

It is intuitive that users close to each other experience similar wireless
channels to the same transmitter. The feature that models such behavior,
called spatial consistency, was first introduced in [5] and further discussed
in [1] and researchers have been focusing on this topic ever since. In [6]
Kurras et. al. provide an evaluation of the 3GPP spatial consistency feature
implemented in the open source channel model quasi-deterministic radio channel
generator (QuaDRiGa) [7], taking into account angular distance, chordal
distance and the correlation matrix distance (CMD). In [8] measurements at
$73\text{\,}\mathrm{GHz}$ are conducted with a single horn antenna at both
ends, where the transmit antennas are fixed and the receive antennas sweep
with a $5\text{\,}\mathrm{\SIUnitSymbolDegree}$ resolution. With such a
measurement setup the angular spread of multi-path component (MPC) can not
sufficiently be measured. Additionally, with a wavelength of
$\approx$0.41\text{\,}\mathrm{mm}$$ and a given measurement density of
$0.2\text{\,}\mathrm{m}$ which is approximately a resolution of 50
wavelengths, the changes in the phases of the multi-path components can not be
captured accurately. [9] provides a purely simulation based study where the
spatial consistency of dominant MPCs in a high-speed train scenario is
evaluated comparing the 3GPP channel model with ray-tracing. In [10], the
spatial consistency in vehicles is compared between the
$3-$11\text{\,}\mathrm{GHz}$$ and $55-$65\text{\,}\mathrm{GHz}$$ bands.
However, the evaluation only focuses on power-delay, without taking any
angular information into account.

In this paper, we use single-input multiple-output (SIMO) measurements
including angular information of the MPCs [11] to provide an evaluation of the
spatial consistency feature. The angular information is obtained with the help
of a cylindrical antenna array at the receive base station (BS). With a
measurement center frequency of $3.675\text{\,}\mathrm{GHz}$ in a small-cell
like deployment, the result shown in this paper can directly be considered for
the upcoming cellular fifth generation (5G) systems111MIMO simulation
assumptions in 3GPP consider $4\text{\,}\mathrm{GHz}$ center frequency, see
Section 7.1.6 in [12] ..

In the remainder of this paper, we provide details on the CMD evaluation
metric for spatial consistency in Section II. In Section III, details of the
underlying measurement campaign are given, which are used for results in
Section IV. Finally, Section V summarizes our findings and concludes this
paper.

##  II Spatial Consistency

Previous geometry-based stochastic channel models (GSCMs) suffer from the lack
of realistic correlation in the small-scale fading (SSF), i.e. even though the
large scale parameters (LSPs) are always spatially consistent, the models fail
to provide correlated SSF, governed by the position of the scattering
clusters. The 3GPP proposed a new channel model for the upcoming 5G of
wireless communications in [1]. This model solves the drawbacks in previous
GSCMs and a part of it is the introduction of the so called spatial
consistency feature. With spatial consistency, two closely located users will
not only have similar LSPs, they will also observe similar angles of received
MPCs.

In order to evaluate the spatial consistency feature, authors in [6] compared
different performance metrics, among which the covariance matrix based ones
have provided reasonable and representative results. The covariance matrix is
assumed to be slow time-varying and is therefore a suitable key performance
indicator (KPI) for evaluating the spatial correlation between users [4, 6].
Thanks to this property, lots of studies on user clustering are established on
covariance matrix based similarity measures (e.g. chordal distance [13, 4] or
CMD [14]). Therefore, this paper focuses on the covariance matrix based CMD
metric for evaluation of the spatial consistency and details on how the
covariance matrix is obtained are provided next.

###  II-A Covariance Matrix

The covariance matrix is obtained by averaging the channel coefficients over
the time duration $\tau$ and the number of orthogonal frequency division
multiplexing subcarriers $N$. Channel coefficients between the transmitter and
the receiver at time $t$ on the $n$-th subcarrier are denoted as
$\mathbf{H}_{t,n}\in\mathbb{C}^{n_{r}\times n_{t}}$, where $n_{r}$ and $n_{t}$
are the number of antennas at the receiver and the transmitter, respectively.
The covariance matrix at the receiver side
$\mathbf{R}\in\mathbb{C}^{n_{r}\times n_{r}}$ is often defined as

$\mathbf{R}^{(\mathrm{Lit})}=\mathbb{E}\left[\mathbf{H}_{t,n}\mathbf{H}^{\mathrm{H}}_{t,n}\right],$
(1)

without further explanation on how exactly the covariance matrix can be
obtained [4, 14], because the covariance matrix is directly generated in the
simulations and Gaussian noise is added for individual channel realizations.
In Eq. 1, $\mathbb{E}[\cdot]$ denotes the expectation value. However, in our
case the covariance matrix has to be obtained from discrete measurements and
is calculated as

$\mathbf{R}^{(\mathrm{Meas})}=\frac{1}{\tau
N}\sum_{t=1}^{\tau}\sum_{n=1}^{N}\mathbf{H}_{t,n}\mathbf{H}^{\mathrm{H}}_{t,n}.$
(2)

It can be seen from Eq. 2 that the covariance matrix depends on the selection
of the averaging time $\tau$ and the averaging bandwidth represented by $N$.

Based on the covariance matrix according to Eq. 2, details on the CMD to study
the spatial correlation between two user is given next.

###  II-B Correlation Matrix Distance

The CMD was proposed in [13] and served as a novel measure to track the
changes in spatial structure of non-stationary MIMO channels. Results in [6]
have shown a strong correlation between the physical distance and the CMD.
Given the covariance matrices of two users
($\mathbf{R}_{1}$,$\mathbf{R}_{2}$), the similarity measure based on CMD,
according to [13], can be obtained by

$d_{\mathrm{CMD}}\left(\mathbf{R}_{1},\mathbf{R}_{2}\right)=\frac{\mathrm{Tr}(\mathbf{R}_{1}^{\mathrm{H}}\mathbf{R}_{2})}{\|\mathbf{R}_{1}\|_{\mathrm{F}}\cdot\|\mathbf{R}_{2}\|_{\mathrm{F}}}\\\
$ (3)

where $\mathrm{Tr}(\cdot)$ denotes the “trace” operator. The CMD based
similarity measure is a normalized metric which is upper-bounded by $1$ in the
case of $\mathbf{R}_{1}$ and $\mathbf{R}_{2}$ being collinear, and lower-
bounded by $0$ in the case of $\mathbf{R}_{1}$ and $\mathbf{R}_{2}$ being
orthogonal.

##  III Measurements

Figure 1: Map of measurement locations in Berlin. Circles with numbers
represent BS locations and “Tx01” - “Tx26” indicate measurement tracks.

The massive SIMO small cell measurements have been taken in Berlin City close
to Fraunhofer Heinrich Hertz Institute (HHI) main building. Fig. 1 shows a map
of the measurement area, where circles with numbers 1 to 3 and 5 to 7
represent BS locations and the colored lines with text “Tx01” to “Tx26” next
to it denote the measurement tracks. The strokes at one end together with
small arrows indicate the starting point of the tracks and circles at the
other end represent the end point. At each BS location two different BS
heights of $3\text{\,}\mathrm{m}$ and $6\text{\,}\mathrm{m}$ have been
measured. The BS locations 1 to 3 are associated with tracks 1 to 12 and
belong to the “Campus” scenario. The remaining BS locations 5 to 7 with
measured tracks 13 to 26 belong to the “Open Street” scenario. It can be seen
from Fig. 1 that the tracks cover line of sight (LoS) and non-line of sight
(NLoS) situations in both scenarios.

