# Lithium Evolution of Giant Stars Observed by LAMOST and _Kepler_

Jinghua Zhang CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, People’s Republic
of China  Jian-Rong Shi<EMAIL_ADDRESS>CAS Key Laboratory of Optical
Astronomy, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100101, People’s Republic of China  School of Astronomy and Space
Science, University of Chinese Academy of Sciences, Beijing 100101, People’s
Republic of China  Hong-Liang Yan CAS Key Laboratory of Optical Astronomy,
National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100101, People’s Republic of China  School of Astronomy and Space Science,
University of Chinese Academy of Sciences, Beijing 100101, People’s Republic
of China<EMAIL_ADDRESS>Yaguang Li Sydney Institute for Astronomy (SIfA),
School of Physics, University of Sydney, NSW 2006, Australia  Stellar
Astrophysics Centre, Department of Physics and Astronomy, Aarhus University,
Ny Munkegade 120, DK-8000 Aarhus C, Denmark  Qi Gao CAS Key Laboratory of
Optical Astronomy, National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100101, People’s Republic of China  School of Astronomy and
Space Science, University of Chinese Academy of Sciences, Beijing 100101,
People’s Republic of China  Chun-Qian Li CAS Key Laboratory of Optical
Astronomy, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100101, People’s Republic of China  School of Astronomy and Space
Science, University of Chinese Academy of Sciences, Beijing 100101, People’s
Republic of China  Xianfei Zhang Department of Astronomy, Beijing Normal
University, Beijing 100875, People’s Republic of China  Shuai Liu  CAS Key
Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100101, People’s Republic of China  School of
Astronomy and Space Science, University of Chinese Academy of Sciences,
Beijing 100101, People’s Republic of China  Shaolan Bi  Department of
Astronomy, Beijing Normal University, Beijing 100875, People’s Republic of
China  Gang Zhao CAS Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101,
People’s Republic of China  School of Astronomy and Space Science, University
of Chinese Academy of Sciences, Beijing 100101, People’s Republic of China
Yan Li School of Astronomy and Space Science, University of Chinese Academy of
Sciences, Beijing 100101, People’s Republic of China  Yunnan Observatories,
Chinese Academy of Sciences, Kunming 650216, People’s Republic of China  Key
Laboratory for Structure and Evolution of Celestial Objects, Chinese Academy
of Sciences, Kunming 650216, People’s Republic of China  Center for
Astronomical Mega-Science, Chinese Academy of Sciences, Beijing, 100012,
People’s Republic of China

###### Abstract

Mapping lithium evolution for evolved stars will provide restrictions and
constraints on the fundamental stellar interior physical processes, which
further shed light on our understanding of the theory of stellar structure and
evolution. Based on a sample of 1,848 giants with known evolutionary phases
and lithium abundances from the LAMOST-_Kepler_ and LAMOST-_K_ 2 fields, we
construct mass-radius diagrams to characterize the evolutionary features of
lithium. The stars at red giant branch (RGB) phase show natural depletion
along with their stellar evolution, particularly, there is no obvious crowd
stars with anomalously high Li abundances near the bump. Most of the low-mass
stars reaching their zero-age sequence of core-helium-burning (ZAHeB) have Li
abundances around $\sim 1.0$ dex, which show an increase of Li abundance by
$\sim 0.6$ dex compared to the stars above the bump of RGB. This suggests the
helium flash can be responsible for moderate Li production. While for super
Li-rich stars, some special mechanisms should be considered during helium
flash. Other scenarios, such as merger, could also be interpretations given
the Li-rich stars can be found at anytime during the steady state phase of
core He-burning. During the core He-burning (HeB) phase, there is no
indication of obvious lithium depletion.

Stellar evolution(1599), Stellar abundances(1577), Lithium stars(927),
Chemically peculiar stars(226), Red giant stars(1372), Red giant clump(1370),
Low-mass star(2050), Asteroseismology(73)

##  1 Introduction

The surface lithium abundances display various patterns for stars of different
types as well as at different evolutionary stages. Lithium is easily destroyed
by nuclear reactions in stars, and it is sensitive to mixing events connecting
the envelope with warm layers where it easily undergoes p-captures. Thus, the
surface Li has experienced depletion over time for stars with convective
envelope (Twarog et al., 2020; Randich & Magrini, 2021). Therefore, the
signatures of Li are good manifestations of subtle physical processes beneath
the surface, and provide key information on inside stellar structure and
evolution.

During the past few decades, the increasing observational results have
confirmed the theoretical predictions on the depletions of Li in giant stars
(e.g. Lind et al., 2009; Deliyannis et al., 2019; Twarog et al., 2020).
However, due to the difficulties of classifying evolutionary stages,
especially separating core-helium-burning (HeB) from red giant branch (RGB)
bump stars by traditional approach, the evolutionary features of Li from RGB
to HeB stage are still in the dark. Moreover, the nature of Li-rich giants
found in a small number of objects is still beyond the standard stellar
evolution theory (e.g. Wallerstein & Sneden, 1982; Yan et al., 2018; Smiljanic
et al., 2018; Gao et al., 2019; Casey et al., 2019; Deepak & Reddy, 2019;
Martell et al., 2021). The physical mechanisms responsible for those Li-rich
giants are under debate (e.g. Charbonnel & Lagarde, 2010; Kumar et al., 2020;
Schwab, 2020; Zhang et al., 2020; Mori et al., 2021; Yan et al., 2021; Singh
et al., 2021), which further complicates Li evolution in evolved stars.

Fortunately, the asteroseismic analysis for a mass of giants in the _Kepler_
(Borucki et al., 2010) and K2 (Howell et al., 2014) fields has uncovered their
evolutionary stages (e.g. Stello et al., 2013; Mosser et al., 2015; Vrard et
al., 2016; Hon et al., 2018; Zinn et al., 2020). Particularly, based on the
asteroseismic scaling relations, the positions of RGB bump as well as the
zero-age sequence of HeB (ZAHeB) stars could be well defined in mass-radius
diagram (Li et al., 2021). Deepak & Lambert (2021) explored evolution of
lithium in giant stars based on $\sim$200 seismic data. Based on the
distribution of $A_{\rm Li}$ as a function of mean large frequency separation
($\Delta\nu$), they found no indication of Li enrichment near the luminosity
bump. Martell et al. (2021) explored the properties of 1262 Li-rich giant
stars from the GALAH (De Silva et al., 2015) and K2 surveys, and found that
the less massive primary red clump stars are 1.5 times as likely to be super
lithium-rich as the more massive secondary red clump stars. Differing from the
result shown in Deepak & Lambert (2021), Martell et al. (2021) showed that
there is a concentration of lithium-rich stars near the luminosity of the red
giant branch bump. Recently, there are thousands of stars in the _Kepler_ and
K2 fields have been observed by the Large Sky Area Multi-Object Fibre
Spectroscopic Telescope (LAMOST) spectroscopic survey (Zhao et al., 2006; Cui
et al., 2012; Zhao et al., 2012), and their surface Li abundances have been
presented recently (Gao et al., 2019, 2021). Combining these data sets, it is
possible for us to investigate the signatures of Li abundances for stars
evolved from RGB to HeB phase. Also, it would shed light on the Li production
in evolved stars, and give more detailed insight into the nature of the
stellar evolution theory.

In this letter, we characterized signatures of Li abundances for giant stars
at different evolutionary stages which were observed by both LAMOST and
_Kepler_ (K2 included). In Section 2, we introduce the sample selection. In
Section 3, we estimate the mass and radius for our sample stars, and
characterize the evolutionary features of Li abundances in different stages.
We summarise our results in Section 4.

##  2 Stellar samples

Gao et al. (2019) searched for Li-rich giants using the LAMOST low-resolution
spectra, and they found 10,535 giant stars with $A_{\rm Li}\geq 1.5$ dex. This
catalog provided the updated largest sample of Li-rich giants. Very recently,
Gao et al. (2021) further determined Li abundances for a sample of 165,479
stars based on the LAMOST medium-resolution spectra. We thus considered the
two catalogs as our initial targets, and cross-matched the combined catalog
with these of Yu et al. (2018) and Zinn et al. (2020). The latter two catalogs
provided the frequency of maximum power ($\nu_{\rm max}$), the mean large
frequency separation ($\Delta\nu$) and the classification of evolutionary
stages for samples of 16,094 giants in the _Kepler_ field and of 4,395 giants
in the K2 fields, respectively. A total of 1,848 giant stars are finally
selected, including 904 RGB stars and 944 HeB stars. Among them, there are 30
stars that have been detected Li abundances both in the low- and medium-
resolution observational modes, and the Li results from medium-resolution
modes are adopted in this work. Because of the way these stars were selected,
we do not try to address the detection fractions of Li-rich stars here.
However, we do note that Li was detected in $\sim$80% of the sample, with no
strong trends in detection fraction as a function of mass, metallicity, or
evolutionary state likely to bias our analysis.

##  3 Li abundances in _M-R_ diagram

It has been proved that the asteroseismic parameters ($\Delta\nu$, $\nu_{\rm
max}$) are tightly related with stellar fundamental parameters, and the
stellar masses and radii can be estimated by the asteroseismic parameters with
an addition of effective temperature ($T_{\rm eff}$), which are also known as
the asteroseismic scaling relations (see, e.g. Stello et al., 2008; Kallinger
et al., 2010, and references therein). These relations have also been
confirmed having remarkable constraining powers with intrinsic scatter of mass
and radius being limited to $\sim 1.7$ and $\sim 0.4$ per cent, respectively
(Li et al., 2021). Thereby, taking advantage of these asteroseismic scaling
relations, we can construct a mass-radius diagram (hereafter, _M-R_ diagram)
instead of traditional Hertzsprung–Russell diagram to investigate the
evolutionary features of Li abundances in evolved stars. We calculate the mass
and radius of our sample following

$\frac{M}{M_{\odot}}\approx\bigg{(}\frac{\nu_{\rm max}}{f_{\nu_{\rm
max}}\nu_{\rm
max,\odot}}\bigg{)}^{3}\bigg{(}\frac{\Delta\nu}{f_{\Delta\nu}\Delta\nu_{\odot}}\bigg{)}^{-4}\bigg{(}\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\bigg{)}^{3/2},$ (1)
$\frac{R}{R_{\odot}}\approx\bigg{(}\frac{\nu_{\rm max}}{f_{\nu_{\rm
max}}\nu_{\rm
max,\odot}}\bigg{)}\bigg{(}\frac{\Delta\nu}{f_{\Delta\nu}\Delta\nu_{\odot}}\bigg{)}^{-2}\bigg{(}\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\bigg{)}^{1/2},$ (2)

where, $\nu_{\rm max,\odot}=3090\,\mu$Hz, $\Delta\nu_{\odot}=135.1\,\mu$Hz,
and $T_{\rm eff,\odot}=5777$ K (Huber et al., 2011). The parameters
$f_{\nu_{\rm max}}$ and $f_{\Delta\nu}$ are correction factors. Following
Sharma et al. (2016), the factor $f_{\Delta\nu}$ is a function of stellar
atmospheric parameters ($T_{\rm eff}$, $\log g$, [Fe/H]) and evolutionary
phase, and can be determined by interpolation in grids of models for $-3<$
[Fe/H] $<0.4$ and $0.8<M/M_{\odot}<4.0$. Since the factor $f_{\nu_{\rm max}}$
is difficult to be determined theoretically (Belkacem et al., 2011), it was
set to be 1.0 in this work. The stellar atmospheric parameters were adopted
from Gao et al. (2019) and Gao et al. (2021). The $\Delta\nu$ and $\nu_{\rm
max}$ for stars in the K2 fields were adopted from the mean values of
different pipelines (Zinn et al., 2020), while for stars in the _Kepler_ field
they were determined via SYD pipeline (Yu et al., 2018). The small differences
in mass and radius determined by different pipelines results would not affect
us mapping lithium evolution.

###  3.1 Li abundances in RGB stars

To better understand the evolutionary features of Li abundances and the origin
of Li enhancement in evolved stars, we separate HeB from RGB for our sample
stars, and characterize Li abundances for each population. Figure 1 shows the
_M-R_ diagrams of the RGB (panel a) and HeB (panel c) stars in our sample with
color-coded by Li abundances, as well as the Li abundance versus radius for
low-mass RGB stars (panel b). The red line in panel a defines the RGB bump,
while the gold spline in panel c indicates the zero-age HeB (ZAHeB) edge (see
Li et al., 2021, for details).

