# Study of $\psi(3686)\to\pi^{0}h_{c},h_{c}\to\gamma\eta_{c}$ via $\eta_{c}$
exclusive decays

M. Ablikim1, M. N. Achasov5, O. Albayrak3, D. J. Ambrose39, F. F. An1, Q.
An40, J. Z. Bai1, Y. Ban27, J. Becker2, J. V. Bennett17, M. Bertani18A, J. M.
Bian38, E. Boger20,a, O. Bondarenko21, I. Boyko20, R. A. Briere3, V. Bytev20,
X. Cai1, O. Cakir35A, A. Calcaterra18A, G. F. Cao1, S. A. Cetin35B, J. F.
Chang1, G. Chelkov20,a, G. Chen1, H. S. Chen1, J. C. Chen1, M. L. Chen1, S. J.
Chen25, X. Chen27, Y. B. Chen1, H. P. Cheng14, Y. P. Chu1, D. Cronin-
Hennessy38, H. L. Dai1, J. P. Dai1, D. Dedovich20, Z. Y. Deng1, A. Denig19, I.
Denysenko20,b, M. Destefanis43A,43C, W. M. Ding29, Y. Ding23, L. Y. Dong1, M.
Y. Dong1, S. X. Du46, J. Fang1, S. S. Fang1, L. Fava43B,43C, F. Feldbauer2, C.
Q. Feng40, R. B. Ferroli18A, C. D. Fu1, J. L. Fu25, Y. Gao34, C. Geng40, K.
Goetzen7, W. X. Gong1, W. Gradl19, M. Greco43A,43C, M. H. Gu1, Y. T. Gu9, Y.
H. Guan6, A. Q. Guo26, L. B. Guo24, Y. P. Guo26, Y. L. Han1, F. A. Harris37,
K. L. He1, M. He1, Z. Y. He26, T. Held2, Y. K. Heng1, Z. L. Hou1, H. M. Hu1,
T. Hu1, G. M. Huang15, G. S. Huang40, J. S. Huang12, X. T. Huang29, Y. P.
Huang1, T. Hussain42, C. S. Ji40, Q. Ji1, Q. P. Ji26,c, X. B. Ji1, X. L. Ji1,
L. L. Jiang1, X. S. Jiang1, J. B. Jiao29, Z. Jiao14, D. P. Jin1, S. Jin1, F.
F. Jing34, N. Kalantar-Nayestanaki21, M. Kavatsyuk21, W. Kuehn36, W. Lai1, J.
S. Lange36, C. H. Li1, Cheng Li40, Cui Li40, D. M. Li46, F. Li1, G. Li1, H. B.
Li1, J. C. Li1, K. Li10, Lei Li1, Q. J. Li1, S. L. Li1, W. D. Li1, W. G. Li1,
X. L. Li29, X. N. Li1, X. Q. Li26, X. R. Li28, Z. B. Li33, H. Liang40, Y. F.
Liang31, Y. T. Liang36, G. R. Liao34, X. T. Liao1, B. J. Liu1, C. L. Liu3, C.
X. Liu1, C. Y. Liu1, F. H. Liu30, Fang Liu1, Feng Liu15, H. Liu1, H. H. Liu13,
H. M. Liu1, H. W. Liu1, J. P. Liu44, K. Y. Liu23, Kai Liu6, P. L. Liu29, Q.
Liu6, S. B. Liu40, X. Liu22, Y. B. Liu26, Z. A. Liu1, Zhiqiang Liu1, Zhiqing
Liu1, H. Loehner21, G. R. Lu12, H. J. Lu14, J. G. Lu1, Q. W. Lu30, X. R. Lu6,
Y. P. Lu1, C. L. Luo24, M. X. Luo45, T. Luo37, X. L. Luo1, M. Lv1, C. L. Ma6,
F. C. Ma23, H. L. Ma1, Q. M. Ma1, S. Ma1, T. Ma1, X. Y. Ma1, Y. Ma11, F. E.
Maas11, M. Maggiora43A,43C, Q. A. Malik42, Y. J. Mao27, Z. P. Mao1, J. G.
Messchendorp21, J. Min1, T. J. Min1, R. E. Mitchell17, X. H. Mo1, C. Morales
Morales11, C. Motzko2, N. Yu. Muchnoi5, H. Muramatsu39, Y. Nefedov20, C.
Nicholson6, I. B. Nikolaev5, Z. Ning1, S. L. Olsen28, Q. Ouyang1, S.
Pacetti18B, J. W. Park28, M. Pelizaeus37, H. P. Peng40, K. Peters7, J. L.
Ping24, R. G. Ping1, R. Poling38, E. Prencipe19, M. Qi25, S. Qian1, C. F.
Qiao6, X. S. Qin1, Y. Qin27, Z. H. Qin1, J. F. Qiu1, K. H. Rashid42, G. Rong1,
X. D. Ruan9, A. Sarantsev20,d, B. D. Schaefer17, J. Schulze2, M. Shao40, C. P.
Shen37,e, X. Y. Shen1, H. Y. Sheng1, M. R. Shepherd17, X. Y. Song1, S.
Spataro43A,43C, B. Spruck36, D. H. Sun1, G. X. Sun1, J. F. Sun12, S. S. Sun1,
Y. J. Sun40, Y. Z. Sun1, Z. J. Sun1, Z. T. Sun40, C. J. Tang31, X. Tang1, I.
Tapan35C, E. H. Thorndike39, D. Toth38, M. Ullrich36, G. S. Varner37, B.
Wang9, B. Q. Wang27, D. Wang27, D. Y. Wang27, K. Wang1, L. L. Wang1, L. S.
Wang1, M. Wang29, P. Wang1, P. L. Wang1, Q. Wang1, Q. J. Wang1, S. G. Wang27,
X. L. Wang40, Y. D. Wang40, Y. F. Wang1, Y. Q. Wang29, Z. Wang1, Z. G. Wang1,
Z. Y. Wang1, D. H. Wei8, J. B. Wei27, P. Weidenkaff19, Q. G. Wen40, S. P.
Wen1, M. Werner36, U. Wiedner2, L. H. Wu1, N. Wu1, S. X. Wu40, W. Wu26, Z.
Wu1, L. G. Xia34, Z. J. Xiao24, Y. G. Xie1, Q. L. Xiu1, G. F. Xu1, G. M. Xu27,
H. Xu1, Q. J. Xu10, X. P. Xu32, Z. R. Xu40, F. Xue15, Z. Xue1, L. Yan40, W. B.
Yan40, Y. H. Yan16, H. X. Yang1, Y. Yang15, Y. X. Yang8, H. Ye1, M. Ye1, M. H.
Ye4, B. X. Yu1, C. X. Yu26, H. W. Yu27, J. S. Yu22, S. P. Yu29, C. Z. Yuan1,
Y. Yuan1, A. A. Zafar42, A. Zallo18A, Y. Zeng16, B. X. Zhang1, B. Y. Zhang1,
C. Zhang25, C. C. Zhang1, D. H. Zhang1, H. H. Zhang33, H. Y. Zhang1, J. Q.
Zhang1, J. W. Zhang1, J. Y. Zhang1, J. Z. Zhang1, S. H. Zhang1, X. J. Zhang1,
X. Y. Zhang29, Y. Zhang1, Y. H. Zhang1, Y. S. Zhang9, Z. P. Zhang40, Z. Y.
Zhang44, G. Zhao1, H. S. Zhao1, J. W. Zhao1, K. X. Zhao24, Lei Zhao40, Ling
Zhao1, M. G. Zhao26, Q. Zhao1, Q. Z. Zhao9,f, S. J. Zhao46, T. C. Zhao1, X. H.
Zhao25, Y. B. Zhao1, Z. G. Zhao40, A. Zhemchugov20,a, B. Zheng41, J. P.
Zheng1, Y. H. Zheng6, B. Zhong1, J. Zhong2, Z. Zhong9,f, L. Zhou1, X. K.
Zhou6, X. R. Zhou40, C. Zhu1, K. Zhu1, K. J. Zhu1, S. H. Zhu1, X. L. Zhu34, Y.
C. Zhu40, Y. M. Zhu26, Y. S. Zhu1, Z. A. Zhu1, J. Zhuang1, B. S. Zou1, J. H.
Zou1
(BESIII Collaboration)
1 Institute of High Energy Physics, Beijing 100049, P. R. China
2 Bochum Ruhr-University, 44780 Bochum, Germany
3 Carnegie Mellon University, Pittsburgh, PA 15213, USA
4 China Center of Advanced Science and Technology, Beijing 100190, P. R. China
5 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
6 Graduate University of Chinese Academy of Sciences, Beijing 100049, P. R.
China
7 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
8 Guangxi Normal University, Guilin 541004, P. R. China
9 GuangXi University, Nanning 530004,P.R.China
10 Hangzhou Normal University, Hangzhou 310036, P. R. China
11 Helmholtz Institute Mainz, J.J. Becherweg 45,D 55099 Mainz,Germany
12 Henan Normal University, Xinxiang 453007, P. R. China
13 Henan University of Science and Technology, Luoyang 471003, P. R. China
14 Huangshan College, Huangshan 245000, P. R. China
15 Huazhong Normal University, Wuhan 430079, P. R. China
16 Hunan University, Changsha 410082, P. R. China
17 Indiana University, Bloomington, Indiana 47405, USA
18 (A)INFN Laboratori Nazionali di Frascati, Frascati, Italy; (B)INFN and
University of Perugia, I-06100, Perugia, Italy
19 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45, 55099
Mainz, Germany
20 Joint Institute for Nuclear Research, 141980 Dubna, Russia
21 KVI/University of Groningen, 9747 AA Groningen, The Netherlands
22 Lanzhou University, Lanzhou 730000, P. R. China
23 Liaoning University, Shenyang 110036, P. R. China
24 Nanjing Normal University, Nanjing 210046, P. R. China
25 Nanjing University, Nanjing 210093, P. R. China
26 Nankai University, Tianjin 300071, P. R. China
27 Peking University, Beijing 100871, P. R. China
28 Seoul National University, Seoul, 151-747 Korea
29 Shandong University, Jinan 250100, P. R. China
30 Shanxi University, Taiyuan 030006, P. R. China
31 Sichuan University, Chengdu 610064, P. R. China
32 Soochow University, Suzhou 215006, China
33 Sun Yat-Sen University, Guangzhou 510275, P. R. China
34 Tsinghua University, Beijing 100084, P. R. China
35 (A)Ankara University, Ankara, Turkey; (B)Dogus University, Istanbul,
Turkey; (C)Uludag University, Bursa, Turkey
36 Universitaet Giessen, 35392 Giessen, Germany
37 University of Hawaii, Honolulu, Hawaii 96822, USA
38 University of Minnesota, Minneapolis, MN 55455, USA
39 University of Rochester, Rochester, New York 14627, USA
40 University of Science and Technology of China, Hefei 230026, P. R. China
41 University of South China, Hengyang 421001, P. R. China
42 University of the Punjab, Lahore-54590, Pakistan
43 (A)University of Turin, Turin, Italy; (B)University of Eastern Piedmont,
Alessandria, Italy; (C)INFN, Turin, Italy
44 Wuhan University, Wuhan 430072, P. R. China
45 Zhejiang University, Hangzhou 310027, P. R. China
46 Zhengzhou University, Zhengzhou 450001, P. R. China
a also at the Moscow Institute of Physics and Technology, Moscow, Russia
b on leave from the Bogolyubov Institute for Theoretical Physics, Kiev,
Ukraine
c Nankai University, Tianjin,300071,China
d also at the PNPI, Gatchina, Russia
e now at Nagoya University, Nagoya, Japan
f Guangxi University,Nanning,530004,China