For the BS we used a uniform cylindrical array (UCA) with 16 columns and 4
stacked dual-polarized patch antennas resulting in 128 antenna elements in
total, shown in Fig. 2. Inside the UCA a “128 to 1” radio frequency (RF) self-
developed switch was used making it possible to measure all antenna with the
same RF receiver chain sequentially. For transmit antenna at the mobile device
an omni-directional vertically polarized single antenna from Huber+Suhner
model “1399.17.0111” was used.

(a) Side view

(b) Top View

Figure 2: Manufactured switched cylindrical array antenna with 128 elements
and the switched measurement timing [11].

(a) 6 m

(b) 3 m

(c) Transmitter moving on cart

Figure 3: Measurement equipment. The cylindrical receive antenna is mounted on
a tripod at (a) 6～m and (b) 3 m height. (c) The transmitter is mounted on a
dolly and moving at a speed of
$\approx$1.8\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{h}$$. [11]

The complete measurement setup is shown in Fig. 3. During the measurements the
transmit antenna was moved along the $40\text{\,}\mathrm{m}$ tracks at a
height of $1.5\text{\,}\mathrm{m}$ with a constant speed of
$\approx$1.8\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{h}$$. The BS
antenna was receiving Chadoff-Zho like sounding sequences of 1024 sample-
length with a duration of $4.1\text{\,}\mathrm{\SIUnitSymbolMicro s}$.
Switching all 128 antenna elements at the BS, called a complete SIMO short-
term time sample (STTS), was measured every $1.08\text{\,}\mathrm{ms}$. This
includes already some real-time averaging. The signal bandwidth was
$250\text{\,}\mathrm{MHz}$ with a center frequency of
$3.675\text{\,}\mathrm{GHz}$. Transmitter and receiver have been synchronized
by a self-developed Rubidium clock. A long-term time sample (LTTS) consists of
numerous short-term time samples (STTSs), where $\tau$ out of them are used
for covariance matrix averaging, and has a time interval of
$0.66\text{\,}\mathrm{s}$, see Fig. 4. Each track is divided into 120 LTTSs.
Further details of the measurement campaign are given by Raschkowski et. al.
in [11] and by Pitakdumrongkija et. al. in [15].

Figure 4: Illustration of a sample track. The first $\tau$ STTSs are used for
covariance matrix calculation.

##  IV Numerical Results

Parameter |  Value
---|---
Measurement direction |  Uplink
Center frequency |  $3.675\text{\,}\mathrm{GHz}$
Measured scenarios |  Open Square, Campus
Propagation scenarios |  LoS, NLoS
Number BS antennas $n_{r}$ |  128
Distribution BS antennas |  UCA, 16 columns, 4 rows, 2 polarizations, see Fig. 2 in [11]
BS antenna pattern |  $65^{\circ}$ HPBW in azimuth and elevation
Number MS antennas $n_{t}$ |  1
MS antenna type |  Omni-directional vertically polarized
SIMO STTS resolution |  $1.08\text{\,}\mathrm{ms}$
Covariance matrix window $\tau$ |  $10\text{\,}\mathrm{S}\mathrm{T}\mathrm{T}\mathrm{S}$
LTTS resolution |  $0.66\text{\,}\mathrm{s}$, see Fig. 4
Channel bandwidth |  $18\text{\,}\mathrm{MHz}$
Subcarrier bandwidth |  $180\text{\,}\mathrm{kHz}$
Number of OFDM subcarriers $N$ |  100
TABLE I: Measurement and data processing parameters.

The main measurement and data processing parameters are listed in Table I. For
covariance matrix calculation according to Eq. 2, a window length of $\tau=10$
STTSs is applied. Since no studies are available on reliable covariance matrix
calculation with respect to (w.r.t.) $\tau$, the choice of $\tau$ in our work
is heuristic and requires further investigation. Next, we provide a
classification of the measurement tracks, followed by a thorough analysis of
the results.

###  IV-A Track Classification

After a thorough examination of each track with its serving BS and
corresponding surroundings shown in Fig. 1, we sort each measurement into one
of the following classes:

  * •

LoS radial: The track is approximately radial/perpendicular to the BS and LoS
is available throughout the whole track.

  * •

LoS tangential: The track is approximately tangential w.r.t. the position of
the BS with a LoS condition over the complete track.

  * •

Far away: The track is located far away (at least $100\text{\,}\mathrm{m}$)
from the BS w.r.t. the track distance of $\approx$40\text{\,}\mathrm{m}$$.
Only available in the open street scenario.

  * •

Uncorrelated: Tracks with low correlation, with either NLoS condition or
obstructed LoS condition222Obstructed LoS means objects with a size similar to
the wavelength between transmitter and receiver, e.g. three branches..

  * •

Other: The curve shape of the spatial consistency does not follow the other
classes.

Table II maps the measurement tracks from Fig. 1 to the classes given above.
Note that due to the highly dynamic transmission environment, e.g. transition
from LoS to NLoS or vice versa, and obstacles that are not shown in the
measurement map, e.g. parking cars, the above-mentioned classes can not cover
all measurements. Therefore, we leave out evaluation of the measurements in
the “Other” class for future work. We can see from the table that except for
the “Far away” class, each class contains tracks from both “Campus” and “Open
Street” scenario.

In order to study the behavior of the CMD similarity w.r.t. distance, we set
the starting LTTS of each measurement as user 1, while treating each of the
remaining LTTSs as user 2 moving away from user 1. Therefore, the
$d_{\mathrm{CMD}}$ between two users can be calculated according to Eq. 3 for
each track. Fig. 5 shows the CMD similarity of class “LoS radial” over
distance. For visualizing purpose, the area between the minimum and maximum
$d_{\mathrm{CMD}}$ over all tracks in the class at each LTTS is shown as a
gray cover plot. Otherwise, the many lines within the same figure would be
hard to distinguish. Instead, two representative $d_{\mathrm{CMD}}$ curves are
given to show the dynamic range of the measurement tracks. We can see in Fig.
5 that in this class the average $d_{\mathrm{CMD}}$ drops steadily over
distance but overall remains high, e.g. mostly above 0.5 up to
$20\text{\,}\mathrm{m}$. This high correlation can be explained by the LoS
condition and that most of the power is received by the LoS path.

Similarly, Fig. 6 shows results for the “LoS tangential” class. Here
$d_{\mathrm{CMD}}$ decreases almost proportionally with the distance down to
0.1. In comparison to the “LoS radial” class, the “LoS tangential” class
decreases faster. This can be explained by the change in angles. In the “LoS
radial” class, where the tracks are perpendicular to the BS, moving along
tracks only changes the elevation angle of the MPCs, whereas in class “LoS
tangential”, where the tracks are tangential to the BS, moving along tracks
changes both azimuth and elevation angles of the MPCs. The additional change
in the azimuth angle of the MPCs causes the steeper decrease in
$d_{\mathrm{CMD}}$.

Next, the “Far away” class is shown in Fig. 7. It is of interest to note that
the CMD similarity fluctuates over distance and no significant decrease is
observed for tracks in this class. This is due to the fact that distance
between users has a smaller impact on the channel for users located far away
from the BS than the ones which are closer to the BS. To further elaborate,
the change in channel coefficients is dependent on the ratio of the relative
distance between users to the total distance from the BS to the track. The
smaller the ratio, the smaller the changes in $d_{\mathrm{CMD}}$.