As shown in Figure 1b, for the majority of low-mass
($\emph{M}<2\emph{M}_{\odot}$) RGB stars, the Li abundances decrease gradually
by $\sim 2.0$ dex when their radii expand by a factor of $\sim 5$. This
feature indicates the consequence of Li depletion along with stellar
evolution, which has been predicted by the standard stellar evolution theory
(e.g. Iben, 1967). As shown in Figure 1a, there is no obvious crowd stars with
anomalously high Li abundances near the bump. Considering the definition of
$A_{\rm Li}\geq 1.5$ dex for Li-rich stars, the total number of these stars
among low-mass RGB accounts for 45, while only five of them evolve past the
bump. This result implies that the RGB bump may be not the key to Li
enhancement of RGB stars. The similar result was reported by Deepak & Lambert
(2021), who found no indication of Li enrichment near the luminosity bump. As
suggested by previous studies (e.g. Yan et al., 2021), these Li-rich stars may
possess higher $A_{\rm Li}$ than their counterparts at main-sequence stage,
they could still reserve relatively high $A_{\rm Li}$ after finishing the FDU
process. Therefore, the Li-rich RGB stars might be a natural consequence of Li
depletion along with stellar evolution. Given that the depletion of Li by FDU
process is generally with a factor of $\sim 60$ (Iben, 1967), if we take a
low-mass sample star with the highest $A_{\rm Li}$ of $\sim 2.4$ dex below the
bump, the $A_{\rm Li}$ would be up to $\sim 4.2$ dex at the main-sequence
stage. Such stars have been reported by previous studies (e.g. Li et al.,
2018; Gao et al., 2021). The external scenario, on the other hand, such as
engulfment of a substellar component (e.g. Alexander, 1967; Siess & Livio,
1999), can not be excluded, which is in accordance with the observed Li
abundances for our Li-rich RGB stars. A simulation on engulfment of a Jovian
planet could result in a typical upper limit of $A_{\rm Li}\sim 2.2$ dex
(Aguilera-Gómez et al., 2016), which is sufficient for Li production in most
of Li-rich star below the bump in our sample.

Figure 1: Radius versus mass for RGB (upper) and HeB (lower) stars with color-
coded by Li abundance, as well as radius versus Li abundance for low-mass RGB
stars (middle). The red line in upper panel denotes the RGB bump. the gold
spine in lower panel denotes the edge of ZAHeB stars and the red triangles
denote the stars locating within reach of ZAHeB edge.

As the mass increases, we found that Li abundances are comparably higher.
Although the intermediate-mass RGB stars show Li depletion as that of the low-
mass RGB objects, the depletion is moderate. In fact, the thinner convective
envelope in intermediate-mass RGB stars usually accompany with relatively
lower temperature in the deep convection zone and shorter time available for
deep mixing that the destruction of Li is dependent on. As a result, high Li
abundance can be easily found among the intermediate-mass RGB stars.

###  3.2 Li abundances in HeB stars

After leaving the tip of the RGB (tip-RGB) phase, stars evolve down in
luminosity and begin core helium burning (HeB), which are also known as the
red clump phase. The low-luminosity edge defines the beginning of the red
clump, which is also called the zero-age HeB (ZAHeB) phase. The edge is
remarkable in H-R diagram as well as in _M-R_ diagram (Girardi, 2016; Li et
al., 2021), which makes it feasible to look into the evolutionary features of
Li abundance among HeB stars. In Figure 1c, this edge is visible as the lower
boundary.

Figure 2: Li abundances versus mass for stars locating above the RGB bump and
HeB stars within reach of ZAHeB edge. The horizontal line denotes the lower
limit of Li enhancement.

As shown in Figure 1c, the HeB stars possess high frequency of Li-rich stars
whose Li abundances are higher than the primordial abundance (Cyburt et al.,
2016). Moreover, most of these stars are dominated by low-mass stars.
Interestingly, the radii of these stars are distributed over a wide range and
do not show any significant feature, which indicates that the super Li-rich
stars can be found at anytime during the steady state phase of core-helium
burning.

According to some recent studies, the helium-flash-induced mixing event occurs
between tip-RGB and HeB phases, which is proposed to be responsible for Li
enhancement among low-mass HeB stars (see, Schwab, 2020; Kumar et al., 2020,
and references therein). Since the low-mass ZAHeB stars could be well defined
for our sample, one would expect to see a significant increase of $A_{\rm Li}$
for these stars in comparison with that for stars at or near tip-RGB based on
the scenario of helium-flash-induced mixing. Considering the asteroseismic
measurements and inferences become less certain at large radii (Pinsonneault
et al., 2018), we further select a subset of low-mass RGB stars with their
radii greater than $15\emph{R}_{\odot}$ (Figure 1a), and we consider them as
above-bump RGB stars. Then, we select the other subset of low-mass HeB stars
locating within reach of ZAHeB edge (Figure 1c). We present the Li abundance
versus mass for the two subsets in Figure 2. It can be seen that most ZAHeB
stars have higher Li abundances than that of stars above the bump. The median
$A_{\rm Li}$ increases by $\sim 0.6$ dex from post-bump to ZAHeB phase. This
increment is very close to the result given by the model prediction after
including the helium-flash-induced mixing (see their Fig 3 in Schwab, 2020),
which means that the scenario of helium-flash-induced mixing indeed
contributes Li production for most low-mass HeB stars. The increment shown in
our sample provides an observational evidence for the Li production due to
helium-flash-induced mixing. However, the $A_{\rm Li}$ of the majority of
ZAHeB stars are close to 1.0 dex, which hardly meet the traditional standard
of Li-rich giants. For these Li-rich low-mass HeB stars, we suggest that some
special physical mechanisms should be needed during the helium flash so that
abnormal enhancement of Li could be triggered. Another possible interpretation
for these Li-rich low-mass HeB stars is the merger of a helium white dwarf
(HeWD) with an RGB star (Zhang et al., 2020). The HeWD–RGB merger could result
a Li-rich HeB star at anytime during the HeB stars’ steady state phase of core
He-burning (Zhang et al., 2020), which is compatible with the observational
distribution of our sample. Yan et al. (2021) showed that the HeWD–RGB merger
scenario generally agrees with the observed Li abundances in their sample
stars. Our results do not support the proposal of Singh et al. (2021), who
used variations of asymptotic gravity-mode period spacing ($\Delta\Pi$) as
tracer of duration time for core helium-burning and found that the super Li-
rich giants are almost exclusively young HeB stars. Although in this work we
could not define the upper edge of HeB in the _M-R_ diagram, the Li-rich low-
mass HeB stars with large radii are supposed to be evolved HeB.

Figure 3: Same as Figure 1c, but showing HeB stars with $A_{\rm Li}<1.5$.

The features of Li-rich HeB stars indeed provide clues to understand stellar
internal physical processes, however, they are just a small fraction of
evolved stars. To give a more explicit picture for Li evolution, we here
focused on those HeB stars with $A_{\rm Li}$ less than 1.5 dex, which we
referred as Li-normal HeB stars. We plot the _M-R_ diagram only for the Li-
normal HeB stars in Figure 3. On one hand, it can be seen that the Li
abundances strongly relate to the stellar masses, with higher mass stars
possessing higher Li abundances. As a whole, the intermediate-mass stars have
a median $A_{\rm Li}$ of 1.3 dex, which is 0.3 dex higher than that of the
low-mass stars. This relation is also attributed to the thinner convective
envelope in higher mass stars, which weakens the depletion of Li. On the other
hand, there is no obvious sign of decreasing $A_{\rm Li}$ along with radii,
this feature can be justified for that stars at red clump phase would not be
affected by further mixing, because the retreat of convective envelope make it
remaining rather far from the H-burning shell at this evolutionary stage
(Palmerini et al., 2011).

##  4 conclusions

In this letter, we investigate the evolutionary features of Li for a sample of
stars evolved from RGB to HeB stage in the LAMOST-_Kepler_ /K2 project. The
RGB stars show a Li depletion feature along with stellar evolution. The Li-
rich RGB stars mainly locate below the bump, and there is no obvious crowd
stars with anomalously high Li abundances near the bump, which indicates that
Li-rich RGB stars might be a natural consequence of Li depletion along with
stellar evolution. We found an immediate increase of Li abundance by $\sim
0.6$ dex for stars between post-bump of RGB and ZAHeB phase, which suggests
that the helium flash can be responsible for moderate production of Li. This
increment provides an observational evidence for the Li production due to
helium-flash-induced mixing events. Furthermore, we found that Li-rich HeB
stars locate anywhere during the HeB stars’ steady state phase of core He-
burning. We suggest that either some special physical mechanisms should be
required in helium flash for vigorous Li production or they are formed via
other scenarios that could result in random distributions in HeB phase. In
addition, during the HeB stage, we found the depletion feature of Li is not
obvious for stars with $A_{\rm Li}$ less than 1.5 dex, which would be a
visible manifestation of retreat of convective envelope as well as the
inefficient deep-mixing process inside HeB stars.

Acknowledgments We thank the referee for the thorough reviews which have
helped us to improve the manuscript. This work is supported by National Key
R&D Program of China No.2019YFA0405502, the National Natural Science
Foundation of China under grant Nos. 12090040, 12090044, 12022304, U2031203,
11833006, 11973052, 11973042, 11988101, 11890694 and U1931102. H.-L.Y.
acknowledges support from the Youth Innovation Promotion Association of the
CAS (id. 2019060), and NAOC Nebula Talents Program. We acknowledge the entire
_Kepler_ team and everyone involved in the _Kepler_ mission. Funding for the
_Kepler_ Mission is provided by NASA’s Science Mission Directorate.
Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic
Telescope, LAMOST) is a National Major Scientific Project built by the Chinese
Academy of Sciences. Funding for the project has been provided by the National
Development and Reform Commission. LAMOST is operated and managed by the
National Astronomical Observatories, Chinese Academy of Sciences.