###### Abstract

The process $\psi(3686)\to\pi^{0}h_{c},h_{c}\to\gamma\eta_{c}$ has been
studied with a data sample of $106\pm 4$ million $\psi(3686)$ events collected
with the BESIII detector at the BEPCII storage ring. The mass and width of the
$P$-wave charmonium spin-singlet state $h_{c}(^{1}P_{1})$ are determined by
simultaneously fitting distributions of the $\pi^{0}$ recoil mass for 16
exclusive $\eta_{c}$ decay modes. The results, $M(h_{c})=3525.31\pm
0.11~{}{\rm(stat.)}\pm 0.14~{}{\rm(syst.)}$ MeV/$c^{2}$ and
$\Gamma(h_{c})=0.70\pm 0.28\pm 0.22$ MeV, are consistent with and more precise
than previous measurements. We also determine the branching ratios for the 16
exclusive $\eta_{c}$ decay modes, five of which have not been measured
previously. New measurements of the $\eta_{c}$ line-shape parameters in the
$E1$ transition $h_{c}\to\gamma\eta_{c}$ are made by selecting candidates in
the $h_{c}$ signal sample and simultaneously fitting the hadronic mass spectra
for the 16 $\eta_{c}$ decay channels. The resulting $\eta_{c}$ mass and width
values are $M(\eta_{c})=2984.49\pm 1.16\pm 0.52$ MeV/$c^{2}$ and
$\Gamma(\eta_{c})=36.4\pm 3.2\pm 1.7$ MeV.

###### pacs:

14.40.Pq, 13.25.Gv, 12.38.Qk

##  I INTRODUCTION

Studies of charmonium states have played an important role in understanding
Quantum Chromodynamics (QCD) because of their relative immunity from
complications like relativistic effects and the large value of the strong
coupling constant $\alpha_{s}$. In the QCD potential model Eichten:1978tg ,
the spin-independent one-gluon exchange part of the $c\bar{c}$ interaction has
been defined quite well by existing experimental data. The spin dependence of
the $c\bar{c}$ potential is not as well understood. Until recently, the only
well-measured hyperfine splitting was that for the $1S$ states of charmonium,
$\Delta M_{hf}(1S)=M(J/\psi)-M(\eta_{c})=116\pm 1$ MeV/$c^{2}$ ref:PDG_2012 .
In the past several years BaBar etac_prime_babar , Belle etac_prime_belle ,
CLEO etac_prime_cleo , and BESIII etac_prime_bes  have succeeded in
identifying $\eta_{c}(2S)$ and have measured $\Delta
M_{hf}(2S)=M(\psi(3686))-M(\eta_{c}(2S))=47\pm 1$ MeV/$c^{2}$.

Of the charmonium states below $D\bar{D}$ threshold, the $h_{c}(1^{1}P_{1})$
is experimentally the least accessible because it cannot be produced directly
in $e^{+}e^{-}$ annihilation or in the electric-dipole transition of a
$J^{PC}=1^{--}$ charmonium state. Limited statistics and photon-detection
challenges also were major obstacles to the observation of $h_{c}$ in
charmonium transitions. The precise measurement of $h_{c}$ properties is
important because a comparison of its mass with the masses of the $3P$ states
($\chi_{cJ}$) provides much-needed information about the spin dependence of
the $c\bar{c}$ interaction. According to QCD potential models, a small spin-
spin interaction would lead to hyperfine splitting of the $P$ states that is
close to zero swanson .

The first evidence of the $h_{c}$ state was reported by the Fermilab E760
experiment ref:E760hc  and was based on the process
$p\bar{p}\to\pi^{0}J/\psi$. This result was subsequently excluded by the
successor experiment E835 ref:E835hc , which investigated the same reaction
with a larger data sample. E835 also studied $p\bar{p}\to
h_{c}\to\gamma\eta_{c}$, in this case finding an $h_{c}$ signal. Soon after
this the CLEO collaboration observed the $h_{c}$ and measured its mass
ref:cleohc05 ; ref:cleohc08  by studying the decay chain
$\psi(3686)\to\pi^{0}h_{c},h_{c}\to\gamma\eta_{c}$ in $e^{+}e^{-}$ collisions.
CLEO subsequently presented evidence for $h_{c}$ decays to multi-pion final
states ref:cleohc09 . Recently, the BESIII collaboration used inclusive
methods to make the first measurements of the absolute branching ratios
$\mathcal{B}(\psi(3686)\to\pi^{0}h_{c})=(8.4\pm 1.3\pm 1.0)\times 10^{-4}$ and
$\displaystyle\mathcal{B}(h_{c}\to\gamma\eta_{c})=(54.3\pm 6.7\pm 5.2)\%$
ref:bes3hc10 . CLEO has confirmed the BESIII results Ge:2011kq  and also
observed $h_{c}$ in $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ at $\sqrt{s}=4170$ MeV,
demonstrating a new prolific source of $h_{c}$ CLEO:2011aa .

$\eta_{c}(1S)$ is the lowest-lying $S$-wave spin-singlet charmonium state.
Although it has been known for about thirty years Himel:1980dj , its resonance
parameters are still interesting. For a long time, measurements of the
$\eta_{c}$ width from B-factories and from charmonium transitions were
inconsistent ref:PDG_2012 . The discrepancies can be attributed to poor
statistics and inadequate consideration of interference between $\eta_{c}$
decays and non-resonant backgrounds. Besides, the $\eta_{c}$ line shape also
could be distorted by photon energy dependence in the $M1$(or $E1$)
transition, which will affect the resonance parameters measurement. Recent
studies by Belle, Babar, CLEO, and BESIII Vinokurova:2011dy ;
delAmoSanchez:2011bt ; Mitchell:2008aa ; BESIII:2011ab , with large data
samples and careful consideration of interference, obtained similar $\eta_{c}$
width and mass results in two-photon-fusion production and $\psi(3686)$
decays. The $h_{c}\to\gamma\eta_{c}$ transition can provide a new laboratory
to study $\eta_{c}$ properties. The $\eta_{c}$ line shape in the $E1$
transition $h_{c}\to\gamma\eta_{c}$ should not be as distorted as in other
charmonium decays, because non-resonant interfering backgrounds to the
dominant transition are small.