At last, Fig. 8 shows the “Uncorrelated” class. In this case, low correlation
is observed in the measurements, since $d_{\mathrm{CMD}}$ drops below 0.4
after the first few LTTSs. A cross-check with Fig. 1 shows that most of these
measurements are taken on tracks without direct LoS to the serving BS. This
indicates that the angles of the received MPCs at the BS change rapidly even
by moving the transmitter for $1\text{\,}\mathrm{m}$. One explanation for this
could be a fast change of scattering objects that results in non-continuous
phase and amplitude jumps of MPCs. This is also called “death” and “birth”, or
lifetime, of scatterers. In our measurements this relates to scatterers
observed by the BS which was a UCA deployed on a pole and therefore able to
receive MPCs from all azimuth directions.

###  IV-B Result Analysis

These 4 classifications show that the spatial consistency depends less on the
LSPs of a given scenario and more on the individual geometry, as both the
“Campus” and “Open Street” scenarios appear in the same classes. In general,
the similarity of covariance matrices decreases over distance. However,
depending on each classification, the similarity can have large variation
locally, e.g. the similarity remains high over $40\text{\,}\mathrm{m}$
distances in the “Far-Away” category, whereas in the “Uncorrelated” category,
the similarity is barely seen. Furthermore, the similarity of the covariance
matrices is decreasing within a very short distance in many of the NLoS
scenarios. This indicates that clustering of users based on covariance matrix
similarity can be difficult to achieve in NLoS scenarios.

The above described effects are not captured by the current 3GPP proposal of
the spatial consistency feature where only a single dependency on the
“correlation distance” and the distance between users is captured, see [6].
Therefore, an extension of the spatial consistency feature in [1] is required.

Track Class |  BS ids |  Track ids
---|---|---
LoS radial |  1 |  2 ($6\text{\,}\mathrm{m}$)
|  2 |  4 ($6\text{\,}\mathrm{m}$)
|  5 |  13, 25
|  6 |  24 ($6\text{\,}\mathrm{m}$)
LoS tangential |  1 |  4 ($3\text{\,}\mathrm{m}$)
|  2 |  3
|  3 |  4 ($6\text{\,}\mathrm{m}$)
|  5 |  23, 24
|  6 |  14, 15 ($3\text{\,}\mathrm{m}$), 22
|  7 |  17 ($6\text{\,}\mathrm{m}$), 18 ($3\text{\,}\mathrm{m}$), 19 ($3\text{\,}\mathrm{m}$), 20 ($3\text{\,}\mathrm{m}$)
Far away |  5 |  15 ($3\text{\,}\mathrm{m}$), 16, 18 ($3\text{\,}\mathrm{m}$), 19, 20 ($3\text{\,}\mathrm{m}$), 21, 22
|  6 |  17 ($6\text{\,}\mathrm{m}$), 18 ($6\text{\,}\mathrm{m}$), 19 ($6\text{\,}\mathrm{m}$)
|  7 |  13 ($6\text{\,}\mathrm{m}$), 15 ($3\text{\,}\mathrm{m}$),23 ($6\text{\,}\mathrm{m}$), 25 ($3\text{\,}\mathrm{m}$)
Un-correlated Tracks |  1 |  5 ($6\text{\,}\mathrm{m}$), 6, 7, 9, 10, 11, 12
|  2 |  1 ($6\text{\,}\mathrm{m}$), 2 ($6\text{\,}\mathrm{m}$), 5, 6, 7, 8, 10 ($6\text{\,}\mathrm{m}$)
|  3 |  2, 5 ($6\text{\,}\mathrm{m}$), 9, 10 ($3\text{\,}\mathrm{m}$), 11, 12
|  5 |  26 ($3\text{\,}\mathrm{m}$)
|  6 |  21 ($6\text{\,}\mathrm{m}$)
TABLE II: Mapping of measurement tracks to track-classes Figure 5: CMD
similarity over distance for the tracks in class “LoS radial” according to
Table II.  Figure 6: CMD similarity over distance for the tracks in class “LoS
tangential” according to Table II.  Figure 7: CMD similarity over distance for
the tracks in class “Far away” according to Table II.  Figure 8: CMD
similarity over distance for the tracks in class “Uncorrelated” according to
Table II.

##  V Conclusion

In this paper, we evaluate the spatial consistency feature based on SIMO
measurements with angular information of the MPCs. By studying the behavior of
$d_{\mathrm{CMD}}$ over distance we categorize each track into one of the “LoS
radial”, “LoS tangential”, “Far away” and “Uncorrelated” classes. In general,
the similarity decreases over distance, however, the speed of decreasing
varies form class to class due to individual geometry. Moreover, low
similarity is observed in the NLoS scenarios, proving it difficult to cluster
users in such scenarios. These findings could be beneficial for future 3GPP
proposal in channel model.

This work is a first step to motivate a further and deeper analysis of the
measurement data with respect to spatial consistency. As a next step, we will
analyze the dependency of the spatial consistency from angular spread and
K-factor to better understand in which radio environments massive MIMO schemes
that utilize similarity in covariance matrices can be applied. This will also
include a direct comparison with the 3GPP spatial consistency feature.

## Acknowledgment

A part of this work has been performed in the framework of the Horizon 2020
project ONE5G (ICT-760809), funded by the European Union. The authors would
like to acknowledge the contributions of their colleagues in the project,
although the views expressed in this contribution are those of the authors and
do not necessarily represent the project.

Moreover, the authors thank their Fraunhofer HHI colleagues for conducting the
measurements, which is done by Fabian Undi, Leszek Raschkowski and Stephan
Jaeckel. Stephan Jaeckel also did the data post-processing. We also thank
Boonsarn Pitakdumrongkija333While Boonsarn was with NEC at the time of the
measurements, in the meantime he left NEC., Xiao Peng and Masayuki Ariyoshi
from NEC Japan for enabling these measurements in the first place.

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# Arithmetical properties at the level of idempotence

Fethi Bouzeffour a, Wissem Jedidi b a Department of mathematics, College of
Sciences, King Saud University, P. O Box 2455 Riyadh 11451, Saudi Arabia.
<EMAIL_ADDRESS><EMAIL_ADDRESS>b Department of Statistics &
OR, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia and
Université de Tunis El Manar, Faculté des Sciences de Tunis, LR11ES11
Laboratoire d’Analyse Mathématiques et Applications, 2092, Tunis, Tunisia

###### Abstract.

In this paper we give an attempt to extend some arithmetic properties such as
multiplicativity, convolution products to the setting of operators theory. We
provide a significant examples which are of interest in number theory. We also
give a representation of the Euler differential operator by means of the Euler
totient arithmetic function and idempotent elements of some associative unital
algebra.

###### Key words and phrases:

Arithmetic functions, idempotent.

###### Key words and phrases:

Arithmetic functions, convolution products, idempotent.

###### 2000 Mathematics Subject Classification:

11A25, 16U99.