## References

  * Aguilera-Gómez et al. (2016) Aguilera-Gómez, C., Chanamé, J., Pinsonneault, M. H., & Carlberg, J. K. 2016, ApJ, 833, L24
  * Alexander (1967) Alexander, J. B. 1967, The Observatory, 87, 238
  * Belkacem et al. (2011) Belkacem, K., Goupil, M. J., Dupret, M. A., et al. 2011, A&A, 530, A142
  * Borucki et al. (2010) Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977
  * Casey et al. (2019) Casey, A. R., Ho, A. Y. Q., Ness, M., et al. 2019, ApJ, 880, 125
  * Charbonnel & Lagarde (2010) Charbonnel, C., & Lagarde, N. 2010, A&A, 522, A10
  * Cui et al. (2012) Cui, X.-Q., Zhao, Y.-H., Chu, Y.-Q., et al. 2012, Research in Astronomy and Astrophysics, 12, 1197
  * Cyburt et al. (2016) Cyburt, R. H., Fields, B. D., Olive, K. A., & Yeh, T.-H. 2016, Reviews of Modern Physics, 88, 015004
  * De Silva et al. (2015) De Silva, G. M., Freeman, K. C., Bland-Hawthorn, J., et al. 2015, MNRAS, 449, 2604
  * Deepak & Lambert (2021) Deepak, & Lambert, D. L. 2021, MNRAS, 505, 642
  * Deepak & Reddy (2019) Deepak, & Reddy, B. E. 2019, MNRAS, 484, 2000
  * Deliyannis et al. (2019) Deliyannis, C. P., Anthony-Twarog, B. J., Lee-Brown, D. B., & Twarog, B. A. 2019, AJ, 158, 163
  * Gao et al. (2019) Gao, Q., Shi, J.-R., Yan, H.-L., et al. 2019, ApJS, 245, 33
  * Gao et al. (2021) —. 2021, ApJ, 914, 116
  * Girardi (2016) Girardi, L. 2016, ARA&A, 54, 95
  * Hon et al. (2018) Hon, M., Stello, D., & Yu, J. 2018, MNRAS, 476, 3233
  * Howell et al. (2014) Howell, S. B., Sobeck, C., Haas, M., et al. 2014, PASP, 126, 398
  * Huber et al. (2011) Huber, D., Bedding, T. R., Stello, D., et al. 2011, ApJ, 743, 143
  * Iben (1967) Iben, Icko, J. 1967, ApJ, 147, 624
  * Kallinger et al. (2010) Kallinger, T., Weiss, W. W., Barban, C., et al. 2010, A&A, 509, A77
  * Kumar et al. (2020) Kumar, Y. B., Reddy, B. E., Campbell, S. W., et al. 2020, Nature Astronomy, 4, 1059
  * Li et al. (2018) Li, H., Aoki, W., Matsuno, T., et al. 2018, ApJ, 852, L31
  * Li et al. (2021) Li, Y., Bedding, T. R., Stello, D., et al. 2021, MNRAS, 501, 3162
  * Lind et al. (2009) Lind, K., Primas, F., Charbonnel, C., Grundahl, F., & Asplund, M. 2009, A&A, 503, 545
  * Martell et al. (2021) Martell, S. L., Simpson, J. D., Balasubramaniam, A. G., et al. 2021, MNRAS, 505, 5340
  * Mori et al. (2021) Mori, K., Kusakabe, M., Balantekin, A. B., Kajino, T., & Famiano, M. A. 2021, MNRAS, 503, 2746
  * Mosser et al. (2015) Mosser, B., Vrard, M., Belkacem, K., Deheuvels, S., & Goupil, M. J. 2015, A&A, 584, A50
  * Palmerini et al. (2011) Palmerini, S., Cristallo, S., Busso, M., et al. 2011, ApJ, 741, 26
  * Pinsonneault et al. (2018) Pinsonneault, M., Elsworth, Y., Silva Aguirre, V., et al. 2018, in American Astronomical Society Meeting Abstracts, Vol. 231, American Astronomical Society Meeting Abstracts #231, 450.13
  * Randich & Magrini (2021) Randich, S., & Magrini, L. 2021, Frontiers in Astronomy and Space Sciences, 8, 6
  * Schwab (2020) Schwab, J. 2020, ApJ, 901, L18
  * Sharma et al. (2016) Sharma, S., Stello, D., Bland-Hawthorn, J., Huber, D., & Bedding, T. R. 2016, ApJ, 822, 15
  * Siess & Livio (1999) Siess, L., & Livio, M. 1999, MNRAS, 308, 1133
  * Singh et al. (2021) Singh, R., Reddy, B. E., Campbell, S. W., Kumar, Y. B., & Vrard, M. 2021, ApJ, 913, L4
  * Smiljanic et al. (2018) Smiljanic, R., Franciosini, E., Bragaglia, A., et al. 2018, A&A, 617, A4
  * Stello et al. (2008) Stello, D., Bruntt, H., Preston, H., & Buzasi, D. 2008, ApJ, 674, L53
  * Stello et al. (2013) Stello, D., Huber, D., Bedding, T. R., et al. 2013, ApJ, 765, L41
  * Twarog et al. (2020) Twarog, B. A., Anthony-Twarog, B. J., Deliyannis, C. P., & Steinhauer, A. 2020, Mem. Soc. Astron. Italiana, 91, 74
  * Vrard et al. (2016) Vrard, M., Mosser, B., & Samadi, R. 2016, A&A, 588, A87
  * Wallerstein & Sneden (1982) Wallerstein, G., & Sneden, C. 1982, ApJ, 255, 577
  * Yan et al. (2018) Yan, H.-L., Shi, J.-R., Zhou, Y.-T., et al. 2018, Nature Astronomy, 2, 790
  * Yan et al. (2021) Yan, H.-L., Zhou, Y.-T., Zhang, X., et al. 2021, Nature Astronomy, 5, 86
  * Yu et al. (2018) Yu, J., Huber, D., Bedding, T. R., et al. 2018, ApJS, 236, 42
  * Zhang et al. (2020) Zhang, X., Jeffery, C. S., Li, Y., & Bi, S. 2020, ApJ, 889, 33
  * Zhao et al. (2006) Zhao, G., Chen, Y.-Q., Shi, J.-R., et al. 2006, Chinese J. Astron. Astrophys., 6, 265
  * Zhao et al. (2012) Zhao, G., Zhao, Y.-H., Chu, Y.-Q., Jing, Y.-P., & Deng, L.-C. 2012, Research in Astronomy and Astrophysics, 12, 723
  * Zinn et al. (2020) Zinn, J. C., Stello, D., Elsworth, Y., et al. 2020, ApJS, 251, 23


# Multiple-correction and Faster Approximation

Xiaodong Cao, Hongmin Xu and Xu You

###### Abstract

In this paper, we formulate a new _multiple-correction method_. The goal is to
accelerate the rate of convergence. In particular, we construct some sequences
to approximate the Euler-Mascheroni and Landau constants, which are faster
than the classical approximations in literature.

0002010 Mathematics Subject Classification. 11Y60, 41A25, 34E05, 26D15.000Key
words and phrases. Euler-Mascheroni constant, Landau constants, rate of
convergence, multiple-correction, Taylor’s formula, continued fraction.000This
work is supported by the National Natural Science Foundation of China (Grant
No.11171344) and the Natural Science Foundation of Beijing (Grant No.1112010).

##  1 Introduction

Euler constant was first introduced by Leonhard Euler (1707-1783) in 1734 as
the limit of the sequence

(1.1) $\displaystyle\gamma(n):=\sum_{m=1}^{n}\frac{1}{m}-\ln n.$

It is also known as the Euler-Mascheroni constant. There are many famous
unsolved problems about the nature of this constant. For example, it is a
long-standing open problem if it is a rational number. See e.g. the survey
papers or books of Brent and Zimmermann [3], Dence and Dence [15], Havil [22]
and Lagarias [23]. A good part of its mystery comes from the fact that the
known algorithms converging to $\gamma$ are not very fast, at least, when they
are compared to similar algorithms for $\pi$ and $e$.

The sequence $\left(\gamma(n)\right)_{n\in\mathbb{N}}$ converges very slowly
toward $\gamma$, like $(2n)^{-1}$. To evaluate it more accurately, we need to
accelerate the convergence. This can be done using the Euler-Maclaurin
summation formula, Stieltjes approach , exponential integral methods, Bessel
function method, etc. See e.g. Gourdon and Sebah [19].

Up to now, many authors are preoccupied to improve its rate of convergence.
See e.g. Chen and Mortici [11], DeTemple [16], Gavrea and Ivan [18], Lu [25,
26], Mortici [27], Mortici and Chen [31] and references therein. We list some
main results as follows: as $n\rightarrow\infty$,

(1.2)
$\displaystyle\sum_{m=1}^{n}\frac{1}{m}-\ln\left(n+\frac{1}{2}\right)=\gamma+O(n^{-2}),\quad\mbox{(DeTemple~{}\cite[cite]{[\@@bibref{}{De}{}{}]},
1993)},$ (1.3)
$\displaystyle\sum_{m=1}^{n}\frac{1}{m}-\ln\frac{n^{3}+\frac{3}{2}n^{2}+\frac{227}{240}+\frac{107}{480}}{n^{2}+n+\frac{97}{240}}=\gamma+O(n^{-6}),\quad\mbox{(Mortici~{}\cite[cite]{[\@@bibref{}{Mor1}{}{}]},
2010)},$ (1.4)
$\displaystyle\sum_{m=1}^{n}\frac{1}{m}-\ln\rho(n)=\gamma+O(n^{-5}),\quad\mbox{(Chen
and Mortici~{}\cite[cite]{[\@@bibref{}{CM}{}{}]}, 2012)},$

where
$\rho(n)=1+\frac{1}{2n}+\frac{1}{24n^{2}}-\frac{1}{48n^{3}}+\frac{23}{5760n^{4}}$.
Recently, Mortici and Chen [31] provided a very interesting sequence

$\displaystyle\nu(n)=$
$\displaystyle\sum_{m=1}^{n}\frac{1}{m}-\frac{1}{2}\ln\left(n^{2}+n+\frac{1}{3}\right)$
$\displaystyle-\left(\frac{-\frac{1}{180}}{\left(n^{2}+n+\frac{1}{3}\right)^{2}}+\frac{\frac{8}{2835}}{\left(n^{2}+n+\frac{1}{3}\right)^{3}}+\frac{\frac{5}{1512}}{\left(n^{2}+n+\frac{1}{3}\right)^{4}}+\frac{\frac{592}{93555}}{\left(n^{2}+n+\frac{1}{3}\right)^{5}}\right),$

and proved

(1.5)
$\displaystyle\lim_{n\rightarrow\infty}n^{12}\left(\nu(n)-\gamma\right)=-\frac{796801}{43783740}.$

Hence, the rate of convergence of the sequence
$\left(\nu(n)\right)_{n\in\mathbb{N}}$ is $n^{-12}$.

Let $R_{1}(n)=\frac{a_{1}}{n}$ and for $k\geq 2$

(1.6) $\displaystyle
R_{k}(n):=\frac{a_{1}}{n+\frac{a_{2}n}{n+\frac{a_{3}n}{n+\frac{a_{4}n}{\frac{\ddots}{n+a_{k}}}}}},$

where
$(a_{1},a_{2},a_{4},a_{6},a_{8},a_{10},a_{12})=\left(\frac{1}{2},\frac{1}{6},\frac{3}{5},\frac{79}{126},\frac{7230}{6241},\frac{4146631}{3833346},\frac{306232774533}{179081182865}\right)$,
$a_{2k+1}=-a_{2k}$ for $1\leq k\leq 6$, and

(1.7) $\displaystyle r_{k}(n):=\sum_{m=1}^{n}\frac{1}{m}-\ln n-R_{k}(n).$

Lu [25] introduced a continued fraction method to investigate this problem,
and showed

(1.8)
$\displaystyle\frac{1}{120(n+1)^{4}}<r_{3}(n)-\gamma<\frac{1}{120(n-1)^{4}}.$

In fact, Lu [25] determined the constants $a_{1}$ to $a_{4}$. Xu and You [39]
continued Lu’s work to find $a_{5},\cdots,a_{13}$ with the help of
_Mathematica_ software, and obtained

(1.9)
$\displaystyle\lim_{n\rightarrow\infty}n^{k+1}\left(r_{k}(n)-\gamma\right)=C_{k}^{\prime},$

where
$(C_{1}^{\prime},\cdots,C_{13}^{\prime})=\left(-\frac{1}{12},-\frac{1}{72},\frac{1}{120},\frac{1}{200},-\frac{79}{25200},-\frac{6241}{3175200},\frac{241}{105840},\frac{58081}{22018248},-\frac{262445}{91974960},\right.$

$\left.-\frac{2755095121}{892586949408},\frac{20169451}{3821257440},\frac{406806753641401}{45071152103463200},-\frac{71521421431}{5152068292800}\right)$.
Moreover, they improved (1.8) to

(1.10) $\displaystyle C_{10}^{\prime}\frac{1}{(n+1)^{11}}<\gamma-
r_{10}(n)<C_{10}^{\prime}\frac{1}{n^{11}},$ (1.11) $\displaystyle
C_{11}^{\prime}\frac{1}{(n+1)^{12}}<r_{11}(n)-\gamma<C_{11}^{\prime}\frac{1}{n^{12}}.$

However, it seems difficult for us to find more constants $a_{k}$. One of the
main reasons is due to the recursive algorithm. The other reason is that the
parameter $a_{j}$ appears many times in the coefficients of polynomials
$P_{l}(x)$ and $Q_{m}(x)$, and this causes that expanding function
$\frac{P_{l}(x)}{Q_{m}(x)}$ as power series in the terms of $1/x$ needs a huge
of computations. To overcome this difficulty, the purpose of this paper is to
formulate a new _multiple-correction method_ to accelerate the convergence. In
addition, we will use this method to study the sharp bounds for the constants
of Landau.