In this paper, we report new measurements of the mass and width of the $h_{c}$
and $\eta_{c}$, and of the branching ratios
$\mathcal{B}_{1}(\psi(3686)\to\pi^{0}h_{c})\times\mathcal{B}_{2}(h_{c}\to\gamma\eta_{c})\times\mathcal{B}_{3}(\eta_{c}\to
X_{i})$ and $\mathcal{B}_{3}(\eta_{c}\to X_{i})$, via the sequential process
$\psi(3686)\to\pi^{0}h_{c},~{}h_{c}\to\gamma\eta_{c},~{}\eta_{c}\to X_{i}$. In
this reaction $X_{i}$ signifies 16 exclusive hadronic final states:
$p\bar{p}$, $2(\pi^{+}\pi^{-})$, $2(K^{+}K^{-})$, $K^{+}K^{-}\pi^{+}\pi^{-}$,
$p\bar{p}\pi^{+}\pi^{-}$, $3(\pi^{+}\pi^{-})$, $K^{+}K^{-}2(\pi^{+}\pi^{-})$,
$K^{+}K^{-}\pi^{0}$, $p\bar{p}\pi^{0}$, $K^{0}_{S}K^{\pm}\pi^{\mp}$,
$K^{0}_{S}K^{\pm}\pi^{\mp}\pi^{\pm}\pi^{\mp}$, $\pi^{+}\pi^{-}\eta$,
$K^{+}K^{-}\eta$, $2(\pi^{+}\pi^{-})\eta$, $\pi^{+}\pi^{-}\pi^{0}\pi^{0}$, and
$2(\pi^{+}\pi^{-})\pi^{0}\pi^{0}$. Here $K_{S}^{0}$ is reconstructed in its
$\pi^{+}\pi^{-}$ decays, and $\eta$ in its $\gamma\gamma$ final state. The
data sample of $\psi(3686)$ events was collected with the BESIII detector at
the BEPCII $e^{+}e^{-}$ storage ring.

The remainder of this paper is structured as follows: Sect. II describes the
experiment and data sample; Sect. III presents the event selection and
background analysis; Sect. IV discusses the extraction of $h_{c}$ and
$\eta_{c}$ results; Sect. V describes the estimation of systematic
uncertainties; and Sec. VI provides a summary and discussion of the results.

##  II Experiment and Data Sample

BEPCII is a two-ring $e^{+}e^{-}$ collider designed for a peak luminosity of
$10^{33}$ cm${}^{-2}s^{-1}$ at a beam current of 0.93 A per beam. The
cylindrical core of the BESIII detector consists of a helium-gas-based drift
chamber (MDC) for charged-particle tracking and particle identification by
d$E$/d$x$, a plastic scintillator time-of-flight system (TOF) for additional
particle identification, and a 6240-crystal CsI(Tl) Electromagnetic
Calorimeter (EMC) for electron identification and photon detection. These
components are all enclosed in a superconducting solenoidal magnet providing a
1.0-T magnetic field. The solenoid is supported by an octagonal flux-return
yoke with resistive-plate-counter muon detector modules (MUC) interleaved with
steel. The geometrical acceptance for charged tracks and photons is $93\%$ of
$4\pi$, and the resolutions for charged-track momentum and photon energy at 1
GeV are $0.5\%$ and $2.5\%$, respectively. More details on the features and
capabilities of BESIII are provided in Ref. ref:bes3 .

The data sample for this analysis consists of 156.4 pb-1 of $e^{+}e^{-}$
annihilation data collected at a center-of-mass energy of 3.686 GeV, the peak
of the $\psi(3686)$ resonance. By measuring the production of multihadronic
events we determine the number of $\psi(3686)$ decays in the sample to be
$(1.06\pm 0.04)\times 10^{8}$, where the uncertainty is dominated by
systematics ref:psiptotnumber . An additional 42 $\rm pb^{-1}$ of data were
collected at a center-of-mass energy of 3.65 GeV to determine non-resonant
continuum background contributions.

The optimization of the event selection and the estimation of physics
backgrounds are performed with simulated Monte Carlo (MC) samples. A
GEANT4-based Agostinelli:2002hh ; Allison:2006ve  detector simulation package
is used to model the detector response. Signal and background processes are
generated with specialized models that have been packaged and customized for
BESIII ref:bes3gen . The $\psi(3686)$ resonance is generated by KKMC ref:kkmc
, and EvtGen ref:evtgen  is used to model events for
$\psi(3686)\to\pi^{0}h_{c}$ and for exclusive backgrounds in $\psi(3686)$
decays. An inclusive sample (100 million events) is used to simulate hadronic
background processes. Known $\psi(3686)$ decay modes are generated with
EvtGen, using branching ratios set to world-average values ref:PDG_2012 . The
remaining $\psi(3686)$ decay modes are generated by LUNDCHARM ref:bes3gen ,
which is based on JETSET Sjostrand:1993yb  and tuned for the charm-energy
region. The decays $\psi(3686)\to\pi^{0}h_{c}$ are excluded from this sample.

The $\psi(3686)\to\pi^{0}h_{c}$ events are generated with an $h_{c}$ mass of
$3525.28$ MeV/$c^{2}$ and a width equal to that of the $\chi_{c1}$ (0.9 MeV).
The $E1$ transition $h_{c}\to\gamma\eta_{c}$ is generated with an angular
distribution in the $h_{c}$ rest frame of $1+\cos^{2}\theta^{*}$, where
$\theta^{*}$ is the angle of the E1 photon with respect to the beam direction
in the $h_{c}$ rest frame. Multi-body $\eta_{c}$ decays are generated
according to phase space.

##  III Event selection and background analysis

For $\psi(3686)\to\pi^{0}h_{c}$, $h_{c}\to\gamma\eta_{c}$, the expected
$\pi^{0}$ momentum is $P_{\pi^{0}}\simeq 84$ MeV/$c$, and the $E1$ transition
photon emitted in $h_{c}\to\gamma\eta_{c}$ has an expected energy of
$E(\gamma_{\rm E1})\simeq 503$ MeV in the $h_{c}$ rest frame. We select signal
candidates by demanding consistency with these expectations. For the selected
candidates we then compute the $\pi^{0}$ recoil mass. Fitting the distribution
of this recoil mass for the full event sample gives the results for the
$h_{c}$ resonance parameters and signal yields.

Charged tracks in BESIII are reconstructed from MDC hits within a polar-angle
($\theta$) acceptance range of $|\cos\theta|<0.93$. To optimize the momentum
measurement, we require that these tracks be reconstructed to pass within 10
cm of the interaction point in the beam direction and within 1 cm in the plane
perpendicular to the beam. Tracks used in reconstructing $K^{0}_{S}$ decays
are exempted from these requirements.

A vertex fit constrains charged tracks to a common production vertex, which is
updated on a run-by-run basis. For each charged track, time-of-flight and
d$E$/d$x$ information is combined to compute particle identification (PID)
confidence levels for the pion, kaon, and proton hypotheses. The track is
assigned to the particle type with the highest confidence level.

Electromagnetic showers are reconstructed by clustering EMC crystal energies.
Efficiency and energy resolution are improved by including energy deposits in
nearby TOF counters. A photon candidate is defined as a shower with an energy
deposit of at least 25 MeV in the “barrel” region ($|\cos\theta|<0.8$), or of
at least 50 MeV in the “end-cap” region ($0.86<|\cos\theta|<0.92$). Showers at
angles intermediate between the barrel and the end-cap are not well measured
and are rejected. An additional requirement on the EMC hit timing suppresses
electronic noise and energy deposits unrelated to the event.

A candidate $\pi^{0}$($\eta$) is reconstructed from pairs of photons with an
invariant mass in the range $|M_{\gamma\gamma}-m_{\pi^{0}}|<15\,\rm MeV/c^{2}$
($|M_{\gamma\gamma}-m_{\eta}|<15\,\rm MeV/c^{2}$) ref:PDG_2012 . A one-
constraint (1-C) kinematic fit is performed to improve the energy resolution,
with the $M(\gamma\gamma)$ constrained to the known $\pi^{0}$($\eta$) mass.

We reconstruct $K^{0}_{S}\to\pi^{+}\pi^{-}$ candidates using pairs of
oppositely charged tracks with an invariant mass in the range
$|M_{\pi\pi}-m_{K^{0}_{S}}|<20\,\rm MeV/c^{2}$, where $m_{K^{0}_{S}}$ is the
known $K^{0}_{S}$ mass ref:PDG_2012 . To reject random $\pi^{+}\pi^{-}$
combinations, a secondary-vertex fitting algorithm is employed to impose the
kinematic constraint between the production and decay vertices
ref:ks0-reconstruction . Accepted $K^{0}_{S}$ candidates are required to have
a decay length of at least twice the vertex resolution.

The $\eta_{c}$ candidate is reconstructed in 16 exclusive decay modes, and the
event is accepted or rejected based on consistency with the
$h_{c}\to\gamma\eta_{c}$ hypothesis. Specifically, the reconstructed mass
$M(\eta_{c})$ is required to be between 2.900 GeV/$c^{2}$ and 3.050
GeV/$c^{2}$, and the transition-photon energy is required to be between 0.450
GeV and 0.550 GeV. Events passing this selection are subjected to a
4-constraint (4-C) kinematic fit to take advantage of energy-momentum
conservation. Because of differing signal/background characteristics, we
individually optimize requirements on $\chi^{2}_{4C}$, the $\chi^{2}$ of the
4-C fit, for the 16 $\eta_{c}$ channels. If multiple $\eta_{c}$ candidates are
found in an event, the one with the smallest value of $\chi^{2}=\chi^{2}_{\rm
4C}+\chi^{2}_{\rm 1C}+\chi^{2}_{\rm pid}+\chi^{2}_{\rm vertex}$ is accepted,
where $\chi^{2}_{\rm 1C}$ is the $\chi^{2}$ of the 1-C fit of the
$\pi^{0}$($\eta$), $\chi^{2}_{\rm pid}$ is the PID $\chi^{2}$ summation for
all charged tracks included in the $h_{c}$ candidate, and $\chi^{2}_{\rm
vertex}$ is the $\chi^{2}$ of the $K_{S}^{0}$ vertex fit. If there is no
$\pi^{0}/\eta$ ($K_{S}^{0}$) in an event, the corresponding $\chi^{2}_{\rm
1C}$ ($\chi^{2}_{\rm vertex}$) is set to zero.