###### 2010 Mathematics Subject Classification:

Primary 16U99; Secondary 11A05

##  1\. Introduction

In number theory, an arithmetical, or number-theoretic function is a function
$\alpha:\,\mathbb{N}\rightarrow\mathbb{C}$. Their various properties were
investigated by several authors and they represent an important research topic
up to now. An important property shared by many number-theoretic functions is
multiplicativity: an arithmetical function $\alpha$ is said to be
multiplicative, if for all relatively prime positive integers $n,\,m$, we have

$\alpha(nm)=\alpha(n)\alpha(m).$

Examples of important arithmetical functions include: the Euler totient
function, denoted $\varphi$, and defined as the number of positive integers
less than and relatively prime to $n$. The Möbius function given by

$\mu(n)=\begin{cases}(-1)^{\omega(n)}&\mbox{ if }n\mbox{ is a square-free
integer}\\\ 0&\mbox{ otherwise },\end{cases}$

where $\omega(n)$ is the number of prime factors of $n$ is also a
multiplicative functions. Another important multiplicative function is the
Ramanujan sum’s, which is defined by [8]

$c_{n}(j):=\sum_{\begin{subarray}{c}\gcd(k,n)=1\\\ 1\leq k\leq
n\end{subarray}}\varepsilon_{n}^{jk},$

where $\varepsilon_{n}$ denotes a primitive $n$-th root of unity and
$\gcd(k,n)$ to denote the greatest common divisor of the positive integers $k$
and $n.$ These sums fit naturally with other number-theoretic functions. For
instance, one has

(1.1) $c_{n}(1)=\mu(n),\quad c_{n}(n)=\varphi(n),$

For a more elaborate account on the multiplicative functions, we refer the
reader to the texts [1, 2] and the survey articles [6].

In this work we suggest to extend the rang of the number-theoretic function
and consider functions $f$ such that:

\- their domain are the positive integers and whose range is a subset of an
unital associative algebra $A$ over $\mathbb{C}$;

\- they satisfy the property

(1.2) $f(nm)=f(n)f(m),\,\,\text{when}\,\,\gcd(n,m)=1.$

Our first objective is to give a variety of significant examples, which are of
interest in number theory. Notice that if $\mathcal{A}$ is an unital algebra
then every number-theoretic function $\alpha:\mathbb{N}\rightarrow\mathbb{C}$
can be identified with the function $n\rightarrow\alpha(n)\,e$ ($e$ is the
unit of $\mathcal{A}$).

Recall that an element $P\in\mathcal{A}$ such that $P^{2}=P$ is called
idempotent. A set of idempotent elements $P_{1},\dots,P_{n}$ is called
orthogonal if,

$P_{i}\,P_{j}=\delta_{ij}P_{i}\quad\mbox{and}\quad\sum_{i=1}^{n}P_{i}=e.$

In this paper, we consider a set of orthogonal idempotent $P_{j}(n)$, indexed
by two integers $n\geq 1$ and $j\geq 0$, satisfying:

\- for every fixed integer $n,$ the sequence $j\rightarrow P_{j}(n)$ is
periodic with period $n$;

\- for every arithmetic progression $j+n,\dots j+rn$, we have

$P_{j}(n)=\sum_{k=1}^{r}P_{j+kn}(nr).$

Under these conditions and for every fixed $j\geq 0$, the function
$n\rightarrow P_{j}(n)$ is multiplicative.
Another example considered in this note, consists to replace the root of unity
in the Ramanujan sum’s by an element $s$ of $\mathcal{A}$ satisfying
$s^{n}=e$. Then, the following sum

(1.3) $\displaystyle\sum_{\begin{subarray}{c}\gcd(k,n)=1\\\ 1\leq k\leq
n\end{subarray}}s^{k}$

is a multiplicative function.

##  2\. Arithmetic properties of Idempotents

###  2.1. Multiplicative functions

Throughout, $\mathcal{A}$ will be an associative unital algebra over
$\mathbb{C}$. We recall the following definitions and basic facts

######  Definition 2.1.

A function $f:\mathbb{N}\rightarrow\mathcal{A}$ is called multiplicative if

$f(nm)=f(n)f(m),\,\,\text{when}\,\,\gcd(n,m)=1.$

######  Lemma 2.1.

If $f:\mathbb{N}\rightarrow\mathcal{A}$ is multiplicative, then $f(1)$ is
idempotent.

###### Proof.

Taking $n=m=1$, we have $f(1)=f(1.1)=f(1)f(1).$ So, $f(1)$ is idempotent. ∎

The following proposition is a characterization of multiplicative functions.

######  Proposition 2.2.

For an arithmetical function $f:\mathbb{N}\rightarrow\mathcal{A}$, the
following are equivalent:

1\. $f$ is multiplicative,

2\. for all $n=p_{1}^{\alpha_{1}}\dots p_{k}^{\alpha_{k}}$ (the standard form
factorization of $n$), we have

$f(n)=f(p_{1}^{\alpha_{1}})\dots f(p_{k}^{\alpha_{k}}).$

We naturally extend the convolution product such as Drichlet product,
$\operatorname{lcm}$ product and unitary product (see, [1, 14, 12]) as
follows:

(2.1) $\displaystyle(f\,*\,g)(n)=\sum_{kl=n}f(k)g(l)\quad(\text{Dirichlet
product}).$ (2.2)
$\displaystyle(f\,\Box\,g)(n)=\sum_{\operatorname{lcm}(k,l)=n}f(k)g(l)\quad(\text{lcm-
product)}.$ (2.3)
$\displaystyle(f\,\sqcup\,g)(n)=\sum_{kl=n\,\,\gcd(k,l)=1}f(k)g(l)\quad(\text{unitary
product)}.$

where $f,\,g:\mathbb{N}\rightarrow\mathcal{A}$.

The convolution products defined in (2.1), (2.2) and (2.3) are associative but
not commutative in general. If, in addition, if for all integers $n,m$, and
every prime numbers $p$ and $q,$ $f(p^{n})$ commutes with $g(q^{m})$, then we
have

$f\,*\,g=g\,*\,f,\quad f\,\Box\,g=g\,\Box\,f,\quad f\,\sqcup\,g=g\,\sqcup\,f.$

The following propositions are easy to prove.

######  Proposition 2.3.

1) If $f$ is an arithmetic function on $\mathcal{A}$, then

$f\,*\,I=I\,*\,f=f,\quad f\,\Box\,I=I\,\Box\,f=f,\quad
f\,\sqcup\,I=I\,\sqcup\,f=f,$

where

$I(n)=\left\\{\begin{array}[]{l l}e\quad\text{if}\quad n=1\\\
0\quad\text{otherwise}.\end{array}\right.$

2) If $f$ is an arithmetical function such that $f(1)$ is invertible element
in $\mathcal{A}$, then $f$ has a unique inverse with respect to the Dirichlet
product.

3) If $f$ is a multiplicative function on $\mathcal{A}$ and $b\in\mathcal{A}$
invertible element , then the function $n\mapsto bf(n)b^{-1}$ is
multiplicative.

4) If $f$ and $g$ are multiplicative functions, then $f*g$ is also a
multiplicative function.

######  Proposition 2.4.

1)If $\mathcal{A}$ is equipped with a norm $\|.\|$ and $f$ a multiplicative
function on $A$, then $n\rightarrow\|f(n)\|$ is a multiplicative function.
2)If $\mathcal{A}$ is the algebra of $N\times N$-matrix and $f$ is
multiplicative on $\mathbb{C}^{N}$, then function $n\mapsto\det(f(n))$ is
multiplicative.

###  2.2. Idempotents

######  Definition 2.2.

Let $\mathcal{A}$ be an algebra with unity $e$. We say that a collection
$\,\\{P_{j}(n)\\}_{j=0,n=1}^{\infty}\,$ is an arithmetic system of idempotent
if the following conditions are satisfied:

I) for all $i,\,j=1,\,2,\dots,$ we have
$P_{i}(n)P_{j}(n)=\delta_{ij}P_{i}$(n),

II) for all $n,\,j=1,\,2,\dots,$ we have $P_{j+n}(n)=P_{j}(n),$

III) for every arithmetic progression $\\{j+n,\dots,j+nr\\}$ we have

$P_{j}(n)=\sum_{k=1}^{r}P_{j+kn}(nr).$

The integer $n$ in $P_{j}(n)$ is called level of the idempotence.