The Landau’s constants are defined for all integers $n\geq 0$ by

(1.12) $\displaystyle G(n)=\sum_{k=0}^{n}\frac{1}{16^{k}}\binom{2k}{k}^{2}.$

The constants $G(n)$ are important in complex analysis. In 1913, Landau [24]
proved that if $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ is an analytic function in
the unit disc which satisfies $|f(z)|<1$ for $|z|<1$, then
$\left|\sum_{k=0}^{n}a_{k}\right|\leq G(n)$, and that this bound is optimal.
Landau [24] showed that

(1.13) $\displaystyle G(n)\sim\frac{1}{\pi}\ln n,(n\rightarrow\infty).$

In 1930, Watson [37] obtained the following more precise asymptotic formula

(1.14) $\displaystyle
G(n)\sim\frac{1}{\pi}\ln(n+1)+c_{0}-\frac{1}{4\pi(n+1)}+O\left(\frac{1}{n^{2}}\right),(n\rightarrow\infty),$

where

(1.15) $\displaystyle c_{0}=\frac{1}{\pi}(\gamma+4\ln
2)=1.0662758532089143543\ldots.$

The work of Watson opened up a novel insight into the asymptotic behavior of
the Landau sequences $(G(n))_{n\geq 0}$. Inspired by formula (1.14), many
authors investigated the upper and lower bounds of $G(n)$. Some of the main
results are listed as follows:

(1.16) $\displaystyle\frac{1}{\pi}\ln(n+1)+1\leq
G(n)<\frac{1}{\pi}\ln(n+1)+c_{0}\quad(n\geq
0),\quad\mbox{(Brutman~{}\cite[cite]{[\@@bibref{}{Br}{}{}]},1982)},$ (1.17)
$\displaystyle\frac{1}{\pi}\ln\left(n+\frac{3}{4}\right)+c_{0}<G(n)\leq\frac{1}{\pi}\ln\left(n+\frac{3}{4}\right)+1.0976\quad(n\geq
0),\quad\mbox{(Falaleev~{}\cite[cite]{[\@@bibref{}{Fal}{}{}]}, 1991)},$ (1.18)
$\displaystyle\frac{1}{\pi}\ln\left(n+\frac{3}{4}\right)+c_{0}<G(n)<\frac{1}{\pi}\ln\left(n+\frac{3}{4}+\frac{11}{192n}\right)+c_{0}\quad(n\geq
1),\quad\mbox{(Mortici~{}\cite[cite]{[\@@bibref{}{Mor4}{}{}]}. 2011)}$

Very recently, Chen [9] presented the following better approximation to
$G(n)$: as $n\rightarrow\infty$,

(1.19) $\displaystyle G(n)=$ $\displaystyle
c_{0}+\frac{1}{\pi}\ln\left(n+\frac{3}{4}+\frac{11}{192(n+\frac{3}{4})}-\frac{2009}{184320(n+\frac{3}{4})^{3}}+\frac{2599153}{371589(n+\frac{3}{4})^{5}}\right)$
$\displaystyle+O\left(\frac{1}{(n+\frac{3}{4})^{8}}\right),$

and the better upper bound:

(1.20) $\displaystyle
G(n)<c_{0}+\frac{1}{\pi}\ln\left(n+\frac{3}{4}+\frac{11}{192(n+\frac{3}{4})}\right),(n\geq
0).$

Another direction for developing the approximation to $G(n)$ was initiated by
Cvijović and Klinowski [12], who established estimates for $G(n)$ in terms of
the Psi(or Digamma) function $\psi(z):=\frac{\Gamma^{\prime}(z)}{\Gamma(z)}$:

(1.21)
$\displaystyle\frac{1}{\pi}\psi\left(n+\frac{5}{4}\right)+c_{0}<G(n)<\frac{1}{\pi}\psi\left(n+\frac{5}{4}\right)+1.0725,\quad(n\geq
0),$ (1.22)
$\displaystyle\frac{1}{\pi}\psi\left(n+\frac{3}{2}\right)+0.9883<G(n)<\frac{1}{\pi}\psi\left(n+\frac{3}{2}\right)+c_{0},\quad(n\geq
0).$

Since then, many authors have made significant contributions to sharper
inequalities and asymptotic expansions for $G(n)$. See e.g. Alzer [1], Chen
[8], Cvijović and Srivastava [14], Granath [21], Mortici [30], Nemes [32, 33],
Popa [34], Popa and Secelean [35], Zhao [41], Gavrea and M. Ivan [18], Chen
and Choi [5, 7], etc. To the best knowledge of authors, the latest upper bound
is due to Chen [9], who proved

(1.23) $\displaystyle
G(n)<c_{0}+\frac{1}{\pi}\psi\left(n+\frac{5}{4}+\frac{1}{64(n+\frac{3}{4})}\right),(n\geq
0).$

Here, the authors would like to thank Alzer, Chen, Choi, DeTemple, Granath,
Lu, Mortici, etc., it is their important works that makes the present work
becomes possible.

Notation. Throughout the paper, the notation $P_{k}(x)$(or $Q_{k}(x)$) as
usual denotes a polynomial of degree $k$ in terms of $x$. The notation
$\Psi(k;x)$ means a polynomial of degree $k$ in terms of $x$ with all of its
non-zero coefficients being positive, which may be different at each
occurrence. Notation $\Phi(k;x)$ denotes a polynomial of degree $k$ in terms
of $x$ with the leading coefficient being equal to one, which may be different
at different subsection.

##  2 Some Lemmas

The following lemma gives a method for measuring the rate of convergence, for
the proof of which, see Mortici [27, 28].

######  Lemma 1.

If the sequence $(x_{n})_{n\in\mathbb{N}}$ is convergent to zero and there
exists the limit

(2.1)
$\displaystyle\lim_{n\rightarrow+\infty}n^{s}(x_{n}-x_{n+1})=l\in[-\infty,+\infty]$

with $s>1$, then

(2.2) $\displaystyle\lim_{n\rightarrow+\infty}n^{s-1}x_{n}=\frac{l}{s-1}.$

In the study of Landau constants, we need to apply a so-called Brouncker’s
continued fraction formula.

######  Lemma 2.

For all integer $n\geq 0$, we have

(2.3) $\displaystyle
q(n):=\left(\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)}\right)^{2}=\frac{4}{1+4n+\frac{1^{2}}{2+8n+\frac{3^{2}}{2+8n+\frac{5^{2}}{2+8n+\ddots}}}}.$

In 1654 Lord William Brouncker found this remarkable fraction formula, when
Brouncker and Wallis collaborated on the problem of squaring the circle.
Formula (2.3) was not published by Brouncker himself, but first appeared in
[36]. For a general $n$, Formula (2.3) follows from Entry 25 in Chapter 12 in
Ramanujan s notebook [2], which gives a more general continued fraction
formula for quotients of gamma functions, and which have several proofs
published by different authors.

Writing continued fractions in this way of (2.3) takes a lot of space. So
instead we use the following shorthand notation

(2.4) $\displaystyle
q(n)=\frac{4}{1+4n+}\frac{1^{2}}{2+8n+}\frac{3^{2}}{2+8n+}\frac{5^{2}}{2+8n+}\cdots,$

and its $k$-th approximation $q_{k}(n)$ is defined by

(2.5) $\displaystyle q_{1}(n)$ $\displaystyle=\frac{4}{1+4n},$ (2.6)
$\displaystyle q_{k}(n)$
$\displaystyle=\frac{4}{1+4n+}\frac{1^{2}}{2+8n+}\frac{3^{2}}{2+8n+}\cdots\frac{(2k-3)^{2}}{2+8n},\quad(k\geq
2).$

In the proof of our inequalities for the constants of Euler-Mascheroni and
Landau, we also use the following simple inequality.

######  Lemma 3.

Let $f^{\prime\prime}(x)$ be a continuous function. If
$f^{\prime\prime}(x)>0$, then

(2.7) $\displaystyle\int_{a}^{a+1}f(x)dx>f(a+1/2).$

###### Proof.

By letting $x_{0}=a+1/2$ and Taylor’s formula, we have

$\displaystyle\int_{a}^{a+1}f(x)dx=$
$\displaystyle\int_{a}^{a+1}\left(f(x_{0})+f^{\prime}(x_{0})(x-x_{0})+\frac{1}{2}f^{\prime\prime}(\theta_{x})(x-x_{0})^{2}\right)dx$
$\displaystyle>$
$\displaystyle\int_{a}^{a+1}\left(f(x_{0})+f^{\prime}(x_{0})(x-x_{0})\right)dx=f(a+1/2).$

This completes the proof of Lemma 3. Also see Lemma 2 in Xu and You [39].∎

##  3 Two Examples for Euler-Mascheroni Constant

In this section, to illustrate quickly and clearly the main ideas of this
paper, we consider the simplest case of Euler-Mascheroni constant by using the
correction-process again.

Example 1. We choose an initial-correction function $\theta_{0}(n)$ given by

(3.1) $\displaystyle\theta_{0}(n)=\frac{13+30n}{6(1+6n+10n^{2})},$

and define

(3.2) $\displaystyle\nu_{0}(n)=\sum_{m=1}^{n}\frac{1}{m}-\ln n-\theta_{0}(n).$

Applying Lemma 1, one can check(See Theorem 1 in [39] , or (1.9))

(3.3)
$\displaystyle\lim_{n\rightarrow\infty}n^{6}\left(\nu_{0}(n)-\nu_{0}(n+1)\right)=\frac{1}{40},$
(3.4)
$\displaystyle\lim_{n\rightarrow\infty}n^{5}\left(\nu_{0}(n)-\gamma\right)=\frac{1}{200}.$

By using the similar idea of Kummer’s acceleration method and inserting the
correction function $-\frac{1}{200}\frac{1}{n^{5}}$ in (3.2) again, one can
use Lemma 1 again to show

(3.5)
$\displaystyle\lim_{n\rightarrow\infty}n^{6}\left(\nu_{0}(n)-\frac{1}{200}\frac{1}{n^{5}}-\gamma\right)=-\frac{773}{126000}.$

Furthermore, we try to obtain an algorithm with a faster convergent rate by
using $\Phi(5;n)$ instead of $n^{5}$. To do that, let

(3.6) $\displaystyle\theta(n)=$
$\displaystyle\frac{\frac{1}{200}}{\Phi(5;n)}=\frac{\frac{1}{200}}{n^{5}+a_{4}n^{4}+a_{3}n^{3}+a_{2}n^{2}+a_{1}n+a_{0}},$
(3.7) $\displaystyle\nu(n)=$ $\displaystyle\sum_{m=1}^{n}\frac{1}{m}-\ln
n-\theta_{0}(n)-\theta(n).$

First, we use the method of undetermined coefficients to find $a_{j}(0\leq
j\leq 4)$. By using the _Mathematica_ software, we expand the difference
$\nu(n)-\nu(n+1)$ into a power series in terms of $n^{-1}$:

(3.8) $\displaystyle\nu(n)-\nu(n+1)$ $\displaystyle=$
$\displaystyle\frac{-773+630a_{4}}{21000}\frac{1}{n^{7}}+\frac{4033+1050a_{3}-3150a_{4}-1050a_{4}^{2}}{30000}\frac{1}{n^{8}}$
$\displaystyle+\frac{-37657+4500a_{2}-15750a_{3}+31500a_{4}-9000a_{3}a_{4}+15750a_{4}^{2}+4500a_{4}^{3}}{112500}\frac{1}{n^{9}}$
$\displaystyle+\frac{\varphi_{10}}{500000}\frac{1}{n^{10}}+\frac{\varphi_{11}}{4125000}\frac{1}{n^{11}}+\frac{\varphi_{12}}{75000000}\frac{1}{n^{12}}+O\left(\frac{1}{n^{13}}\right),$

where

$\displaystyle\varphi_{10}=$ $\displaystyle
350191+22500a_{1}-90000a_{2}+210000a_{3}-22500a_{3}^{2}-315000a_{4}-45000a_{2}a_{4}$
$\displaystyle+180000a_{3}a_{4}-210000a_{4}^{2}+67500a_{3}a_{4}^{2}-90000a_{4}^{3}-22500a_{4}^{4},$
$\displaystyle\varphi_{11}=$
$\displaystyle-5465923+206250a_{0}-928125a_{1}+2475000a_{2}-4331250a_{3}-412500a_{2}a_{3}$
$\displaystyle+928125a_{3}^{2}+5197500a_{4}-412500a_{1}a_{4}+1856250a_{2}a_{4}-4950000a_{3}a_{4}$
$\displaystyle+618750a_{3}^{2}a_{4}+4331250a_{4}^{2}+618750a_{2}a_{4}^{2}-2784375a_{3}a_{4}^{2}+2475000a_{4}^{3}$
$\displaystyle-825000a_{3}a_{4}^{3}+928125a_{4}^{4}+206250a_{4}^{5},$
$\displaystyle\varphi_{12}=$ $\displaystyle
175990871-20625000a_{0}+61875000a_{1}-123750000a_{2}-4125000a_{2}^{2}$
$\displaystyle+173250000a_{3}-8250000a_{1}a_{3}+41250000a_{2}a_{3}-61875000a_{3}^{2}+4125000a_{3}^{3}$
$\displaystyle-173250000a_{4}-8250000a_{0}a_{4}+41250000a_{1}a_{4}-123750000a_{2}a_{4}+247500000a_{3}a_{4}$
$\displaystyle+24750000a_{2}a_{3}a_{4}-61875000a_{3}^{2}a_{4}-173250000a_{4}^{2}+12375000a_{1}a_{4}^{2}-61875000a_{2}a_{4}^{2}$
$\displaystyle+185625000a_{3}a_{4}^{2}-24750000a_{3}^{2}a_{4}^{2}-123750000a_{4}^{3}-16500000a_{2}a_{4}^{3}$
$\displaystyle+82500000a_{3}a_{4}^{3}-61875000a_{4}^{4}+20625000a_{3}a_{4}^{4}-20625000a_{4}^{5}-4125000a_{4}^{6}.$