Based on studies of the inclusive MC sample, we identified several background
processes with potential to reduce the precision of measurements made with
specific $\eta_{c}$ exclusive channels because of sizable low-energy $\pi^{0}$
production. The processes and suppression procedures are as follows:

  * •

$\psi(3686)\to\pi^{+}\pi^{-}J/\psi$

The mass $M_{X}$ of the system recoiling against the $\pi^{+}\pi^{-}$ in
$\psi(3686)\to\pi^{+}\pi^{-}X$ is calculated and the candidate is rejected if
$M_{X}$ is within $\pm 12$ MeV/$c^{2}$ of the known $J/\psi$ mass.

  * •

$\psi(3686)\to\pi^{0}\pi^{0}J/\psi$

The mass $M_{X}$ of the system recoiling against the $\pi^{0}\pi^{0}$ in
$\psi(3686)\to\pi^{0}\pi^{0}X$ is calculated and the candidate is rejected if
$M_{X}$ is within $\pm 15$ MeV/$c^{2}$ of the known $J/\psi$ mass for all
$\eta_{c}$ final states except $\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{0}\pi^{0}$.
For this mode the lower $\pi^{0}$ momentum leads to recoil masses near 3.1
GeV/$c^{2}$, so the exclusion window is narrowed to $\pm 10$ MeV/$c^{2}$.

  * •

$\psi(3686)\to\gamma\chi_{c2}$

A candidate is rejected if it includes a $\pi^{0}$ for which either daughter
photon has an energy within $\pm 5$ MeV of that expected for the $\psi(3686)$
radiative transition to $\chi_{c2}$ (128 MeV).

  * •

$E1$ photon candidates that are $\pi^{0}$ decay products

A candidate is rejected if its $E1$ photon can be combined with another photon
in the event to form a $\pi^{0}$ within a mass window of $\pm 10$ MeV/$c^{2}$.

  * •

$\pi^{0}$ candidates that are from $\eta\to\pi^{+}\pi^{-}\pi^{0}$

Masses $M(\pi^{+}\pi^{-}\pi^{0})$ are calculated for all possible combinations
in the event and the candidate is rejected if any combination has a mass
within $\pm 15$ MeV/$c^{2}$ of the known $\eta$ mass.

Decisions about whether to apply a requirement to a particular $\eta_{c}$ mode
and the optimization of the $\chi^{2}_{4C}$ and PID requirements were made on
a channel-by-channel basis. The figure-of-merit used was
$\mathcal{S}=N_{S}/\sqrt{N_{S}+N_{B}}$, where $N_{S}$ is the number of signal
and $N_{B}$ the number of background candidates. PDG values ref:PDG_2012  are
used for the input $\eta_{c}$ branching ratios, and for channels not tabulated
by the PDG we estimate branching ratios based on conjugate channels or other
similar modes. The optimized selection criteria are listed in Table 1, in
which the $N(p),~{}N(\pi)~{}{\rm and}~{}N(K)$ denote the numbers of identified
protons, pions and kaons in a event.

Table 1: Event-selection requirements for each exclusive channel.

Mode | $\chi^{2}_{\rm~{}4C}$ | PID |  $\pi^{+}\pi^{-}J/\psi$ veto |  $\pi^{0}\pi^{0}J/\psi$ veto |  $\gamma\chi_{c2}$ veto |  $\pi^{0}$ veto for $E1$ photon |  $\eta\to\pi^{+}\pi^{-}\pi^{0}$ veto
---|---|---|---|---|---|---|---
$p\bar{p}$ | 30 | $N(p)\geq 1$ | no | no | yes | no | no
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ | 60 | $N(\pi)\geq 3$ | yes | yes | yes | yes | yes
$K^{+}K^{-}K^{+}K^{-}$ | 60 | $N(K)\geq 3$ | no | no | no | yes | no
$K^{+}K^{-}\pi^{+}\pi^{-}$ | 40 | $N(K)\geq 2,N(\pi)\geq 0$ | yes | yes | yes | yes | yes
$p\bar{p}\pi^{+}\pi^{-}$ | 30 | $N(p)\geq 2,N(\pi)\geq 0$ | yes | yes | yes | yes | yes
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{-}\pi^{-}$ | 50 | $N(\pi)\geq 4$ | yes | yes | no | yes | yes
$K^{+}K^{-}\pi^{+}\pi^{-}\pi^{-}\pi^{-}$ | 70 | $N(K)\geq 2,N(\pi)\geq 2$ | yes | no | no | no | no
$K^{+}K^{-}\pi^{0}$ | 50 | $N(K)\geq 1$ | no | yes | no | no | no
$p\bar{p}\pi^{0}$ | 40 | $N(p)\geq 1$ | no | yes | yes | yes | no
$K^{0}_{S}K^{\pm}\pi^{\mp}$ | 70 | $-$ | no | no | no | no | yes
$K^{0}_{S}K^{\pm}\pi^{\mp}\pi^{\pm}\pi^{\mp}$ | 50 | $-$ | no | no | yes | no | no
$\pi^{+}\pi^{-}\eta$ | 50 | $-$ | no | no | no | yes | no
$K^{+}K^{-}\eta$ | 70 | $N(K)\geq 1$ | no | no | yes | yes | no
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\eta$ | 30 | $-$ | yes | no | no | yes | no
$\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ | 40 | $-$ | yes | yes | yes | yes | yes
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ | 60 | $-$ | yes | yes | no | yes | no

The $\pi^{0}$ recoil mass spectra for events passing these requirements show
clear $h_{c}$ signals in the expected range, as can be seen in Fig. 1. No
peaking backgrounds in the signal region are found in the 100-million-event
inclusive MC sample, in the continuum data sample taken at $\sqrt{s}=3.65$
GeV, or in $\eta_{c}$-candidate-mass side-band distributions.

Figure 1: The $\pi^{0}$ recoil mass spectrum in
$\psi(3686)\to\pi^{0}{h}_{c},{h}_{c}\to\gamma\eta_{c}$, $\eta_{c}\to X_{i}$
summed over the 16 final states $X_{i}$. The dots with error bars represent
the $\pi^{0}$ recoil mass spectrum in data. The solid line shows the total fit
function and the dashed line is the background component of the fit.

##  IV EXTRACTION OF YIELDS AND RESONANCE PARAMETERS

We obtain the $h_{c}$ mass, width and branching ratios from simultaneous fits
to the $\pi^{0}$ recoil mass distributions for the 16 exclusive $\eta_{c}$
decay modes. Here only 1-C kinematic fits with $\pi^{0}$ mass hypothesis are
used to improve the energy resolution. The 4C-fits used in event selection are
not used in the $\pi^{0}$ recoil mass reconstruction, because the energy
resolution of the signal $\pi^{0}$ in 4C-fits is not as good as in the
1C-fits, according to a MC study. From the same data sample we also determine
the $\eta_{c}$ resonance parameters by fitting the 16 invariant-mass spectra
of the hadronic system accompanying the transition photon in
$h_{c}\to\gamma\eta_{c}$.

###  IV.1 Fitting the $h_{c}$ signal

To extract the $h_{c}$ resonant parameters and the yield for each $\eta_{c}$
decay channel, the 16 $\pi^{0}$ recoil mass distributions are fitted
simultaneously with a binned maximum likelihood method. A Breit-Wigner
function convolved with the instrumental resolution is used to describe the
signal shape. An efficiency correction is not needed because of the small
$h_{c}$ width and the good $\pi^{0}$ mass resolution. The resolution function
is channel-dependent and is obtained from MC simulation. The parameters
$M(h_{c})$ and $\Gamma(h_{c})$ of the Breit-Wigner function are constrained to
be the same for all 16 channels, which is essential for the decay modes with
low statistics. For the recoil mass fit to each channel, the background shape
is obtained from the $\eta_{c}$ mass side bands (2300$-$2700, 3070$-$3200
MeV/$c^{2}$), and the signal and the background normalizations for each mode
are allowed to float. The summed and mode-by-mode fit results are shown in
Figs. 1 and 2, respectively. The $\chi^{2}$ per degree of freedom for this fit
is 1.60, where sparsely populated bins are combined so that there are at least
seven counts per bin in the $\chi^{2}$ calculation. The parameters of the
$h_{c}$ resonance are determined to be $M(h_{c})=3525.31\pm 0.11$ MeV/$c^{2}$
and $\Gamma(h_{c})=0.70\pm 0.28$ MeV, where the errors are statistical only.

Figure 2: The simultaneously fitted $\pi^{0}$ recoil mass spectra in
$\psi(3686)\to\pi^{0}{h}_{c},{h}_{c}\to\gamma\eta_{c}$, $\eta_{c}\to X_{i}$
for the 16 final states $X_{i}$.

The MC-determined selection efficiency $\epsilon_{i}$ and yield $N_{i}$ for
each $\eta_{c}$ decay mode are listed in Table 2.