######  Theorem 2.5.

Let $\\{P_{j}(n)\\}_{j=0,n=1}^{\infty}$ be an arithmetic system of idempotents
of $\mathcal{A}$. Then for arbitrary positive integers $n$, $m$ $k$ and $l$,
we have

(2.4) $P_{k}(n)P_{l}(m)=\left\\{\begin{array}[]{l
l}P_{j}(\operatorname{lcm}(n,m))\quad\text{if}\quad\gcd(n,m)\mid l-k\\\
0\quad\text{otherwise}.\end{array}\right.$

where $j$ is the unique solution $\mod(\operatorname{lcm}(n,m))$ of the system
of congruences

(2.5) $\left\\{\begin{array}[]{l l}j\equiv k\mod(n)\\\ j\equiv
l\mod(m).\end{array}\right.$

Here $\operatorname{lcm}(n,m)$ is the least common multiple of the integer
$n,\,m.$

######  Lemma 2.6.

(The Chinese Remainder Theorem)
The system of congruences

(2.6) $\left\\{\begin{array}[]{l l}x\equiv a\mod(n)\\\ x\equiv
b\mod(m).\end{array}\right.$

is solvable if, and only if $\gcd(n,m)|a-b,$ any two solutions of the system
are incongruent mod $\operatorname{lcm}(n,m)$.

###### Proof.

From condition $III)$ we can write

$\displaystyle P_{k}(n)P_{l}(m)$
$\displaystyle=\sum_{s=0}^{m-1}\sum_{r=0}^{n-1}P_{k+sn}(nm)P_{l+rm}(nm).$

If $\gcd(n,m)\nmid l-k$, then $k+sn\neq l+rm$ and $P_{k}(n)P_{l}(m)=0$.
If $\gcd(n,m)\mid l-k,$ then by Chinese remainder theorem there exist unique
integer $j$ $\mod(\operatorname{lcm}(n,m))$ such that

(2.7) $\left\\{\begin{array}[]{l l}j\equiv k\mod(n)\\\ j\equiv
l\mod(m).\end{array}\right.$

Then from $II)$ we get

$\displaystyle P_{k}(n)P_{l}(m)$
$\displaystyle=\sum_{s=0}^{m-1}\sum_{r=0}^{n-1}P_{j+sn}(nm)P_{j+rm}(nm).$

On the other hand, from $I)$ we see that for $0\leq r\leq n-1$ and $0\leq
s\leq m-1$ the term $P_{j+rn}(nm)P_{j+sm}(nm)$ is different from zero, if and
only if $rn=sm$, which is equivalent to $r=r^{\prime}\frac{n}{\gcd(n,m)}$ and
$s=r^{\prime}\frac{m}{\gcd(n,m)}$, with $0\leq r^{\prime}\leq\gcd(n,m)-1$.
Then

$\displaystyle P_{k}(n)P_{l}(m)$
$\displaystyle=\sum_{r^{\prime}=0}^{\gcd(n,m)-1}P_{j+r^{\prime}\operatorname{lcm}(n,m)}(nm)=P_{j}(\operatorname{lcm}(n,m)).$

Therefore

$\displaystyle P_{k}(n)P_{l}(m)=\left\\{\begin{array}[]{l
l}P_{j}(\operatorname{lcm}(n,m))\quad\text{if}\quad\gcd(n,m)\mid l-k\\\
0\quad\text{otherwise}\end{array}\right.$

where $j$ is the unique integer $\mod(\operatorname{lcm}(n,m))$ such that

(2.8) $\left\\{\begin{array}[]{l l}j\equiv l\mod(n)\\\ j\equiv
k\mod(m).\end{array}\right.$

∎

######  Corollary 2.7.

The following hold
1\. If $n$ and $m$ are relatively prime, we have

$P_{j}(n)P_{j}(m)=P_{j}(nm).$

2\. If $n\mid m$, we have

$P_{j}(n)P_{k}(m)=\left\\{\begin{array}[]{l
l}P_{k}(m)\,\,\,\text{if}\,\,\,k\equiv j\mod(n)\\\
0\,\,\,\text{otherwise}.\end{array}\right.$

######  Proposition 2.8.

Let $\alpha,\,\beta:\,\mathbb{N}\rightarrow\mathbb{C}$ two arithmetic
functions and $j=0,1\dots$, we have

(2.9) $\displaystyle\alpha P_{j}\,\square\,\beta
P_{j}=(\alpha\square\beta)P_{j},$ (2.10) $\displaystyle\alpha P_{j}\sqcup\beta
P_{j}=(\alpha\sqcup\beta)\,P_{j},$ (2.11)
$\displaystyle\nu_{0}*(\alpha\square\beta)P_{j}=(\nu_{0}*\alpha
P_{j})\,(\nu_{0}*\beta P_{j}).$

In particular,

(2.12) $\displaystyle P_{j}\,\square\,P_{j}(n)=M_{2}(n)P_{j}(n),\quad
P_{j}\sqcup P_{j}(n)=2^{\omega(n)}P_{j}(n)$

where

(2.13) $M_{s}(n)=\left\\{\begin{array}[]{l l}1\quad\text{if}\,\,\,n=1,\\\
\prod_{k=1}^{r}\big{(}(a_{s}+1)^{s}-a_{k}^{s}\big{)}\quad\,\text{if}\quad
n=\prod_{k=1}^{r}p_{k}^{a_{k}},\end{array}\right.$

and $\nu_{0}(n)=1.$

###### Proof.

Let $n$ be a positive integer, we have

$\displaystyle\alpha P_{j}\,\square\,\beta P_{j}(n)$
$\displaystyle=\sum_{\operatorname{lcm}(k,l)=n}\alpha(k)\beta(l)P_{j}(k)P_{j}(l)$
$\displaystyle=\sum_{\operatorname{lcm}(k,l)=n}\alpha(k)\beta(l)P_{j}(\operatorname{lcm}(k,l))$
$\displaystyle=(\alpha\,\square\,\beta)(n)P_{j}(n).$

To prove (2.10)

$\displaystyle\alpha P_{j}\,\sqcup\,\beta P_{j}(n)$
$\displaystyle=\sum_{kl=n,\gcd(k,l)=1}\alpha(k)\beta(l)P_{j}(k)P_{j}(l)$
$\displaystyle=\sum_{kl=n,\gcd(k,l)=1}\alpha(k)\beta(l)P_{j}(n)$
$\displaystyle=(\alpha\sqcup\beta)(n)P_{j}(n).$

The equation (2.11) follows from the following identity [9]

(2.14) $(\nu_{0}\ast\alpha)(\nu_{0}\ast\alpha)=\nu_{0}\ast(\alpha\Box\beta).$

∎

##  3\. Ramanujan sum’s

The function $\alpha:\,\mathbb{N}\rightarrow\,\mathbb{C}$ is called even
function ($\mod\,d$ ) if $\alpha(n)=\alpha(\gcd(n,d))$ for all $n$, that is if
the value $\alpha(n)$ depends only on the $\gcd(n,d)$. Hence if $\alpha$ is
even($\mod\,d$ ), then it is sufficient to know the values $f(r)$, where
$r\mid d$.
The Ramanujan sum’s