According to Lemma 1, we have five parameters $a_{4},a_{3},a_{2},a_{1}$ and
$a_{0}$ which produce the fastest convergence of the sequence from (3.8)

$\displaystyle\begin{cases}-773+630a_{4}=0,\\\
4033+1050a_{3}-3150a_{4}-1050a_{4}^{2}=0\\\
-37657+4500a_{2}-15750a_{3}+31500a_{4}-9000a_{3}a_{4}+15750a_{4}^{2}+4500a_{4}^{3}=0\\\
\varphi_{10}=0,\\\ \varphi_{11}=0,\end{cases}$

namely if

(3.9)
$\displaystyle\theta(n)=\frac{\frac{1}{200}}{n^{5}+\frac{773}{630}n^{4}+\frac{21361}{15876}n^{3}+\frac{1348075}{2000376}n^{2}-\frac{91207415}{252047376}n-\frac{178345771979}{1746688315680}}.$

Thus, we get

(3.10)
$\displaystyle\nu(n)-\nu(n+1)=-\frac{10992878936527}{160060165655040}\frac{1}{n^{12}}+O\left(\frac{1}{n^{13}}\right).$

We can apply another approach to find $a_{4},a_{3},a_{2},a_{1}$ and $a_{0}$
step by step, which is achieved by using
$n^{5}+a_{4}n^{4},n^{5}+a_{4}n^{4}+a_{3}n^{3},n^{5}+a_{4}n^{4}+a_{3}n^{3}+a_{2}n^{2},n^{5}+a_{4}n^{4}+a_{3}n^{3}+a_{2}n^{2}+a_{1}n$,
$n^{5}+a_{4}n^{4}+a_{3}n^{3}+a_{2}n^{2}+a_{1}n+a_{0}$ instead of $\Phi(5;n)$
in turn. For the reader’s convenience, here we give an example to explain how
_Mathematica_ software generates $\nu(n)-\nu(n+1)$ into power series in the
terms of $\frac{1}{n}$. For example, find $a_{3}$. We manipulate _Mathematica_
program

$\displaystyle\mbox{Normal[Series[}\left(-\frac{1}{n+1}+\log[1+\frac{1}{n}]-\left(\frac{13+30n}{6(1+6n+10n^{2})}+\frac{1/200}{n^{5}+(773/630)n^{4}+a_{3}n^{3}}\right)\right.$
$\displaystyle\left.+\left(\frac{13+30n}{6(1+6n+10n^{2})}+\frac{1/200}{n^{5}+(773/630)n^{4}+a_{3}n^{3}}\right)/.n\rightarrow
n+1\right)/.n\rightarrow 1/x,\\{x,0,10\\}]]/.x\rightarrow 1/n$

to generate

(3.11)
$\displaystyle\nu(n)-\nu(n+1)=\frac{-\frac{21361}{453600}+\frac{7a3}{200}}{n^{8}}+\frac{\frac{9173092}{31255875}-\frac{3751a3}{15750}}{n^{9}}+O\left(\frac{1}{n^{10}}\right).$

By solving the equation $-\frac{21361}{453600}+\frac{7a3}{200}=0$, we also
find $a_{3}=\frac{21361}{15876}$. In what follows, we always use this
approach.

By Lemma 1 again, we obtain finally

(3.12)
$\displaystyle\lim_{n\rightarrow\infty}n^{11}\left(\nu(n)-\gamma\right)=-\frac{10992878936527}{1760661822205440}.$

We observe that the above twice-correction improves the rate of convergence
from $n^{-6}$ to $n^{-11}$.∎

######  Remark 1.

The main idea of twice-correction is that from
$n^{5},n^{5}+a_{4}n^{4},n^{5}+a_{4}n^{4}+a_{3}n^{3}+a_{2}n^{2},n^{5}+a_{4}n^{4}+a_{3}n^{3}+a_{2}n^{2}+a_{1}n$
to $n^{5}+a_{4}n^{4}+a_{3}n^{3}+a_{2}n^{2}+a_{1}n+a_{0}$, their approximations
in turn become better and better.

######  Remark 2.

It should be noted that once we find the exact values of the parameters
$a_{4}$ to $a_{0}$, it is not very difficult for us to check the formula
(3.10) with the help of _Mathematica_ software.

Example 2. We would like to give another example. Now we take the initial-
correction function $\eta_{0}(n)=\frac{6n-1}{12n^{2}}$(see Theorem 1.1 in Lu
[25] or (1.9), which is found by the continued fraction method), and define

(3.13) $\displaystyle\mu_{0}(n)=$ $\displaystyle\sum_{m=1}^{n}\frac{1}{m}-\ln
n-\eta_{0}(n).$

One may check by using Lemma 1

(3.14)
$\displaystyle\lim_{n\rightarrow\infty}n^{4}\left(\mu_{0}(n)-\gamma\right)=\frac{1}{120}.$

Similarly, we insert a correction function $-\eta(n)$ in (3.13) again, which
has the form of

(3.15)
$\displaystyle\eta(n)=\frac{\frac{1}{120}}{n^{4}+b_{3}n^{3}+b_{2}n^{2}+b_{1}n+b_{0}}.$

Applying the same method as Example 1, we can find
$(b_{3},b_{2},b_{1},b_{0})=(0,\frac{10}{21},0,-\frac{241}{882})$. The details
is omitted here. Now we define

(3.16) $\displaystyle\mu(n)=$ $\displaystyle\sum_{m=1}^{n}\frac{1}{m}-\ln
n-\eta_{0}(n)-\eta(n),$

where

(3.17)
$\displaystyle\eta_{0}(n)=\frac{6n-1}{12n^{2}}\quad\mbox{and}\quad\eta(n)=\frac{\frac{1}{120}}{n^{4}+\frac{10}{21}n^{2}-\frac{241}{882}}.$

By using _Mathematica_ software and Lemma 1, we can attain

(3.18)
$\displaystyle\mu(n)-\mu(n+1)=-\frac{13775}{305613}\frac{1}{n^{11}}+O\left(\frac{1}{n^{12}}\right),$
(3.19)
$\displaystyle\lim_{n\rightarrow\infty}n^{10}\left(\mu(n)-\gamma\right)=-\frac{13775}{3056130}.$

######  Remark 3.

We observe that the above twice-correction improves the rate of convergence
from $n^{-4}$ to $n^{-10}$, which is the desired result. However, it is
interesting to note that both $b_{3}$ and $b_{1}$ equal zero. The reason of
why inserting the sub-correction term $b_{3}n^{3}$(or $b_{1}n$) does not
improves the rate of convergence(i.e. compare $n^{4}+b_{3}n^{3}$ with $n^{4}$,
or $n^{4}+b_{3}n^{3}+b_{2}n^{2}+b_{1}n$ with $n^{4}+b_{3}n^{3}+b_{2}n^{2}$)
may be that the function $n^{3}$(or $n$) _changes_ too rapidly when $n$ tends
to infinity. Fortunately, these losses are made up by the sub-correction terms
$b_{2}n^{2}$ and $b_{0}$.

More precisely, we will improve (3.19), and prove the following double-sides
inequalities.

######  Theorem 1.

Let $\mu(n)$ be defined by (3.16). Then for all positive integer $n$, we have

(3.20)
$\displaystyle\frac{13775}{3056130}\frac{1}{(n+\frac{3}{4})^{10}}<\gamma-\mu(n)<\frac{13775}{3056130}\frac{1}{(n-\frac{1}{4})^{10}}.$

######  Remark 4.

In fact, Theorem 1 implies that $\mu(n)$ is a strictly increasing function of
$n$.

###### Proof.

It follows from (3.16)

(3.21)
$\displaystyle\mu(n)-\mu(n+1)=-\frac{1}{n+1}+\ln(1+\frac{1}{n})+\eta_{0}(n+1)+\eta(n+1)-\eta_{0}(n)-\eta(n).$

We write $D=\frac{13775}{27783}$, and define for $x\geq 1$

(3.22)
$\displaystyle-\omega(x)=-\frac{1}{x+1}+\ln(1+\frac{1}{x})+\eta_{0}(x+1)+\eta(x+1)-\eta_{0}(x)-\eta(x).$

By _Mathematica_ software, it is not difficult to check

(3.23) $\displaystyle-\omega^{\prime}(x)-\frac{D}{(x+\frac{1}{4})^{12}}$
$\displaystyle=$
$\displaystyle-\frac{\Psi_{1}(20;x)(x-1)+4032098201877889488940287625}{277830x^{3}(1+x)^{3}(1+4x)^{12}(-241+420x^{2}+882x^{4})^{2}(1061+4368x+5712x^{2}+3528x^{3}+882x^{4})^{2}}$
$\displaystyle<$ $\displaystyle 0,$

and

(3.24) $\displaystyle-\omega^{\prime}(x)-\frac{D}{(x+\frac{3}{4})^{12}}$
$\displaystyle=$
$\displaystyle\frac{\Psi_{2}(20;x)(x-1)+51726219719747325679290363431}{277830x^{3}(1+x)^{3}(3+4x)^{12}(-241+420x^{2}+882x^{4})^{2}(1061+4368x+5712x^{2}+3528x^{3}+882x^{4})^{2}}$
$\displaystyle>$ $\displaystyle 0.$

Note that $\omega(\infty)=0$. From (3.23) and Lemma 3, one has

(3.25) $\displaystyle\omega(n)=$
$\displaystyle\int_{n}^{\infty}-\omega^{\prime}(x)dx<D\int_{n}^{\infty}\frac{dx}{(x+\frac{1}{4})^{12}}$
$\displaystyle=$
$\displaystyle\frac{D}{11}\frac{1}{(n+\frac{1}{4})^{11}}<\frac{D}{11}\int_{n-\frac{1}{4}}^{n+\frac{3}{4}}\frac{dx}{x^{11}}.$

Note that $\mu(\infty)=\gamma$. Combining (3.21),(3.22) and (3.25), we have

(3.26) $\displaystyle\gamma-\mu(n)=$
$\displaystyle\sum_{m=n}^{\infty}\left(\mu(m+1)-\mu(m)\right)<\frac{D}{11}\sum_{m=n}^{\infty}\int_{m-\frac{1}{4}}^{m+\frac{3}{4}}\frac{dx}{x^{11}}$
$\displaystyle=$
$\displaystyle\frac{D}{11}\int_{n-\frac{1}{4}}^{\infty}\frac{dx}{x^{11}}=\frac{D}{110}\frac{1}{(n-\frac{1}{4})^{10}}.$

This finishes the proof of right-hand inequality in (3.20). Similarly, it
follows from (3.24)

(3.27) $\displaystyle\omega(n)=$
$\displaystyle\int_{n}^{\infty}-\omega^{\prime}(x)dx>D\int_{n}^{\infty}\frac{dx}{(x+\frac{3}{4})^{12}}$
$\displaystyle=$
$\displaystyle\frac{D}{11}\frac{1}{(n+\frac{3}{4})^{11}}>\frac{D}{11}\int_{n+\frac{3}{4}}^{n+\frac{7}{4}}\frac{dx}{x^{11}}.$

Finally, by (3.21),(3.22) and (3.27), one has

(3.28) $\displaystyle\gamma-\mu(n)=$
$\displaystyle\sum_{m=n}^{\infty}\left(\mu(m+1)-\mu(m)\right)>\frac{D}{11}\sum_{m=n}^{\infty}\int_{m+\frac{3}{4}}^{m+\frac{7}{4}}\frac{dx}{x^{11}}$
$\displaystyle=$
$\displaystyle\frac{D}{11}\int_{n+\frac{3}{4}}^{\infty}\frac{dx}{x^{11}}=\frac{D}{110}\frac{1}{(n+\frac{3}{4})^{10}}.$

This completes the proof of Theorem 1.∎

##  4 The multiple-correction method

Based on the work of Section 2, we will formulate a new _multiple-correction
method_ to study faster approximation problem for the constants of Euler-
Mascheroni and Landau.