Table 2: MC-determined efficiencies $\epsilon_{i}$ and yields $N_{i}$ for $\psi(3686)\to\pi^{0}h_{c},h_{c}\to\gamma\eta_{c},\eta_{c}\to X_{i}$, where $X_{i}$ refers to the 16 final states . Mode | $\epsilon_{i}(\%)$ | $N_{i}$
---|---|---
$p\bar{p}$ | 22.2 | 15.3 $\pm$ 4.5
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ | 12.6 | 100.3 $\pm$ 11.3
$K^{+}K^{-}K^{+}K^{-}$ | 6.6 | 6.6 $\pm$ 2.6
$K^{+}K^{-}\pi^{+}\pi^{-}$ | 8.7 | 38.4 $\pm$ 7.0
$p\bar{p}\pi^{+}\pi^{-}$ | 7.8 | 19.0 $\pm$ 5.4
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{-}\pi^{-}$ | 5.4 | 50.5 $\pm$ 9.0
$K^{+}K^{-}\pi^{+}\pi^{-}\pi^{-}\pi^{-}$ | 2.7 | 10.3 $\pm$ 4.9
$K^{+}K^{-}\pi^{0}$ | 11.4 | 54.9 $\pm$ 9.2
$p\bar{p}\pi^{0}$ | 8.9 | 14.4 $\pm$ 4.6
$K^{0}_{S}K^{\pm}\pi^{\mp}$ | 8.9 | 107.1 $\pm$ 11.8
$K^{0}_{S}K^{\pm}\pi^{\mp}\pi^{\pm}\pi^{\mp}$ | 3.4 | 43.3 $\pm$ 8.0
$\pi^{+}\pi^{-}\eta$ | 4.3 | 32.9 $\pm$ 6.7
$K^{+}K^{-}\eta$ | 3.0 | 6.7 $\pm$ 3.2
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\eta$ | 1.9 | 38.6 $\pm$ 7.6
$\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ | 5.5 | 118.4 $\pm$ 12.8
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ | 2.2 | 175.2 $\pm$ 17.3
Total | - | 831.9$\pm$35.0

Based on these numbers, we can calculate the product branching ratios
$\mathcal{B}_{1}(\psi(3686)\to\pi^{0}h_{c})\times\mathcal{B}_{2}(h_{c}\to\gamma\eta_{c})\times\mathcal{B}_{3}(\eta_{c}\to
X_{i})$. The branching ratio for $\eta_{c}\to X_{i}$ for each of the 16 final
states $X_{i}$ can then be obtained by combining our measurements with
$\mathcal{B}_{1}(\psi(3686)\to\pi^{0}h_{c})\times\mathcal{B}_{2}(h_{c}\to\gamma\eta_{c})=(4.36\pm
0.42)\times 10^{-4}$, the average of two recent measurements by CLEO
ref:cleohc08  and BESIII ref:bes3hc10 . These branching ratios, with both
statistical and systematic errors, are presented in Section VI.

###  IV.2 Measurement of $\eta_{c}$ resonance parameters

In addition to determining the $h_{c}$ resonance parameters, we can also
measure the $\eta_{c}$ mass and width with the same event sample. The decay
chain $h_{c}\to\gamma\eta_{c}$, $\eta_{c}\to X_{i}$ is reconstructed and
kinematically fitted in the 16 $\eta_{c}$ final states $X_{i}$. For candidates
with satisfactory kinematic fits, we use the resulting track and photon
momenta to compute the hadronic mass. We populate distributions of this
hadronic mass by removing our previous $E1$ photon-energy and $M(\eta_{c})$
requirements and selecting candidates inside a $\pi^{0}$ recoil mass window of
$\pm 5\,\rm MeV/c^{2}$ around the $h_{c}$ mass, keeping all other criteria
unchanged.

The line shape for the $\eta_{c}$ signal for these fits is parameterized as
$(E_{\gamma}^{3}\times BW(m)\times f_{d}(E_{\gamma}))\otimes R_{i}(m)$, where
$BW(m)$ is the Breit-Wigner function for $\eta_{c}$ as a function of the
invariant mass $m$ of the decay products for each channel,
$E_{\gamma}(m)=\frac{M(h_{c})^{2}-m^{2}}{2M(h_{c})}$ is the energy of the
transition photon in the rest frame of $h_{c}$, and $f_{d}(E_{\gamma})$ is a
function that damps the divergent tail due to the $E_{\gamma}^{3}$ factor,
which incorporates the energy dependence of the $E1$ matrix element and the
phase-space factor. $R_{i}(m)$ is the signal resolution function for the $i$th
decay mode, which is parameterized by double Gaussians to account for the
distorting effects of the kinematic fit and detector smearing. The damping
function that we use was introduced by the KEDR collaboration ref:kedretac :

$f_{d}(E_{\gamma})=\frac{E_{0}^{2}}{E_{\gamma}E_{0}+(E_{\gamma}-E_{0})^{2}},$

where $E_{0}=E_{\gamma}(m_{\eta_{c}})$ is the $E1$-transition-photon peak
energy. The $\eta_{c}$-candidate hadronic invariant mass spectra from low and
high side bands in the $h_{c}$ mass (3500$-$3515, 3535$-$3550 MeV/$c^{2}$) are
used to obtain the background functions for the $\eta_{c}$ mass fit. To
mitigate the effects of bin-to-bin fluctuations, these side-band mass spectra
are smoothed before fitting. A toy MC study was performed to test the effect
of the smoothing and it was demonstrated to be a robust procedure that does
not systematically distort the fit results. The channel-by-channel signal and
background normalizations are free parameters determined by the fit.

We ignore the effect of interference between the signal and background, which
was considered in the previous measurement of $\psi(3686)\to\gamma\eta_{c}$
BESIII:2011ab , because the branching ratio of $h_{c}\to\gamma\eta_{c}$ is
about $50\%$ (branching ratio of $M1$ transition $\psi(3686)\to\gamma\eta_{c}$
is about $0.3\%$). The radiative decay of $h_{c}\to\gamma 0^{-}$ should be the
same level of $\psi(3686)\to\gamma 0^{-}$, in this case, the non-$\eta_{c}$
intensity in $h_{c}$ is much smaller than that for
$\psi(3686)\to\gamma\eta_{c}$.

Figs. 3 and 4 show the hadronic-mass-fit results. The $\eta_{c}$ mass and
width are determined to be $M(\eta_{c})=2984.49\pm 1.16$ MeV/$c^{2}$ and
$\Gamma(\eta_{c})=36.4\pm 3.2$ MeV, where the errors are statistical. The
$\chi^{2}$ per degree of freedom for this fit is 1.52, using the same
$\chi^{2}$ calculation method to accommodate low-statistics bins as for the
fit to the $\pi^{0}$ recoil mass spectrum.

(a)
(b)

Figure 3:  (a) The hadronic mass spectrum in
$\psi(3686)\to\pi^{0}{h}_{c},{h}_{c}\to\gamma\eta_{c}$, $\eta_{c}\to X_{i}$
summed over the 16 final states $X_{i}$. The dots with error bars represent
the hadronic mass spectrum in data. The solid line shows the total fit
function and the dashed line is the background component of the fit. (b) The
background-subtracted hadronic mass spectrum with the signal shape overlaid.
Figure 4: The simultaneously fitted hadronic mass spectra for the 16
$\eta_{c}$ decay channels.

##  V Systematic Uncertainties

###  V.1 $h_{c}$ parameter measurements

The systematic uncertainties for the $M(h_{c})$ and $\Gamma(h_{c})$
measurements are summarized in Table 3. All sources are treated as
uncorrelated, so the total systematic uncertainty is obtained by summing them
in quadrature. The following subsections describe the procedures and
assumptions that led to these estimates of the uncertainties.

Table 3:  The systematic errors for the $h_{c}$ mass and width measurements. Sources |  $\Delta M_{h_{c}}$ (MeV$/c^{2}$) |  $\Delta\Gamma_{h_{c}}$ (MeV)
---|---|---
Energy calibration | 0.13 | 0.07
Signal shape | 0.00 | 0.06
Fitting range | 0.04 | 0.16
Binning | 0.02 | 0.01
Background shape | 0.01 | 0.08
Background veto | 0.01 | 0.08
Kinematic fit | 0.03 | 0.03
Mass of $\psi(3686)$ | 0.03 | 0.02
Total | 0.14 | 0.22

####  V.1.1 Energy calibration

The potential inconsistency of the photon-energy measurement between data and
MC is evaluated by studying
$\psi(3686)\to\gamma\chi_{c1,2}~{}(~{}\chi_{c1,2}\to\gamma
J/\psi,J/\psi\to\mu^{+}\mu^{-}$) for photons with low energy and radiative
Bhabha events for photons with high energy. Discrepancies of $0.4\%$ in the
energy scale and $4\%$ in the energy resolution between data and MC are found.
We vary the photon response accordingly and take the changes in the results as
the estimated systematic error. For the $M(h_{c})$ measurement, besides the
above studies, the reconstructed photon position and error matrix are taken
into account as additional sources of uncertainty.

####  V.1.2 Signal shape

The uncertainty associated with the $h_{c}$ signal shape in the $\pi^{0}$
recoil mass spectrum includes contributions from the photon line shape and the
1-C kinematic fit. We estimate these by determining the changes in results
after reasonable adjustments in the photon response. The photon-energy
resolution is estimated with the control sample
$\psi(3686)\to\gamma\chi_{c2}$. As above, the energy resolution in data is
found to be about 4% worse than in the MC simulation. We correct for this
discrepancy by adding single-Gaussian smearing to the energy of the $\pi^{0}$
daughter photons and then using the alternative $\pi^{0}$ shape to redo the
fit. The changes in results are assigned as the systematic errors.