(3.1) $c_{n}(j):=\sum_{\begin{subarray}{c}\gcd(k,n)=1\\\ 1\leq k\leq
n\end{subarray}}\varepsilon_{n}^{jk}.$

If $\alpha$ is even ($\mod\,d$ ), then it has a Ramanujan-Fourier expansion of
the form

$\alpha(n)=\sum_{r\mid
d}(\mathcal{R}_{\alpha})(r)c_{r}(n)\qquad(n\in{\mathbb{N}}),$

where the (Ramanujan-)Fourier coefficients $(\mathcal{R}_{\alpha})(r)$ are
uniquely determined and given by

$(\mathcal{R}_{\alpha})(r)=\sum_{\delta\mid
d}\alpha(\tfrac{d}{\delta})c_{\delta}(\tfrac{d}{r}).$

######  Definition 3.1.

We say that a set $\\{P_{j}(n):n\geq 1,j\geq 0\\}$ is an arithmetic system of
orthogonal idempotent, if
1\. $\\{P_{j}(n)\\}_{j,n=1}^{\infty}$ is an arithmetic system of idempotent,
2\. for every $n$

(3.2) $\sum_{j=0}^{n-1}P_{j}(n)=e.$

Consider the sum

(3.3) $\displaystyle C_{j}(n)=\sum_{\begin{subarray}{c}\gcd(k,n)=1\\\ 1\leq
k\leq n\end{subarray}}\varepsilon_{n}^{-jk}S^{k}(n),\quad j=0,\,1,\,\dots,$

where $S(n)=\sum_{j=1}^{n}\varepsilon_{n}^{j}P_{j}(n).$
For every divisor $r$ of $n$, we denote by

(3.4) $\displaystyle
T_{r,j}(n)=\sum_{\begin{subarray}{c}\gcd(k,n)=\tfrac{n}{r}\\\ 1\leq k\leq
n\end{subarray}}P_{k+j}(n).$

Note that $T_{n,j}(n)$ is denoted simply by $T_{j}(n)$.

######  Proposition 3.1.

The following properties hold

$C_{j}(n)=\mu*\nu_{1}P_{j}(n),\quad\&\quad T_{j}(n)=\nu_{0}*\mu P_{j}(n),$

where $\nu_{k}(n)=n^{k}.$

###### Proof.

We have

$\displaystyle\mu*\nu_{1}P_{j}(n)$ $\displaystyle=\sum_{d\mid
n}\mu(d)\tfrac{n}{d}P_{j}(\tfrac{n}{d})$ $\displaystyle=\sum_{d\mid
n}\mu(d)\sum_{k=1}^{\tfrac{n}{d}}\varepsilon_{n}^{-jkd}S_{n}^{kd}$
$\displaystyle=\sum_{k=1}^{n}\varepsilon_{n}^{-jk}S_{n}^{k}\sum_{\begin{subarray}{c}d\mid
n\\\ d\mid k\end{subarray}}\mu(d)$
$\displaystyle=\sum_{k=1}^{n}\varepsilon_{n}^{-jk}S_{n}^{k}\sum_{\begin{subarray}{c}d\mid\gcd(k,n)\end{subarray}}\mu(d).$

From Theorem 2.1 in [1] we have

(3.5) $\sum_{d\mid\gcd(k,n)}\mu(r)=\left\\{\begin{array}[]{l
l}1\quad\,\text{if}\quad\gcd(k,n)=1,\\\
0\quad\,\text{if}\quad\gcd(k,n)\,>1.\end{array}\right.$

Hence

(3.6) $C_{j}(n)=\mu*\nu_{1}P_{j}(n).$

To prove 2)

$\displaystyle T_{j}(n)$
$\displaystyle=\sum_{\begin{subarray}{c}\gcd(k,n)=1\\\ 1\leq k\leq
n\end{subarray}}P_{j+k}(n)$
$\displaystyle=\sum_{k=1}^{n}\sum_{\begin{subarray}{c}\delta\mid\gcd(k,n)\end{subarray}}\mu(\delta)P_{j+k}(n)$
$\displaystyle=\sum_{\begin{subarray}{c}\delta\mid
n\end{subarray}}\mu(\delta)\sum_{k=1}^{\tfrac{n}{\delta}}P_{j+k\delta}(n)$
$\displaystyle=\sum_{\begin{subarray}{c}\delta\mid
n\end{subarray}}\mu(\delta)P_{j}(\delta)$

∎

######  Corollary 3.2.

$\displaystyle
C_{j}(n)=n\prod_{l=1}^{N}\big{(}P_{j}(p^{\alpha_{l}})-\frac{1}{p}P_{j}(p^{\alpha_{l}-1}),$
$\displaystyle T_{j}(n)=\prod_{\begin{subarray}{c}p\mid n\\\
p\,\,\text{prime}\end{subarray}}(e-P_{j}(p)).$

###### Proof.

The functions $C_{j}(n)$ and $T_{j}$ are multiplicative. So it suffice to
compute $C_{j}(p^{k})$ and $T_{j}(p^{k})$ where $p$ is prime number. Form
proposition 4.1, we have

$\displaystyle C_{j}(p^{k})$
$\displaystyle=\sum_{l=0}^{k}\mu(p^{l})p^{k-l}P_{j}(p^{k-l})$
$\displaystyle=p^{k}P_{j}(p^{k})-p^{k-1}P_{j}(p^{k-l})$
$\displaystyle=p^{k}\big{(}P_{j}(p^{k})-p^{-1}P_{j}(p^{k-1})\big{)},$

and

$T_{j}(p^{k})=\left\\{\begin{array}[]{l l}e-P_{j}(p)\quad\,\text{if}\quad
k=1,\\\ 0\quad\,\text{otherwise}.\end{array}\right.$

Since $C_{j}$ is multiplicative we have

$\displaystyle C_{j}(n)$
$\displaystyle=\prod_{l=1}^{N}C_{j}(p^{\alpha_{l}})=n\prod_{l=1}^{N}\big{(}P_{j}(p^{\alpha_{l}})-\frac{1}{p}P_{j}(p^{\alpha_{l}-1}).$

Similarly,

$T_{j}(n)=\prod_{\begin{subarray}{c}p\mid n\\\
p\,\,\text{prime}\end{subarray}}(e-P_{j}(p))$

∎

######  Theorem 3.3.

For fixed integers $n$ and $j$, $\\{T_{r,j}(n):r\mid n\\}$ is a set of
$\tau(n)$ orthgonal idempotent:

(3.7) $\sum_{r\mid n}T_{r,j}(n)=e\quad\text{and}\quad
T_{r,j}(n)T_{r^{\prime},j}(n)=\delta_{r,r^{\prime}}T_{r,j}(n)$

where $\tau(n)$ is the number of divisor of $n.$

######  Lemma 3.4.

The operators $T_{j}(n)$ and $C_{j}(n)$ are related by

$C_{j}(n)=\sum_{r\mid n}c_{n}(n/r)T_{r,j}(n),\quad
T_{n,j}(n)=\frac{1}{n}\sum_{r\mid n}c_{n}(n/r)\,C_{j}(r).$

###### Proof.

(3.8) $\displaystyle\sum_{r\mid n}c_{n}(n/r)T_{r,j}(n)$
$\displaystyle=\sum_{r\mid
n}c_{n}(n/r)\sum_{\gcd(k,n)=\tfrac{n}{r}}P_{k+j}(n)$ (3.9)
$\displaystyle=\sum_{l=1}^{n}\sum_{r\mid
n}c_{n}(n/r)\sum_{\gcd(k,n)=\tfrac{n}{r}}\varepsilon_{n}^{-(k+j)l}S_{n}^{l}$
(3.10)
$\displaystyle=\frac{1}{n}\sum_{l=1}^{n}\varepsilon_{n}^{-lj}\sum_{r\mid
n}c_{n}(n/r)c_{r}(l)S_{n}^{l}$