Let $(v(n))_{n\geq 1}$ be a sequence to be approximated. Throughout the paper,
we always assume that the following three conditions hold.

Condition (i). The initial-correction function $\eta_{0}(n)$ satisfies

$\displaystyle\lim_{n\rightarrow\infty}\left(v(n)-\eta_{0}(n)\right)=0,$
$\displaystyle\lim_{n\rightarrow\infty}n^{l_{0}}\left(v(n)-v(n+1)-\eta_{0}(n)+\eta_{0}(n+1)\right)=C_{0}\neq
0,$

with some a positive integer $l_{0}\geq 2$.

Condition (ii). The $k$-th correction function $\eta_{k}(n)$ has the form of
$-\frac{C_{k-1}}{\Phi_{k}(l_{k-1};n)}$, where

$\displaystyle\lim_{n\rightarrow\infty}n^{l_{k-1}}\left(v(n)-v(n+1)-\sum_{j=0}^{k-1}\left(\eta_{j}(n)-\eta_{j}(n+1)\right)\right)=C_{k-1}\neq
0,$

Condition (iii). The difference
$\left(v(1/x)-v(1/x+1)-\eta_{0}(1/x)+\eta_{0}(1/x+1)\right)$ is an analytic
function in a neighborhood of point $x=0$.

###  4.1 Euler-Mascheroni Constant

(Step 1) The initial-correction. We choose $\eta_{0}(n)=0$, and let

(4.1) $\displaystyle\nu_{0}(n)=\sum_{m=1}^{n}\frac{1}{m}-\ln
n-\eta_{0}(n)=\sum_{m=1}^{n}\frac{1}{m}-\ln n.$

By lemma 1, it is not difficult to prove that

(4.2)
$\displaystyle\lim_{n\rightarrow\infty}n^{2}\left(\nu_{0}(n)-\nu_{0}(n+1)\right)=\frac{1}{2},$
(4.3)
$\displaystyle\lim_{n\rightarrow\infty}n\left(\nu_{0}(n)-\gamma\right)=\frac{1}{2}=:C_{0}.$

(Step 2) The first-correction. We let

(4.4)
$\displaystyle\eta_{1}(n)=\frac{C_{0}}{\Phi_{1}(1;n)}=\frac{\frac{1}{2}}{n+b_{(1,0)}},$

and define

(4.5) $\displaystyle\nu_{1}(n)=\sum_{m=1}^{n}\frac{1}{m}-\ln
n-\eta_{0}(n)-\eta_{1}(n).$

By the same method as (3.11), we find $b_{(1,0)}=\frac{1}{6}$. Applying Lemma
1 again, one has

(4.6)
$\displaystyle\lim_{n\rightarrow\infty}n^{4}\left(\nu_{1}(n)-\nu_{1}(n+1)\right)=-\frac{1}{24},$
(4.7)
$\displaystyle\lim_{n\rightarrow\infty}n^{3}\left(\nu_{1}(n)-\gamma\right)=-\frac{1}{72}:=C_{1}.$

(Step 3) The second-correction. Similarly, we set the second-correction
function in the form of $\eta_{2}(n)=\frac{C_{1}}{\Phi_{2}(3;n)}$, and define

(4.8) $\displaystyle\nu_{2}(n)=\sum_{m=1}^{n}\frac{1}{m}-\ln
n-\eta_{0}(n)-\eta_{1}(n)-\eta_{2}(n).$

By using similar approach of (3.11), we can find

(4.9)
$\displaystyle\eta_{2}(n)=\frac{-\frac{1}{72}}{n^{3}+\frac{23}{30}n^{2}+\frac{14}{25}n+\frac{1333}{10500}}=\frac{-\frac{1}{72}}{(n+\frac{23}{90})^{3}+\frac{983}{2700}(n+\frac{23}{90})+\frac{2197}{127575}}.$

By Lemma 1, one can obtain

(4.10)
$\displaystyle\lim_{n\rightarrow\infty}n^{8}\left(\nu_{2}(n)-\nu_{2}(n+1)\right)=-\frac{7061}{5400},$
(4.11)
$\displaystyle\lim_{n\rightarrow\infty}n^{7}\left(\nu_{2}(n)-\gamma\right)=-\frac{7061}{3780000}:=C_{2}.$

(Step 4) The third-correction. We set
$\eta_{3}(n)=\frac{C_{2}}{\Phi_{3}(7;n)}$, and define

(4.12) $\displaystyle\nu_{3}(n)=\sum_{m=1}^{n}\frac{1}{m}-\ln
n-\eta_{0}(n)-\eta_{1}(n)-\eta_{2}(n)-\eta_{3}(n).$

By using _Mathematica_ software, we can find

(4.13) $\displaystyle\Phi_{3}(7;n)=$ $\displaystyle
n^{7}+\frac{126901}{70610}n^{6}+\frac{302657774122}{78525910575}n^{5}+\frac{203050524517143511}{52278737145178500}n^{4}$
$\displaystyle+\frac{4586975399311716291806}{2713180197918474605475}n^{3}+\frac{139644786102811402696525439}{298861139889036647352440010}n^{2}$
$\displaystyle+\frac{1335669056713380727335512329403306}{243734857761374337083369364175455}n$
$\displaystyle+\frac{1768275723433572920281319954725767947341}{1032607098391838516487402648265732653000}.$

Now by Lemma 1 again, we obtain

######  Theorem 2.

Let $\nu_{3}(n)$ be defined by (4.12). Then we have

(4.14)
$\displaystyle\lim_{n\rightarrow\infty}n^{16}\left(\nu_{3}(n)-\nu_{3}(n+1)\right)=15C_{3},$

and

(4.15)
$\displaystyle\lim_{n\rightarrow\infty}n^{15}\left(\nu_{3}(n)-\gamma\right)=-\frac{6044981017774921659252823535814990412377703}{102460439337930176798462527774167322493925000}:=C_{3}.$

######  Remark 5.

It could be imagined that if we apply the correction-process many times, then,
we can obtain $k$th-correction sequence

(4.16) $\displaystyle\nu_{k}(n)=\sum_{m=1}^{n}\frac{1}{m}-\ln
n-\eta_{0}(n)-\sum_{j=1}^{k}\frac{C_{j-1}}{\Phi_{j}(l_{j-1};n)}$

with the rate of convergence $\frac{1}{n^{l_{k}}}$, here $l_{k}\geq
2l_{k-1}+1$, i.e.

(4.17)
$\displaystyle\lim_{n\rightarrow\infty}n^{l_{k}}\left(\nu_{k}(n)-\gamma\right)={C_{k}}\neq
0,$ (4.18) $\displaystyle\gamma=\sum_{m=1}^{n}\frac{1}{m}-\ln
n-\eta_{0}(n)-\sum_{j=1}^{k}\frac{C_{j-1}}{\Phi_{j}(l_{j-1};n)}+O\left(\frac{1}{n^{l_{k}}}\right).$

######  Remark 6.

For comparison, the result $\nu_{1}(n)$ in Theorem 2 is the same as $r_{2}(n)$
in (1.9), and
$\lim_{n\rightarrow\infty}\frac{\nu_{2}(n)-\gamma}{r_{6}(n)-\gamma}=\frac{C_{2}}{C_{6}^{\prime}}=0.950367\cdots<1$.
Theoretically at least, for a large $n$ the above formula may reduce or
eliminate numerically computations compared with Euler-Maclaurin summation
formula. For example, if we take $n=2^{15}=32768$ in Theorem 2, then
$-1.09418\cdot 10^{-69}<\nu_{3}(n)-\gamma<0$.

######  Remark 7.

We can take different initial-correction function to find some other simple
faster approximations. For example, we choose the initial-correction function
$\eta_{0}(n)=-\ln n+\frac{1}{2}\ln\left(n^{2}+n+\frac{1}{3}\right)$, see, Chen
and Li [10]. By Lemma 1, it is not very difficult for us to check that

(4.19) $\displaystyle\nu_{0}(n)=\sum_{m=1}^{n}\frac{1}{m}-\ln
n-\eta_{0}(n)=\sum_{m=1}^{n}\frac{1}{m}-\frac{1}{2}\ln\left(n^{2}+n+\frac{1}{3}\right),$
(4.20)
$\displaystyle\lim_{n\rightarrow\infty}n^{4}\left(\nu_{0}(n)-\gamma\right)=-\frac{1}{180}=:C_{0}.$

Let

(4.21)
$\displaystyle\eta_{1}(n)=\frac{C_{0}}{\Phi_{1}(4;n)}\quad\mbox{and}\quad\eta_{2}(n)=\frac{C_{1}}{\Phi_{2}(10;n)},$

where $C_{1}=\frac{457528}{123773265}$,

(4.22) $\displaystyle\eta_{1}(n)=$
$\displaystyle(n+\frac{1}{2})^{4}+\frac{85}{126}(n+\frac{1}{2})^{2}-\frac{18287}{63504},$
(4.23) $\displaystyle\eta_{2}(n)=$
$\displaystyle(n+\frac{1}{2})^{10}+\frac{28038237821}{5995446912}(n+\frac{1}{2})^{8}+\frac{11612938185382451401}{35945383674610335744}(n+\frac{1}{2})^{6}$
$\displaystyle+\frac{163544744039006129564874642269}{8307589279805355451415003136}(n+\frac{1}{2})^{4}$
$\displaystyle-\frac{2762081970439756978947606523226093660107}{15021373006058621011789214048600457216}(n+\frac{1}{2})^{2}$
$\displaystyle+\frac{2771007475606973680958970352585491511233080640189551}{1350897666047614749541649384438829437061829754880}.$

Now define

(4.24) $\displaystyle\nu_{2}(n)=\sum_{m=1}^{n}\frac{1}{m}-\ln
n-\eta_{0}(n)-\eta_{1}(n)-\eta_{2}(n).$

By using Lemma 1, one may check

(4.25)
$\displaystyle\lim_{n\rightarrow\infty}n^{22}\left(\nu_{2}(n)-\gamma\right)={C_{2}},$

where

(4.26) $\displaystyle
C_{2}=\frac{1864841554154123790589398711158437373230857062719102654146029}{18653565767176841210967548892254397636853629986414159462400}.$

Some other interesting correction functions can be found in Gourdon and Sebah
[20].