####  V.1.3 Fitting range and binning

The systematic uncertainties due to the fitting of the $\pi^{0}$ recoil mass
spectrum are evaluated by varying the fitting range and the bin size in the
fit. The spreads of results obtained with the alternative assumptions are used
to assign the systematic errors.

####  V.1.4 Background shape

To estimate the uncertainty associated with the side-band method for assigning
background function shapes, we use an ARGUS function Albrecht:1990am  as an
alternative background description for each channel and record the changes in
the fit results.

####  V.1.5 Background veto

The systematic uncertainties associated with the requirements to suppress
background are estimated by varying the excluded ranges.

####  V.1.6 Kinematic fit

Systematic uncertainties caused by the kinematic fit are studied by tuning the
tracking parameters and error matrices of charged tracks and photons based on
the data. Control samples of $J/\psi\to\phi f_{0}(980),\phi\to
K^{+}K^{-},f_{0}(980)\to\pi^{+}\pi^{-}$, and $\psi(3686)\to\gamma\chi_{cJ}$
are used for this purpose Ablikim:2012pg . Channel-by-channel changes of
$M(\eta_{c})$ and $\Gamma(\eta_{c})$ are calculated after the tuning and then
averaged by yields and taken as systematic errors.

####  V.1.7 $\psi(3686)$ mass

The systematic uncertainties of the $M(h_{c})$ and $\Gamma(h_{c})$
determinations associated with the uncertainty in the $\psi(3686)$ mass are
estimated to be 0.03 MeV/$c^{2}$ and 0.02 MeV, respectively. These are found
by shifting $M_{\psi(3686)}$ by one standard deviation according to the PDG
value ref:PDG_2012  and redetermining the results.

###  V.2 $\eta_{c}$ branching ratio measurements

The systematic errors in the $\eta_{c}$ branching ratio measurements are
listed in Tables 4. All sources are treated as uncorrelated, so the total
systematic uncertainty is obtained by summing them in quadrature. The
following subsections describe the procedures and assumptions that led to the
estimates of these uncertainties.

Table 4: The systematic errors (in %) in the $\eta_{c}$ branching ratio
measurements of the $\eta_{c}$ exclusive decay channels.

Sources | $p\bar{p}$ | $2(\pi^{+}\pi^{-})$ | $2(K^{+}K^{-})$ | $K^{+}K^{-}\pi^{+}\pi^{-}$ | $p\bar{p}\pi^{+}\pi^{-}$ | $3(\pi^{+}\pi^{-})$ | $K^{+}K^{-}2(\pi^{+}\pi^{-})$ | $K^{+}K^{-}\pi^{0}$
---|---|---|---|---|---|---|---|---
N($\psi(3686)$) | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0
Tracking | 4.0 | 8.0 | 8.0 | 8.0 | 8.0 | 12.0 | 12.0 | 4.0
PID ($K^{0}_{S}$) | 2.0 | 6.0 | 6.0 | 4.0 | 4.0 | 8.0 | 8.0 | 2.0
Photon eff | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 5.0
Fit range | 2.2 | 1.2 | 2.6 | 2.9 | 1.5 | 5.3 | 3.3 | 2.7
Bkg shape | 10.3 | 2.5 | 4.7 | 0.9 | 0.3 | 0.2 | 3.5 | 2.8
Signal shape | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3
KmFit eff. | 7.0 | 6.3 | 7.0 | 8.8 | 10.8 | 7.3 | 4.2 | 2.0
Bkg veto | 5.9 | 5.5 | 1.1 | 0.6 | 3.1 | 2.3 | 5.2 | 1.7
Cross feed | 0.0 | 2.5 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.4
$\eta_{c}$ decay models | 0.0 | 2.1 | 3.7 | 0.6 | 2.5 | 0.0 | 3.0 | 4.6
$\eta_{c}$ line shape | 0.7 | 0.8 | 0.6 | 0.9 | 0.6 | 0.6 | 0.6 | 0.7
Sum | 15.7 | 14.8 | 14.9 | 14.1 | 15.7 | 18.0 | 17.8 | 10.6

Sources | $p\bar{p}\pi^{0}$ | $K^{0}_{S}K^{\pm}\pi^{\mp}$ | $K^{0}_{S}K^{\pm}\pi^{\mp}\pi^{\pm}\pi^{\mp}$ | $\pi^{+}\pi^{-}\eta$ | $K^{+}K^{-}\eta$ | $2(\pi^{+}\pi^{-})\eta$ | $\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ | $2(\pi^{+}\pi^{-}\pi^{0})$
---|---|---|---|---|---|---|---|---
N($\psi(3686)$) | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0
Tracking | 4.0 | 8.0 | 12.0 | 4.0 | 4.0 | 8.0 | 4.0 | 8.0
PID ($K^{0}_{S}$) | 2.0 | 1.0 | 1.0 | 0.0 | 2.0 | 0.0 | 0.0 | 0.0
Photon eff | 5.0 | 3.0 | 3.0 | 5.0 | 5.0 | 5.0 | 7.0 | 7.0
Fit range | 7.7 | 2.1 | 1.5 | 0.6 | 6.0 | 1.8 | 0.6 | 2.0
Bkg shape | 0.1 | 4.7 | 4.7 | 0.1 | 5.9 | 0.8 | 3.3 | 1.6
Signal shape | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3
KmFit eff. | 6.8 | 6.8 | 7.3 | 2.0 | 1.2 | 6.7 | 2.4 | 2.4
Bkg veto | 3.7 | 0.7 | 2.8 | 11.8 | 5.4 | 14.7 | 12.8 | 5.5
Cross feed | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.3 | 0.0
$\eta_{c}$ decay models | 5.8 | 2.5 | 5.2 | 5.5 | 8.1 | 0.0 | 0.1 | 0.5
$\eta_{c}$ line shape | 0.6 | 0.6 | 0.8 | 0.8 | 0.7 | 0.7 | 0.7 | 0.7
Sum | 14.8 | 13.2 | 17.0 | 15.4 | 15.3 | 19.4 | 16.4 | 13.3

####  V.2.1 Tracking and photon detection

The uncertainty in the tracking efficiency is 2% per track and the uncertainty
due to photon detection is 1% per photon Ablikim:2011kv . MC studies
demonstrate that the trigger efficiency for signal events is almost 100%, so
that the associated uncertainty in the results is negligible.

####  V.2.2 PID and $K^{0}_{S}$ reconstruction

The systematic uncertainties due to kaon and pion identifications are
determined to be 2% in Ref. Ablikim:2011kv . We choose $J/\psi\to
K^{*0}K^{0}_{S},K^{*0}\to K\pi$ to evaluate the efficiency of $K^{0}_{S}$
reconstruction. The $1\%$ difference between data and MC is assigned as the
systematic error due to this source.

####  V.2.3 Kinematic fitting

The systematic errors associated with kinematic fitting are estimated by using
the control samples of $\psi(3686)\to\pi^{0}\pi^{0}J/\psi$ with $J/\psi$ decay
to hadronic final states, which have similar event topology as
$\psi(3686)\to\pi^{0}h_{c},h_{c}\to\gamma\eta_{c}$. The average efficiency
difference between data and MC, with the same $\chi^{2}$ requirements in the
$h_{c}$ selection, is taken as the systematic uncertainty.

####  V.2.4 Cross-feed

To evaluate the effect of cross-feed among the 16 signal modes, we use samples
of 50,000 MC events per mode. We find that $\eta_{c}\to
2(\pi^{+}\pi^{-}),\eta_{c}\to K^{+}K^{-}\pi^{0}$ and
$\eta_{c}\to\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ are contaminated by $\eta_{c}\to
K^{0}_{S}K^{\pm}\pi^{\mp}$ with levels of $2.5\%$, $1.4\%$, and $1.3\%$,
respectively. These numbers are assigned as the systematic errors associated
with cross-feed. For other channels, this contamination is found to be
negligible.

####  V.2.5 $\eta_{c}$ decay models

We use phase space to simulate $\eta_{c}$ decays in our analysis. To estimate
the systematic uncertainty due to neglecting intermediate states in these
decays, we extract invariant masses of $\eta_{c}$ daughter particles from
$\psi(3686)\to\gamma\eta_{c},\eta_{c}\to X_{i}$. We analyze MC samples
generated according to these invariant masses. To illustrate, Fig. 5 shows the
invariant-mass distribution comparison between the data and MC for the decay
mode $\eta_{c}\to K^{0}_{S}K^{\pm}\pi^{\mp}$. In addition, for channels with
low statistics and well-understood intermediate states, MC samples with these
intermediate states were generated according to the relative branching ratios
given by PDG. The spreads of the efficiencies obtained from the phase-space
and alternative MC are taken as the systematic errors.

Figure 5: The dots show the mass spectra for
$\psi(3686)\to\gamma\eta_{c},\eta_{c}\to K^{0}_{S}K^{\pm}\pi^{\mp}$ in data,
and the solid lines are the corresponding mass spectra from the MC simulation.

####  V.2.6 $\eta_{c}$ line shape

Because of the $\eta_{c}$ mass window requirement in our event selection, the
line shape of $\eta_{c}$ could be a source of systematic error in the
measurement. We vary the input $\eta_{c}$ resonant parameters by one standard
deviation to estimate the uncertainty due to this source.