From formula Exercise 2.23 in [3], we have

(3.11) $\sum_{r\mid n}c_{n}(n/r)c_{r}(l)=\left\\{\begin{array}[]{l
l}n\quad\,\text{if}\quad\gcd(l,n)=1,\\\
0\quad\text{if}\quad\gcd(l,n)>1.\end{array}\right.$

Hence

(3.12) $\displaystyle\sum_{r\mid
n}c_{n}(n/r)T_{r,j}(r)=\sum_{\gcd(l,n)=1}\varepsilon_{n}^{-lj}S_{n}^{l}=C_{j}(n).$

Similarly,

(3.13) $\displaystyle\sum_{r\mid n}c_{n}(n/r)C_{j}(r)$
$\displaystyle=\sum_{r\mid
n}c_{n}(n/r)\sum_{\gcd(k,n)=\frac{n}{r}}\varepsilon_{n}^{-jk}S_{n}^{k}$ (3.14)
$\displaystyle=\sum_{l=1}^{n}\sum_{r\mid n}c_{n}(n/r)c_{r}(l-j)P_{l}(n)$
(3.15) $\displaystyle=\sum_{l=1}^{n}\sum_{r\mid
n}c_{n}(n/r)c_{r}(l)P_{l+j}(n)$

The result follows from formula (see, Exercise 2.23 [3])

(3.16) $\sum_{r\mid n}c_{n}(\tfrac{n}{r})c_{r}(k)=\left\\{\begin{array}[]{l
l}n\quad\,\text{if}\quad\gcd(k,n)=1,\\\
0\quad\text{if}\quad\gcd(k,n)>1.\end{array}\right.$

Hence

(3.17) $\displaystyle\sum_{r\mid
n}c_{n}(n/r)C_{j}(r)=n\sum_{\gcd(k,n)=1}P_{k+j}(n)=nT_{n,j}(n).$

∎

######  Theorem 3.5.

Let $\alpha:\mathbb{N}\rightarrow\mathbb{C}$ be even function $(mod\,\,n)$,
then for all $j=0,\,1,\,\dots,$, we have

(3.18) $\sum_{r\mid n}\alpha(n/r)C_{j}(r)=\sum_{r\mid
n}\mathcal{R}(\alpha)(r)T_{r,j}(n).$

###### Proof.

$\displaystyle\sum_{r\mid n}\alpha(r)C_{j}(r)$ $\displaystyle=\sum_{r\mid
n}\sum_{\delta\mid r}c_{r}(r/\delta)T_{\delta,j}(r)\,\alpha(n/r)$
$\displaystyle=\sum_{r\mid n}\sum_{\delta\mid
r}c_{r}(r/\delta)\sum_{\begin{subarray}{c}\gcd(k,r)=\tfrac{r}{\delta}\\\ 1\leq
k\leq r\end{subarray}}P_{k+j}(r)\,\alpha(n/r)$ $\displaystyle=\sum_{r\mid
n}\alpha(n/r)\sum_{k=1}^{r}c_{r}(k)P_{k+j}(r)$

The Ramanujan sum $c_{r}$ is periodic function with period equal to $r$, then
from condition $III)$ of the Definition 2.2, we can write

$\displaystyle\sum_{k=1}^{r}c_{r}(k)P_{k+j}(r)$
$\displaystyle=\sum_{k=1}^{r}\sum_{l=0}^{n/r-1}c_{r}(k+lr)P_{k+lr+j}(r)$
$\displaystyle=\sum_{k=1}^{n}c_{r}(k)P_{k+j}(n)$
$\displaystyle=\sum_{\delta\mid
n}c_{r}(n/\delta)\sum_{\begin{subarray}{c}\gcd(k,n)=\tfrac{n}{\delta}\\\ 1\leq
k\leq k\end{subarray}}P_{k+j}(n)$ $\displaystyle=\sum_{\delta\mid
n}c_{r}(n/\delta)T_{\delta,j}(n).$

Hence

$\displaystyle\sum_{r\mid n}\alpha(r)C_{j}(r)$ $\displaystyle=\sum_{\delta\mid
n}\sum_{r\mid n}c_{r}(n/\delta)\alpha(n/r)T_{\delta,j}(n)$
$\displaystyle=\sum_{\delta\mid n}\mathcal{R}(\alpha)(\delta)T_{\delta,j}(n).$

∎

##  4\. Example

We denote by $H_{R}$ the vector space of all analytic functions on the open
ball $B(0,R)$ in complex plane $\mathbb{C}$. $H_{R}$ endowed with the topology
of compact convergence, it is a complete locally convex topological vector
spaces.

For $n=2,\,3,\,\dots,$ we denote by $S(n)$ the diagonal operator acting on
monomials $e_{k}(z)=z^{k},\,(k=0,\,1,\dots\,)$ as follows

$S(n)e_{k}=\varepsilon^{k}_{n}e_{k},\quad\varepsilon_{n}=e^{\tfrac{2i\pi}{n}}.$

Its clearly that $S(n)$ is a continuous map from $H_{R}$ into its self
satisfying $S^{n}(n)=i_{H_{R}}$. The primitive idempotents $P_{1}(n),\,\dots
P_{n}(n)$ related to $S(n)$ are given by

(4.1) $P_{j}(n)=\frac{1}{n}\sum_{l=0}^{n-1}\varepsilon_{n}^{-lj}S^{l}(n).$

These obey at the following relations

(4.2)
$i_{H_{R}}=\sum_{j=1}^{n}P_{j}(n),\,\,\,\,\,P_{i}(n)P_{j}(n)=\delta_{ij}P_{j}(n).$

The set $\\{P_{j}(n)\\}_{j,n=1}^{\infty}$ is an arithmetic system of
orthogonal idempotent on the algebra $L(H_{R})$ of all continuous linear maps
of the space $H_{R}.$ To see this it suffice to prove for all positive
integers $n$ and $m$ such that $n$ divide $m$ the following identity

$P_{j}(n)=\sum_{k=1}^{m/n}P_{j+kn}(m).$

Indeed, from (4.1) and the following formula

(4.3) $\sum_{k=1}^{r}\varepsilon_{r}^{-lk}=\left\\{\begin{array}[]{l
l}r,\,\,\text{if}\,\,r|l\\\ 0,\,\,\,\text{otherwise},\end{array}\right.$

we can write

$\displaystyle\sum_{k=1}^{m/n}P_{j+kr}(m)$
$\displaystyle=\frac{1}{m}\sum_{k=1}^{m/n}\sum_{l=1}^{m}\varepsilon_{m}^{-l(j+kr)}S^{l}(m)$
$\displaystyle=\frac{1}{m}\sum_{l=1}^{d}\varepsilon_{m}^{-lj}S^{l}(m)\sum_{k=1}^{m/n}\varepsilon_{m}^{-lkr}$
$\displaystyle=\frac{1}{n}\sum_{l=1}^{r}\varepsilon_{n}^{-lj}S^{l}(n)$
$\displaystyle=P_{j}(n).$

The operators $C_{0}(n)$ and $T_{0}(n)$ acting on the basis $\\{e_{k}\\}$ as
follows