###  4.2 Landau Constants

(Step 1) The initial-correction. Let $c_{0}$ be defined by (1.15). Motivated
by inequalities (1.17) and (1.18), we choose
$\eta_{0}(n)=\frac{1}{\pi}\ln(n+\frac{3}{4})+c_{0}$, and define

(4.27) $\displaystyle
u_{0}(n)=G(n)-\eta_{0}(n)=G(n)-\frac{1}{\pi}\ln(n+\frac{3}{4})-c_{0}.$

Now we consider the difference $u_{0}(n)-u_{0}(n+1)$. It follows immediately
from (4.27)

(4.28) $\displaystyle
u_{0}(n)-u_{0}(n+1)=G(n)-G(n+1)-\frac{1}{\pi}\ln(n+\frac{3}{4})+\frac{1}{\pi}\ln(n+\frac{7}{4}).$

First, from the duplication formula (Legendre, 1809)

(4.29) $\displaystyle
2^{2z-1}\Gamma(z)\Gamma(z+\frac{1}{2})=\sqrt{\pi}\Gamma(2z),$

one can prove

(4.30) $\displaystyle
G(n)-G(n-1)=\frac{(\Gamma(2n+1))^{2}}{16^{2}(\Gamma(n+1))^{4}}=\left(\frac{(2n)!}{4^{n}(n!)^{2}}\right)^{2}=\frac{1}{\pi}q(n),$

where $q(n)$ is defined by (2.3). Also see p.739 in Granath [21] or p.306 in
Chen [9]. By (4.28) and (4.30), one has

(4.31) $\displaystyle
u_{0}(n)-u_{0}(n+1)=-\frac{1}{\pi}q(n+1)-\frac{1}{\pi}\ln(n+\frac{3}{4})+\frac{1}{\pi}\ln(n+\frac{7}{4}).$

From Lemma 2 and (2.6), on one hand, it can be observed that for all positive
integer $j$, one has

(4.32) $\displaystyle
q_{2}(n)<q_{4}(n)<\cdots<q_{2j}(n)<q(n)<q_{2j+1}(n)<\cdots<q_{3}(n)<q_{1}(n).$

On the other hand, we can check by using _Mathematica_ software

(4.33) $\displaystyle q_{9}(n)-q_{8}(n)=O\left(\frac{1}{n^{17}}\right).$

Combining (4.32) and (4.33) gives

(4.34) $\displaystyle q(n+1)=q_{8}(n+1)+O\left(\frac{1}{n^{16}}\right).$

By using the _Mathematica_ software, we expand $q_{8}(n+1)$ into a power
series in terms of $n^{-1}$. Noting formula (4.34), we obtain

(4.35) $\displaystyle q(n+1)=$ $\displaystyle
q_{8}(n+1)+O\left(\frac{1}{n^{16}}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{n}-\frac{5}{4}\frac{1}{n^{2}}+\frac{49}{32}\frac{1}{n^{3}}-\frac{235}{128}\frac{1}{n^{4}}+\frac{4411}{2048}\frac{1}{n^{5}}-\frac{20275}{8192}\frac{1}{n^{6}}+\frac{183077}{65536}\frac{1}{n^{7}}$
$\displaystyle-\frac{815195}{262144}\frac{1}{n^{8}}+\frac{28754131}{8388608}\frac{1}{n^{9}}-\frac{125799895}{33554432}\frac{1}{n^{10}}+\frac{1091975567}{268435456}\frac{1}{n^{11}}$
$\displaystyle-\frac{4702048685}{1073741824}\frac{1}{n^{12}}+\frac{80679143663}{17179869184}\frac{1}{n^{13}}-\frac{346250976095}{68719476736}\frac{1}{n^{14}}$
$\displaystyle+\frac{2947620308941}{549755813888}\frac{1}{n^{15}}+O\left(\frac{1}{n^{16}}\right).$

The above expression is also used in the first and second-correction below. In
addition, it is not difficult to obtain

(4.36)
$\displaystyle-\ln(n+\frac{3}{4})+\ln(n+\frac{7}{4})=\frac{1}{n}-\frac{5}{4}\frac{1}{n^{2}}+\frac{79}{48}\frac{1}{n^{3}}+O\left(\frac{1}{n^{4}}\right).$

Inserting (4.35) and (4.36) into (4.31) yields

(4.37) $\displaystyle
u_{0}(n)-u_{0}(n+1)=\frac{11}{96\pi}\frac{1}{n^{3}}+O\left(\frac{1}{n^{4}}\right).$

Note that the inequalities (1.18) implies $u_{0}(\infty)=0$. Applying Lemma 1,
we obtain

(4.38)
$\displaystyle\lim_{n\rightarrow\infty}n^{2}u_{0}(n)=\frac{11}{192\pi}=C_{0}.$

(Step 2) The first-correction. We let

(4.39)
$\displaystyle\eta_{1}(n)=\frac{C_{0}}{\Phi_{1}(2;n)}=\frac{C_{0}}{n^{2}+a_{1}n+a_{0}},$

and define

(4.40) $\displaystyle u_{1}(n)=G(n)-\eta_{0}(n)-\eta_{1}(n).$

Hence

(4.41) $\displaystyle u_{1}(n)-u_{1}(n+1)=$
$\displaystyle\left(u_{0}(n)-\eta_{1}(n)\right)-\left(u_{0}(n+1)-\eta_{1}(n+1)\right)$
$\displaystyle=$
$\displaystyle\left(u_{0}(n)-u_{0}(n+1)\right)-\left(\eta_{1}(n)-\eta_{1}(n+1)\right).$

Note that the first term of (4.41) can be treated by the same method in (step
1). Here we only need to replace (4.36) by the following more accurate power
series expansion

(4.42) $\displaystyle-\ln(n+\frac{3}{4})+\ln(n+\frac{7}{4})=$
$\displaystyle\frac{1}{n}-\frac{5}{4}\frac{1}{n^{2}}+\frac{79}{48}\frac{1}{n^{3}}-\frac{145}{64}\frac{1}{n^{4}}+\frac{4141}{1280}\frac{1}{n^{5}}-\frac{14615}{3072}\frac{1}{n^{6}}$
$\displaystyle+\frac{205339}{28672}\frac{1}{n^{7}}-\frac{179945}{16384}\frac{1}{n^{8}}+\frac{10083481}{589824}\frac{1}{n^{9}}-\frac{7060405}{262144}\frac{1}{n^{10}}$
$\displaystyle+\frac{494287399}{11534336}\frac{1}{n^{11}}-\frac{865047235}{12582912}\frac{1}{n^{12}}+\frac{24221854021}{218103808}\frac{1}{n^{13}}$
$\displaystyle-\frac{84777286235}{469762048}\frac{1}{n^{14}}+\frac{1186886790259}{4026531840}\frac{1}{n^{15}}+O\left(\frac{1}{n^{16}}\right).$

By applying _Mathematica_ software again, we have

(4.43) $\displaystyle\frac{1}{\Phi_{1}(2;n)}-\frac{1}{\Phi_{1}(2;n+1)}$
$\displaystyle=$
$\displaystyle\frac{2}{n^{3}}+\frac{-3-3a_{1}}{n^{4}}+\frac{4+6a_{1}+4a_{1}^{2}-4a_{0}}{n^{5}}$
$\displaystyle+\frac{-5-10a_{1}-10a_{1}^{2}-5a_{1}^{3}+10a_{0}+10a_{1}a_{0}}{n^{6}}$
$\displaystyle+\frac{6+15a_{1}+20a_{1}^{2}+15a_{1}^{3}+6a_{1}^{4}-20a_{0}-30a_{1}a_{0}-18a_{1}^{2}a_{0}+6a_{0}^{2}}{n^{7}}+O\left(\frac{1}{n^{8}}\right).$

Now combining (4.41), (4.31),(4.35), (4.42) and (4.43), and performing some
simplifications, we can obtain

(4.44) $\displaystyle\pi\left(u_{1}(n)-u_{1}(n+1)\right)=$
$\displaystyle\frac{\frac{235}{128}+\frac{-134+11a_{1}}{64}}{n^{4}}$
$\displaystyle+\frac{-\frac{4411}{2048}+\frac{11543-1320a_{1}-880a_{1}^{2}+880a_{0}}{3840}}{n^{5}}$
$\displaystyle+\frac{\frac{20275}{8192}+\frac{5(-2747+352a_{1}+352a_{1}^{2}+176a_{1}^{3}-352a_{0}-352a_{1}a_{0})}{3072}}{n^{6}}$
$\displaystyle+\frac{-\frac{183077}{65536}+\frac{\sigma}{86016}}{n^{7}}+O\left(\frac{1}{n^{8}}\right),$

where

$\displaystyle\sigma=586449-73920a_{1}-98560a_{1}^{2}-73920a_{1}^{3}-29568a_{1}^{4}+98560a_{0}+147840a_{1}a_{0}+88704a_{1}^{2}a_{0}-29568a_{0}^{2}.$

The fastest sequence $(u_{1}(n))_{n\geq 1}$ is obtained when the first two
coefficients of this power series vanish. In this case

(4.45) $\displaystyle a_{1}=\frac{3}{2},\quad a_{0}=\frac{5501}{7040},$

thus

$\displaystyle
u_{1}(n)-u_{1}(n+1)=\frac{89684299}{3027763200\pi}\frac{1}{n^{7}}+O\left(\frac{1}{n^{8}}\right).$

Finally, by using Lemma 1, one has

(4.46)
$\displaystyle\lim_{n\rightarrow\infty}n^{6}u_{1}(n)=\frac{89684299}{18166579200\pi}=C_{1}.$

(Step 3) The second-correction. We let

(4.47)
$\displaystyle\eta_{2}(n)=\frac{C_{1}}{\Phi_{2}(6;n)}=\frac{C_{1}}{n^{6}+b_{5}n+b_{4}n^{4}+b_{3}n^{3}+b_{2}n^{2}+b_{1}n+b_{0}},$

and define

(4.48) $\displaystyle u_{2}(n)=G(n)-\eta_{0}(n)-\eta_{1}(n)-\eta_{2}(n).$

Thus

(4.49) $\displaystyle
u_{2}(n)-u_{2}(n+1)=\left(u_{0}(n)-u_{0}(n+1)\right)-\left(\eta_{1}(n)+\eta_{2}(n)-\eta_{1}(n+1)-\eta_{2}(n+1)\right).$

We use (4.35) and (4.42) to expand $u_{0}(n)-u_{0}(n+1)$ into a power series
as in terms of $n^{-1}$. In addition, as mentioned already in Section 3, one
can use a similar _Mathematica_ program in Example 1 to find
$b_{5},b_{4},b_{3},b_{2},b_{1}$ and $b_{0}$ in turn. Here we omit the details.
We write

(4.50) $\displaystyle
C_{2}=-\frac{5691942495934169497683736629269380931519449}{65873649616252391923660120676946385934745600\pi}.$

By using Lemma 1 again, it is not very difficult for us to check the following
assertion.

######  Theorem 3.

Let $c_{0}$, $C_{2}$ be defined by (1.15) and (4.50) respectively, and

(4.51) $\displaystyle
u_{2}(n):=G(n)-\left(\frac{1}{\pi}\ln(n+\frac{3}{4})+c_{0}+\frac{\frac{11}{192\pi}}{\Phi_{1}(2;n)}+\frac{\frac{89684299}{18166579200\pi}}{\Phi_{2}(6;n)}\right),$

where

(4.52) $\displaystyle\Phi_{1}(2;n)=$
$\displaystyle(n+\frac{3}{4})^{2}+\frac{1541}{7040},$ (4.53)
$\displaystyle\Phi_{2}(6;n)=$
$\displaystyle(n+\frac{3}{4})^{6}+\frac{1092000370209}{631377464960}(n+\frac{3}{4})^{4}-\frac{111862508515629162375}{181198865117870921728}(n+\frac{3}{4})^{2}$
$\displaystyle+\frac{1824588073050833974528912179250963}{540823069619183303269309779804160}.$

Then we have

(4.54)
$\displaystyle\lim_{n\rightarrow\infty}n^{15}\left(u_{2}(n)-u_{2}(n+1)\right)=14C_{2},$
(4.55) $\displaystyle\lim_{n\rightarrow\infty}n^{14}u_{2}(n)=C_{2}.$

######  Remark 8.

It should be stressed that that a “good” initial-correction is very important
for us to accelerate the convergence. In addition, one may study analogous
question by choosing different initial-correction.

The following Theorem tells us how to improve (1.17) and (1.20).

######  Theorem 4.

Let $c_{0}$ be defined by (1.15). Then for all integer $n\geq 0$, we have

(4.56)
$\displaystyle\frac{C_{1}}{(n+\frac{3}{2})^{6}}<G(n)-\frac{1}{\pi}\ln(n+\frac{3}{4})-c_{0}-\frac{\frac{11}{192\pi}}{(n+\frac{3}{4})^{2}+\frac{1541}{7040}}<\frac{C_{1}}{(n+\frac{1}{2})^{6}},$

where $C_{1}=\frac{89684299}{18166579200\pi}$.

######  Remark 9.

In fact, Theorem 4 implies that $u_{1}(n)$ is a strictly decreasing function
of $n$.