###  V.3 $\eta_{c}$ parameter measurements

Systematic errors for the $M(\eta_{c})$ and $\Gamma(\eta_{c})$ measurements
are summarized in Table 5. All sources are treated as uncorrelated, so the
total systematic uncertainty is obtained by summing in quadrature. The
following subsections describe the procedures and assumptions that led to the
estimates of these uncertainties.

Table 5: The systematic errors for $\eta_{c}$ parameter measurements.  Sources | M($\eta_{c}$) (MeV$/c^{2}$) |  $\Gamma(\eta_{c})$ (MeV)
---|---|---
Background shape | $0.36$ | $1.45$
Fitting range | $0.03$ | $0.33$
Resolution description | $0.10$ | $0.02$
Mass-dependent efficiencies | $0.11$ | $0.27$
Mass-dependent resolutions | $0.00$ | $0.01$
Kinematic fitting | $0.33$ | $0.76$
Fitting method | $0.11$ | $0.40$
Sum | $0.52$ | $1.74$

####  V.3.1 Background shape

Our standard background shape is the smoothed $h_{c}$ side-band shape. To
estimate the systematic uncertainty due to the background procedure, we change
the smoothing level and technique, and vary the $h_{c}$ side-band ranges. The
largest changes in results among these alternatives are assigned as the
systematic errors.

####  V.3.2 Fitting range

The systematic uncertainties due to the fitting range are estimated by
considering several alternatives to the standard fitting range of 2.3-3.2 $\rm
GeV/c^{2}$, 2.4-3.2 $\rm GeV/c^{2}$, 2.5-3.2 $\rm GeV/c^{2}$, 2.6-3.2 $\rm
GeV/c^{2}$, and 2.3-3.15 $\rm GeV/c^{2}$. The systematic uncertainties are
assigned to be the largest differences between the standard fit results and
those from the alternative ranges.

####  V.3.3 Resolution description

In order to estimate the systematic uncertainties associated with the
detector-resolution description, we use MC signal shapes obtained by setting
the $\eta_{c}$ width to zero as alternatives to double Gaussians. The changes
in fit results between these two methods provide the systematic errors.

####  V.3.4 Mass-dependent efficiency and resolution

Since the $\eta_{c}$ signal spreads over a sizable mass range, the
uncertainties due to the use of mass-independent efficiencies and resolutions
need to be estimated. Mass-dependent efficiencies and resolutions are
determined from MC simulation and used as an alternative to the default
assumption, and the resulting differences are taken to be the systematic
errors.

####  V.3.5 Kinematic fitting

The method to evaluate the systematic errors due to the kinematic fitting
procedure and momentum measurement is the same as that in the measurement of
the $h_{c}$ parameters.

####  V.3.6 Fitting method

Because we use the smoothed side-band shape to describe the background, the
potential for bias due to the smoothing technique must be considered. This was
investigated with a toy MC study. We start with a signal sample for each of
the 16 channels selected from our standard MC to have the same statistics as
data. A corresponding background sample for each channel is constructed from
the mass side bands in data. The hadronic-mass distributions for these samples
are then treated with a variety of smoothing procedures and fitted. The ranges
in the fit results are used to set the systematic errors from this source.

##  VI SUMMARY AND DISCUSSION

In summary, we have studied the process $\psi(3686)\to\pi^{0}h_{c}$ followed
by $h_{c}\to\gamma\eta_{c}$ with an exclusive-reconstruction technique. Using
a sample of 106 million $\psi(3686)$ decays we have obtained new measurements
of the mass and width of the $h_{c}$ and $\eta_{c}$ charmonium resonances, and
of the branching ratios for 16 exclusive $\eta_{c}$ hadronic decay modes.

The total yield of events, measured by fitting the $\pi^{0}$ recoil mass
spectrum, is $832\pm 35$ events, where the error is statistical only. With
these events we measure the mass and width of the $h_{c}$:

$\displaystyle M(h_{c})=3525.31\pm 0.11\pm 0.14\,\rm{MeV}/c^{2},~{}and$
$\displaystyle\Gamma(h_{c})=0.70\pm 0.28\pm 0.22\,\rm{MeV},$

where the first errors are statistical and the second are systematic. These
results are consistent with the results of a previous inclusive measurement by
BESIII ref:bes3hc10 :

$\displaystyle M(h_{c})=3525.40\pm 0.13\pm 0.18\,\rm{MeV}/c^{2},~{}and$
$\displaystyle\Gamma(h_{c})<1.44\,\rm{MeV}~{}(at~{}90\%~{}confidence~{}level).$

The branching-ratio results
$\mathcal{B}_{1}(\psi(3686)\to\pi^{0}h_{c})\times\mathcal{B}_{2}(h_{c}\to\gamma\eta_{c})\times\mathcal{B}_{3}(\eta_{c}\to
X_{i})$ and $\mathcal{B}_{3}(\eta_{c}\to X_{i})$ are given in Table 6, quoted
with the statistical and systematic errors of this measurement and, for
$\mathcal{B}_{3}$, an additional systematic error associated with the input
branching-ratio product
$\mathcal{B}_{1}(\psi(3686)\to\pi^{0}h_{c})\times\mathcal{B}_{2}(h_{c}\to\gamma\eta_{c})$.
Most of our $\mathcal{B}_{3}(\eta_{c}\to X_{i})$ branching-fraction results
are consistent with PDG values ref:PDG_2012 , and several branching fractions
are measured for the first time.

Table 6:
$\mathcal{B}_{1}(\psi(3686)\to\pi^{0}h_{c})\times\mathcal{B}_{2}(h_{c}\to\gamma\eta_{c})\times\mathcal{B}_{3}(\eta_{c}\to
X_{i})$ and $\mathcal{B}_{3}(\eta_{c}\to X_{i})$ with systematic errors. The
third errors in $\mathcal{B}_{3}$ measurement are systematic errors due to
uncertainty of
$\mathcal{B}_{1}(\psi(3686)\to\pi^{0}h_{c})\times\mathcal{B}_{2}(h_{c}\to\gamma\eta_{c})$.

$X_{i}$ | $\mathcal{B}_{1}\times\mathcal{B}_{2}\times\mathcal{B}_{3}~{}(\times 10^{-6})$ | $\mathcal{B}_{3}~{}(\%)$ |  $\mathcal{B}_{3}$ in PDG $(\%)$
---|---|---|---
$p\bar{p}$ | 0.65 $\pm$ 0.19 $\pm$ 0.10 | 0.15$\pm$0.04$\pm$0.02$\pm$0.01 | 0.141$\pm$0.017
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ | 7.51 $\pm$ 0.85 $\pm$ 1.11 | 1.72$\pm$0.19$\pm$0.25$\pm$0.17 | 0.86$\pm$0.13
$K^{+}K^{-}K^{+}K^{-}$ | 0.94 $\pm$ 0.37 $\pm$ 0.14 | 0.22$\pm$0.08$\pm$0.03$\pm$0.02 | 0.134$\pm$0.032
$K^{+}K^{-}\pi^{+}\pi^{-}$ | 4.16 $\pm$ 0.76 $\pm$ 0.59 | 0.95$\pm$0.17$\pm$0.13$\pm$0.09 | 0.61$\pm$0.12
$p\bar{p}\pi^{+}\pi^{-}$ | 2.30 $\pm$ 0.65 $\pm$ 0.36 | 0.53$\pm$0.15$\pm$0.08$\pm$0.05 | <1.2 (at 90% C.L.)
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ | 8.82 $\pm$ 1.57 $\pm$ 1.59 | 2.02$\pm$0.36$\pm$0.36$\pm$0.19 | 1.5$\pm$0.50
$K^{+}K^{-}\pi^{+}\pi^{-}\pi^{-}\pi^{-}$ | 3.60 $\pm$ 1.71 $\pm$ 0.64 | 0.83$\pm$0.39$\pm$0.15$\pm$0.08 | 0.71$\pm$0.29
$K^{+}K^{-}\pi^{0}$ | 4.54 $\pm$ 0.76 $\pm$ 0.48 | 1.04$\pm$0.17$\pm$0.11$\pm$0.10 | 1.2$\pm$0.1
$p\bar{p}\pi^{0}$ | 1.53 $\pm$ 0.49 $\pm$ 0.23 | 0.35$\pm$0.11$\pm$0.05$\pm$0.03 | –
$K^{0}_{S}K^{\pm}\pi^{\mp}$ | 11.35 $\pm$ 1.25 $\pm$ 1.50 | 2.60$\pm$0.29$\pm$0.34$\pm$0.25 | 2.4$\pm$0.2
$K^{0}_{S}K^{\pm}\pi^{\mp}\pi^{\pm}\pi^{\mp}$ | 12.01 $\pm$ 2.22 $\pm$ 2.04 | 2.75$\pm$0.51$\pm$0.47$\pm$0.27 | –
$\pi^{+}\pi^{-}\eta$ | 7.22 $\pm$ 1.47 $\pm$ 1.11 | 1.66$\pm$0.34$\pm$0.26$\pm$0.16 | 4.9$\pm$1.8
$K^{+}K^{-}\eta$ | 2.11 $\pm$ 1.01 $\pm$ 0.32 | 0.48$\pm$0.23$\pm$0.07$\pm$0.05 | <1.5 (at 90% C.L.)
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\eta$ | 19.17 $\pm$ 3.77 $\pm$ 3.72 | 4.40$\pm$0.86$\pm$0.85$\pm$0.42 | –
$\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ | 20.31 $\pm$ 2.20 $\pm$ 3.33 | 4.66$\pm$0.50$\pm$0.76$\pm$0.45 | –
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ | 75.13 $\pm$ 7.42 $\pm$ 9.99 | 17.23$\pm$1.70$\pm$2.29$\pm$1.66 | –

Combining our measurement of $M(h_{c})$ with the previously-determined mass of
the centroid of the ${}^{3}P_{J}$ states leads to

$\displaystyle\Delta~{}M_{hf}\equiv\langle
M(1^{3}P)\rangle-M(1^{1}P_{1})=-0.01\pm 0.11~{}(\rm stat.)\pm 0.15~{}(\rm
syst.)\,\rm{MeV/c^{2}},$ (1)

consistent with the lowest-order expectation that the $1P$ hyperfine splitting
is zero.