(4.4) $\displaystyle C_{0}(n)e_{m}=c_{n}(m)e_{m},$ (4.7) $\displaystyle
T_{0}(n)e_{m}=\left\\{\begin{array}[]{l
l}e_{m}\quad\,\text{if}\,\,\gcd(n,m)=1,\\\ 0\quad\,\text{otherwise
}.\end{array}\right.$

Indeed, from Proposition 3.1, we have

$\displaystyle T_{0}(n)e_{m}$ $\displaystyle=\sum_{d\mid
n}\mu(d)P_{0}(d)e_{m}$ $\displaystyle=\sum_{d\mid\gcd(n,m)}\mu(d)\,e_{m}$
$\displaystyle=\left\\{\begin{array}[]{l
l}e_{m}\quad\,\text{if}\,\,\gcd(n,m)=1,\\\ 0\quad\,\text{otherwise
}.\end{array}\right.$

Now, if we compare the trace and the determinant of the matrix
$[T_{0}(n)]_{N}$(resp. $[C_{0}(n)]_{N}$) of the restriction of $C_{0}(n)$
(resp. $T_{0}(n)$) to the subspace generated by $\\{e_{1},\dots e_{N}\\}$, we
get

(4.8) $\prod_{k=1}^{N}c_{n}(k)=\left\\{\begin{array}[]{l
l}\prod_{p|n\,\text{prime}}(1-p)^{[\tfrac{N}{p}]}\quad\text{if}\,\,\,n\,\text{is
squarefree},\\\ 0\quad\,\text{if}\quad\text{otherwise}.\end{array}\right.$
(4.9) $\sum_{\begin{subarray}{c}\gcd(k,n)=1\\\ 1\leq k\leq
n\end{subarray}}[\tfrac{N-k}{n}]=N\omega(n)-\sum_{p|n\,\,\text{prime}}[\tfrac{N}{p}]=\sum_{r\mid
n}\mu(r)[\tfrac{N}{r}],$

and

(4.10)
$\sum_{k=1}^{N}c_{n}(k)=\sum_{l=1}^{k}\big{(}p_{l}^{\alpha_{l}}[\tfrac{N}{p_{l}^{\alpha_{l}}}]-p_{l}^{\alpha_{l}-1}[\tfrac{N}{p_{l}^{\alpha_{l}-1}}]\big{)}=\sum_{d\mid
n}d\mu(n/d)[\tfrac{N}{d}].$

Let $H_{0}$ the subspace of all function $f\in H_{R}$ such that $f(0)=0$ and
let $\alpha:\mathbb{N}\rightarrow\mathbb{C}$. For $f\in H_{0}$, we put

(4.11) $\mathcal{P}(\alpha)f=\sum_{n=1}^{\infty}\alpha(n)P_{0}(n)f.$

######  Proposition 4.1.

Let $\alpha$ be an arithmetic function. Then $\mathcal{P}(\alpha)$ defines a
continuous diagonal map form $H_{0}$ into itself, if and only if
$\limsup_{m\rightarrow\infty}|(\nu_{0}*\alpha)(m)|^{1/m}\leq 1$.
Furthermore,

(4.12)
$\displaystyle\mathcal{P}(\epsilon)=i_{H_{0}},\quad\mathcal{P}(\alpha\Box\beta)=\mathcal{P}(\alpha)\mathcal{P}(\beta).$

where

(4.13) $\epsilon(n)=\left\\{\begin{array}[]{l l}1\quad\text{if}\quad n=1\\\
0\quad\text{otherwise}.\end{array}\right.$

###### Proof.

The operator $\mathcal{P}(\alpha)$ acts on monomials $z^{m}$ as follows

(4.14) $\mathcal{P}(\alpha)z^{m}=\left\\{\begin{array}[]{l
l}\sum_{n=1}^{\infty}\alpha(n)\quad\text{if}\,\,\,m=0,\\\
(\nu_{0}*\alpha)(m)z^{m}\quad\text{otherwise}.\end{array}\right.$

Suppose $\mathcal{P}(\alpha)$ defines a continuous linear map on $H_{0}$. It
is well-known that

$f(z)=\sum_{m=1}^{\infty}a_{m}z^{m}\in
H_{R}\quad\text{iff}\quad\limsup_{m\rightarrow\infty}|a_{m}|^{1/m}\leq R.$

Hence,

$\displaystyle\sum_{m=1}^{\infty}(\nu_{0}*\alpha)(m)a_{m}z^{m}\in H_{R}$
$\displaystyle\quad\Leftrightarrow\quad\limsup_{m\rightarrow\infty}|a_{m}(\nu_{0}*\alpha)(m)|^{1/m}\leq
R$
$\displaystyle\quad\Leftrightarrow\quad\limsup_{m\rightarrow\infty}|(\nu_{0}*\alpha)(m)|^{1/m}\leq
1.$

This show that

$\limsup_{m\rightarrow\infty}|\nu_{0}*\alpha)(m)|^{1/m}\leq 1.$

The converse follows from the Closed Graph Theorem (see, [10] Theorem 2.15, p.
51]). To prove (4.12), we use equation (2.9)

$\displaystyle\mathcal{P}(\alpha\Box\beta)$
$\displaystyle=\sum_{n=1}^{\infty}(\alpha\Box\beta)(n)P_{0}(n)$
$\displaystyle=\sum_{n=1}^{\infty}(\alpha P_{0}\Box\beta P_{0})(n)$
$\displaystyle=\sum_{n=1}^{\infty}\alpha(n)P_{0}(n)\sum_{n=1}^{\infty}\beta(n)P_{0}(n)$
$\displaystyle=\mathcal{P}(\alpha)\mathcal{P}(\beta)$

∎

Many examples of operators acting on the space $H_{0}$ such as the Euler
operator, the integration operator, the shift operator and the backward shift
operator given by

$(\theta
f)(z)=zf^{\prime}(z),\quad(If)(z)=\int_{0}^{z}f(t)\,dt,\quad(Uf)=zf(z),\quad(U^{*}f)(z)=\frac{f(z)-f(0)}{z},$

can be represented by means of the map $\mathcal{P}(\alpha)$ for suitable
arithmetic function $\alpha$. For $\alpha=\varphi$ (where $\varphi$ is the
Euler’s totient function), we have

(4.15)
$\displaystyle\mathcal{P}(\varphi)e_{m}=\nu_{0}*(\mu*\nu_{1})(m)e_{m}=\nu_{1}(m)e_{m}=me_{m}.$

On the other hand, we have

$\limsup_{m\rightarrow\infty}|\nu_{0}*\varphi)(m)|^{1/m}=\limsup_{m\rightarrow\infty}m^{1/m}=1.$

Then, $\mathcal{P}(\varphi)$ is a continuous map on $H_{0}$. From (4.15), the
mapping $\mathcal{P}(\varphi)$ coincides with the Euler operator
$\theta=z\frac{d}{dz}$ on the monomials, hence

(4.16) $\mathcal{P}(\varphi)=\theta.$

Moreover, from Proposition 4.1 and the following identity [3]

$J_{r}=\underbrace{\varphi\,\Box\,\dots\,\Box\,\varphi}_{r\text{-times}},$

we get

$\displaystyle\mathcal{P}(J_{r})=\mathcal{P}(\varphi)\dots\mathcal{P}(\varphi)=\theta^{r}.$

($J_{r}$ called Jordan function)
A similarly arguments show that

$\mathcal{P}(\mu)=e_{1}\otimes e_{1},\quad\mathcal{P}(\mu\ast v_{1})=IU^{*}.$

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