###### Proof.

Although the method used in this section is very similar to that in proof of
Theorem 1, we would like to give a full proof for the sake of completeness.
First, we can see that the inequalities (4.56) are true for $n=0$. Hence, in
the following we only need to prove that these inequalities are also true for
$n\geq 1$. To this end, let

(4.57) $\displaystyle
u_{1}(n)=G(n)-\frac{1}{\pi}\ln(n+\frac{3}{4})-c_{0}-\frac{\frac{11}{192\pi}}{\Phi_{1}(2;n)},$

it follows easily from (4.30)

(4.58) $\displaystyle u_{1}(n)-u_{1}(n+1)=$
$\displaystyle-\frac{1}{\pi}q(n+1)-\frac{1}{\pi}\ln(n+\frac{3}{4})-\frac{\frac{11}{192\pi}}{\Phi_{1}(2;n)}$
$\displaystyle+\frac{1}{\pi}\ln(n+\frac{7}{4})+\frac{\frac{11}{192\pi}}{\Phi_{1}(2;n+1)}.$

Let

(4.59) $\displaystyle f(x)=$
$\displaystyle-\frac{1}{\pi}q_{6}(x+1)-\frac{1}{\pi}\ln(x+\frac{3}{4})-\frac{\frac{11}{192\pi}}{\Phi_{1}(2;x)}+\frac{1}{\pi}\ln(x+\frac{7}{4})+\frac{\frac{11}{192\pi}}{\Phi_{1}(2;x+1)},$
(4.60) $\displaystyle g(x)=$
$\displaystyle-\frac{1}{\pi}q_{5}(x+1)-\frac{1}{\pi}\ln(x+\frac{3}{4})-\frac{\frac{11}{192\pi}}{\Phi_{1}(2;x)}+\frac{1}{\pi}\ln(x+\frac{7}{4})+\frac{\frac{11}{192\pi}}{\Phi_{1}(2;x+1)}.$

From (4.32) and (4.60), one has

(4.61) $\displaystyle g(n)<u_{1}(n)-u_{1}(n+1)<f(n).$

Firstly, we give the lower bound for $g(n)$, and the upper bound for $f(n)$,
respectively. We set $D_{1}=\frac{89684299}{432537600}$. By using the
_Mathematica_ software, we easily obtain

(4.62)
$\displaystyle-f^{\prime}(x)-\frac{D_{1}}{\pi(x+1)^{8}}=-\frac{1}{\pi}\frac{\Psi_{1}(21;n)}{\Psi_{2}(30;n)}<0.$

Noting $f(+\infty)=0$ and utilizing (4.62) and Lemma 3, one has

(4.63) $\displaystyle f(n)$
$\displaystyle=-\int_{n}^{\infty}f^{\prime}(x)dx<\int_{n}^{\infty}\frac{D_{1}}{\pi(x+1)^{8}}dx=\frac{D_{1}}{7\pi}\frac{1}{(n+1)^{7}}$
$\displaystyle<\frac{D_{1}}{7\pi}\int_{n+\frac{1}{2}}^{n+\frac{3}{2}}\frac{1}{x^{7}}dx.$

Similarly, we can check

(4.64)
$\displaystyle-g^{\prime}(x)-\frac{D_{1}}{\pi(x+\frac{3}{2})^{8}}=\frac{1}{\pi}\frac{\Psi_{3}(19;n)}{\Psi_{4}(28;n)}>0.$

Applying $g(+\infty)=0$ and (4.64), we obtain

(4.65) $\displaystyle g(n)$
$\displaystyle=-\int_{n}^{\infty}g^{\prime}(x)dx>\int_{n}^{\infty}\frac{D_{1}}{\pi(x+\frac{3}{2})^{8}}dx=\frac{D_{1}}{7\pi}\frac{1}{(n+\frac{3}{2})^{7}}$
$\displaystyle>\frac{D_{1}}{7\pi}\int_{n+\frac{3}{2}}^{n+\frac{5}{2}}\frac{1}{x^{7}}dx.$

On the other hand, from $u_{1}(\infty)=0$ and (4.63), we have

(4.66) $\displaystyle u_{1}(n)=$
$\displaystyle\sum_{m=n}^{\infty}\left(u_{1}(m)-u_{1}(m+1)\right)<\sum_{m=n}^{\infty}\frac{D_{1}}{7\pi}\int_{m+\frac{1}{2}}^{m+\frac{3}{2}}\frac{1}{x^{7}}dx$
$\displaystyle=$
$\displaystyle\frac{D_{1}}{7\pi}\int_{n+\frac{1}{2}}^{\infty}\frac{1}{x^{7}}dx=\frac{D_{1}}{42\pi}\frac{1}{(n+\frac{1}{2})^{6}}.$

Similarly, it follows from (4.65)

(4.67) $\displaystyle u_{1}(n)=$
$\displaystyle\sum_{m=n}^{\infty}\left(u_{1}(m)-u_{1}(m+1)\right)>\sum_{m=n}^{\infty}\frac{D_{1}}{7\pi}\int_{m+\frac{3}{2}}^{m+\frac{5}{2}}\frac{1}{x^{7}}dx$
$\displaystyle=$
$\displaystyle\frac{D_{1}}{7\pi}\int_{n+\frac{3}{2}}^{\infty}\frac{1}{x^{7}}dx=\frac{D_{1}}{42\pi}\frac{1}{(n+\frac{3}{2})^{6}}.$

This completes the proof of Theorem 4.∎

## References

  * [1] H. Alzer, Inequalities for the constants of Landau and Lebesgue, J. Comput. Appl. Math. 139(2002)215–230.
  * [2] B.C. Berndt, Ramanujan’s Notebooks, Part II, Springer-Verlag, 1999.
  * [3] R. P. Brent and P. Zimmermann, Modern computer arithmetic. Cambridge Monographs on Applied and Computational Mathematics, 18. Cambridge University Press, Cambridge, 2011. xvi+221 pp.
  * [4] L. Brutman, A sharp estimate of the Landau constants, J. Approx. Theory 34 (1982) 217–220.
  * [5] C.-P. Chen and J. Choi, Asymptotic expansions for the constants of Landau and Lebesgue, Advances in mathematics, 254(2014)622–641.
  * [6] C.-P. Chen, J. Choi, Inequalities and asymptotic expansions for the constants of Landau and Lebesgue, RGMIA Res. Rep. Collect. 17 (2014). Article 9, 11. pp. (Available at:http://rgmia.org/papers/v17/v17a09.pdf).
  * [7] C.-P. Chen, J. Choi, Inequalities and asymptotic expansions for the constants of Landau and Lebesgue, Preprint.
  * [8] C.-P. Chen, Approximation formulas for Landau’s constants, J. Math. Anal. Appl. 387 (2012) 916–919.
  * [9] C.-P. Chen, Sharp bounds for the Landau constants, Ramanujan J. 31 (2013) 301–313.
  * [10] C.-P. Chen, L. Li, Two accelerated approximations to the Euler-Mascheroni constant, Sci. Magana, 6 (2010) 161–164.
  * [11] C.P. Chen, C. Mortici, New sequence converging towards the Euler-Mascheroni constant, Comput. Math. Appl. 64 (2012) 391–398.
  * [12] D. Cvijović, J. Klinowski, Inequalities for the Landau constants, Math. Slovaca, 50 (2000) 159–164.
  * [13] D. Cvijović, H.M. Srivastava, Asymptotics of the Landau constants and their relationship with hypergeometric functions, Taiwanese J. Math. 13 (2009) 855–870.
  * [14] Mark W. Coffey and Jonathan Sondow, Rebuttal of Kowalenko s Paper As Concerns the Irrationality of Euler’s Constant $\gamma$, Acta Appl Math (2012) 121:1–3.
  * [15] T.P. Dence, J.B. Dence, A survey of Euler s constant, Math. Mag. 82 (2009) 255–265.
  * [16] D.W. DeTemple, A quicker convergence to Euler’s constant, Amer. Math. Monthly 100 (5) (1993) 468–470.
  * [17] L.P. Falaleev, Inequalities for the Landau constants, Sib. Math. J. 32 (1991) 896–897.
  * [18] I. Gavrea and M. Ivan, Optimal rate of convergence for sequences of a prescribed form. J. Math. Anal. Appl. 402 (2013), no. 1, 35–43.
  * [19] X. Gourdon and P. Sebah, Euler’s constant $\gamma$, http://numbers.computation.free.fr/Constants/constants.html?5000
  * [20] X. Gourdon and P. Sebah, Collection of formulae for the Euler constant. http://numbers.computation.free.fr/Constants/Gamma /gammaFormulas.pdf
  * [21] H. Granath, On inequalities and asymptotic expansions for the Landau constants, J. Math. Anal. Appl. 386 (2012) 738–743.
  * [22] J. Havil, Gamma: Exploring Euler s Constant, Princeton University Press, Princeton, NJ, 2003.
  * [23] Jeffrey C. Lagarias, Euler’s constant: Euler’s work and modern developments. Bull. Amer. Math. Soc. (N.S.) 50 (2013), no. 4, 527–628.
  * [24] E. Landau, Abschätzung der Koeffzientensumme einer Potenzreihe, Arch. Math. Phys. 21 (42-50) (1913) 250–255.
  * [25] Dawei Lu, A new quiker sequence convergent to Euler’s constant, J. Number Theory, 136(2014), 320–329.
  * [26] Dawei Lu , Some quicker classes of sequences convergent to Euler’s constant. Appl. Math. Comput. 232 (2014), 172–177.
  * [27] C. Mortici, On new sequences converging towards the Euler-Mascheroni constant, Comput. Math. Appl. 59 (8) (2010) 2610–2614.
  * [28] C. Mortici, Product approximations via asymptotic integration, Amer. Math. Monthly, 117 (5) (2010) 434–441.
  * [29] C. Mortici, New approximations of the gamma function in terms of the digamma function, Applied Mathematics Letters, 23 (2010) 97–100.
  * [30] C. Mortici, Sharp bounds of the Landau constants, Math. Comp. 80 (2011) 1011–1018.
  * [31] C. Mortici and C.-P. Chen, On the harmonic number expansion by Ramanujan, J. Inequal. Appl. 2013, 2013:222, 10 pp.
  * [32] G. Nemes, A. Nemes, A note on the Landau constants, Appl. Math. Comput. 217 (2011) 8543–8546.
  * [33] G. Nemes, Proofs of two conjectures on the Landau constants, J. Math. Anal. Appl. 388 (2012) 838–844.
  * [34] E.C. Popa, Note of the constants of Landau, Gen. Math. 18 (2010) 113–117.
  * [35] E.C. Popa and N.-A. Secelean, On some inequality for the Landau constants, Taiwanese J. Math. 15 (2011) 1457–1462.
  * [36] J. Wallis, Arithmetica Infinitorum, Oxford, England, 1656; Facsimile of relevant pages available in: J.A. Stedall, Catching Proteus: The collaborations of Wallis and Brouncker. I. Squaring the circle, Notes and Records Roy. Soc. London 54 (3) (2000) 293–316.
  * [37] G.N. Watson, The constants of Landau and Lebesgue, Quart. J. Math. Oxford Ser. 1 (1930) 310–318.
  * [38] R. Wong, Asymptotic Approximations of Integrals, Classics in Applied Mathematics, 34. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. xviii+543 pp. ISBN: 0-89871-497-4
  * [39] Hongmin Xu and Xu You, Continued fraction inequalities for Euler-Mascheroni constant, J. Inequal. Appl. to appear. (Available at: http://arxiv.org/abs/1407.3865v1).
  * [40] Shijun Yang, On an open problem of Chen and Mortici concerning the Euler-Mascheroni constant, J. Math. Anal. Appl. 396 (2012) 689–693.
  * [41] D. Zhao, Some sharp estimates of the constants of Landau and Lebesque, J. Math. Anal. Appl. 349 (2009) 68–73.

Xiaodong Cao

Department of Mathematics and Physics,

Beijing Institute of Petro-Chemical Technology,

Beijing, 102617, P. R. China

e-mail<EMAIL_ADDRESS>

Hongmin Xu

Department of Mathematics and Physics,

Beijing Institute of Petro-Chemical Technology,

Beijing, 102617, P. R. China

e-mail<EMAIL_ADDRESS>

Xu You

Department of Mathematics and Physics,

Beijing Institute of Petro-Chemical Technology,

Beijing, 102617, P. R. China

e-mail<EMAIL_ADDRESS>