The line shape of $\eta_{c}$ was also studied from the $E1$ transition
$h_{c}\to\gamma\eta_{c}$, and the measured resonance parameters are:

$\displaystyle M(\eta_{c})=2984.49\pm 1.16\pm 0.52\,\rm{MeV/c^{2}},and$
$\displaystyle\Gamma(\eta_{c})=36.4\pm 3.2\pm 1.7\,\rm{MeV}.$

These results are consistent with the recent BESIII results from
$\psi(3686)\to\gamma\eta_{c}$ BESIII:2011ab :

$\displaystyle M(\eta_{c})=2984.3\pm 0.6\pm 0.6\,\rm{MeV/c^{2}},and$
$\displaystyle\Gamma(\eta_{c})=32.0\pm 1.2\pm 1.0\,\rm{MeV};$

and B-factory results from $\gamma\gamma\to\eta_{c}$ and $B$ decays
Vinokurova:2011dy ; delAmoSanchez:2011bt . Because of the larger $\psi(3686)$
data sample that will be coming from BESIII and the advantage of negligible
interference effects, we expect that $h_{c}\to\gamma\eta_{c}$ will provide the
most reliable determinations of the $\eta_{c}$ resonance parameters in the
future.

##  VII ACKNOWLEDGMENTS

The BESIII collaboration thanks the staff of BEPCII and the computing center
for their hard efforts. This work is supported in part by the Ministry of
Science and Technology of China under Contract No. 2009CB825200; National
Natural Science Foundation of China (NSFC) under Contracts Nos. 10745001,
10625524, 10821063, 10825524, 10835001, 10935007, 11125525; Joint Funds of the
National Natural Science Foundation of China under Contracts Nos. 11079008,
11179007, 10979058; the Chinese Academy of Sciences (CAS) Large-Scale
Scientific Facility Program; CAS under Contracts Nos. KJCX2-YW-N29,
KJCX2-YW-N45; 100 Talents Program of CAS; Istituto Nazionale di Fisica
Nucleare, Italy; Ministry of Development of Turkey under Contract No.
DPT2006K-120470; U. S. Department of Energy under Contract Nos. DE-
FG02-04ER41291, DE-FG02-91ER40682, DE-FG02-94ER40823; U.S. National Science
Foundation; University of Groningen (RuG); the Helmholtzzentrum fuer
Schwerionenforschung GmbH (GSI), Darmstadt; and WCU Program of National
Research Foundation of Korea under Contract No. R32-2008-000-10155-0.

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# Global numerical bounds for the number-theoretic omega functions

Mehdi Hassani  Mehdi Hassani
Department of Mathematics, University of Zanjan, University Blvd.,
45371-38791, Zanjan, Iran<EMAIL_ADDRESS>
###### Abstract.

We obtain global explicit numerical bounds, with best possible constants, for
the differences $\frac{1}{n}\sum_{k\leqslant n}\omega(k)-\log\log n$ and
$\frac{1}{n}\sum_{k\leqslant n}\Omega(k)-\log\log n$, where $\omega(k)$ and
$\Omega(k)$ refer to the number of distinct prime divisors, and the total
number of prime divisors of $k$, respectively.

###### Key words and phrases:

Arithmetic function, growth of arithmetic functions, prime number,
inequalities.

###### 2010 Mathematics Subject Classification:

11A25, 11N56, 11N05, 11A41, 26D20.

###### Contents

  1. 1 Introduction
  2. 2 Summary of the results
    1. 2.1 Unconditional results
    2. 2.2 Conditional results
  3. 3 Proof of unconditional approximations
  4. 4 Proof of conditional approximations

##  1\. Introduction

For the fixed complex number $s$ the generalized omega function
$\Omega_{s}(k)$ is defined by $\Omega_{s}(k)=\sum_{p^{\ell}\|k}\ell^{s}$,
where $p^{\ell}\|k$ means that $\ell$ is the largest power of $p$ such that
$p^{\ell}|k$. The cases $s=0$ and $s=1$ coincide, respectively, with the well-
known number theoretic omega functions $\omega(k)=\sum_{p|k}1$, the number of
distinct prime divisors of the positive integer $k$, and
$\Omega(k)=\sum_{p^{\ell}\|k}\ell$, the total number of prime divisors of $k$.
Duncan [3] proved that for each arbitrary integer $s\geqslant 0$,

(1.1) $\frac{1}{n}\sum_{k\leqslant n}\Omega_{s}(k)=\log\log
n+M_{s}+O\left(\frac{1}{\log n}\right),$

where $M_{s}$ is a constant depending on $s$, given by
$M_{s}=M+M^{\prime}_{s}$, with $M$ referring to the Meissel–Mertens constant
(see Remark 2.11 for more information), and

$M^{\prime}_{s}=\sum_{p}\sum_{\ell\geqslant
2}\frac{\ell^{s}-(\ell-1)^{s}}{p^{\ell}}.$

Here and through the paper $\sum_{p}$ means that the sum runs over all primes.
Note that $M_{0}=M$. Also, we let $M^{\prime}=M_{1}$ and
$M^{\prime}_{1}=M^{\prime\prime}=\sum_{p}\frac{1}{p(p-1)}$. Thus,
$M^{\prime}=M+M^{\prime\prime}$. Approximation (1.1) is a generalization of
the previously known result of Hardy and Ramanujan [5] concerning the average
of the functions $\omega$ and $\Omega$.

Based on Dirichlet’s hyperbola method and prime number theorem for arithmetic
progressions with error term Saffari [12] obtained a full asymptotic expansion
for the average of $\omega(n)$ where $n$ runs over the arithmetic progression
$a$ modulo $q$ with $\gcd(a,q)=1$. For $a=q=1$, his result reads a follows

(1.2) $\frac{1}{n}\sum_{k\leqslant n}\omega(k)=\log\log
n+M+\sum_{j=1}^{m}\frac{a_{j}}{\log^{j}n}+O\Big{(}\frac{1}{\log^{m+1}n}\Big{)},$

where $m\geqslant 1$ is any fixed integer, and the coefficients $a_{j}$ are
given by

(1.3)
$a_{j}=-\int_{1}^{\infty}\frac{\\{t\\}}{t^{2}}\log^{j-1}t\,\mathrm{d}t=\frac{(-1)^{j-1}}{j}\,\frac{\mathrm{d}^{j}}{\mathrm{d}s^{j}}\left(\frac{1}{s}(s-1)\zeta(s)\right)_{s=1}.$

Diaconis [2] reproved (1.2) using Dirichlet series of $\omega$, Perron’s
formula and complex integration methods. One may obtain similar expansion for
the average of generalized omega function $\Omega_{s}$ for each fixed real
$s\geqslant 0$, replacing $M$ by $M_{s}$ (see [9, Theorem 1] for more
details).

Explicit versions of (1.1) for $s=0$ and $s=1$ is obtained in [8] and [6],
respectively, and then both improved in [7, Theorem 1.2], where it is showed
that for each $n\geqslant 2$ the following double sided approximation holds

(1.4) $-\frac{1.133}{\log n}<\frac{1}{n}\sum_{k\leqslant n}\omega(k)-\log\log
n-M<\frac{1}{2\log^{2}n}.$

Also,

(1.5) $-\frac{1.175}{\log n}<\frac{1}{n}\sum_{k\leqslant n}\Omega(k)-\log\log
n-M^{\prime}<\frac{1}{2\log^{2}n},$

where the left hand side is valid for each $n\geqslant 24$ and the right hand
side is valid for each $n\geqslant 2$.

##  2\. Summary of the results

###  2.1. Unconditional results

In the present paper we are motivated by finding global numerical lower and
upper bounds for the differences $\mathcal{A}_{0}(n)$ and
$\mathcal{A}_{1}(n)$, where $\mathcal{A}_{s}(n)$ defined for any fixed complex
number $s$ as follows

$\mathcal{A}_{s}(n)=\frac{1}{n}\sum_{k\leqslant n}\Omega_{s}(k)-\log\log n.$

The problem for the case $\mathcal{A}_{0}(n)$ is an easy corollary of the
inequalities (1.4). More precisely, we prove the following.

######  Theorem 2.1.

For all natural numbers $n\geqslant 2$, we have

(2.1) $\alpha_{0}\leqslant\mathcal{A}_{0}(n)\leqslant\beta_{0}$

with the best possible constants $\alpha_{0}=\frac{45}{32}-\log\log 32$ and
$\beta_{0}=\frac{1}{2}-\log\log 2$, and the equality in the left hand side
only for $n=32$, and in the right hand side only for $n=2$.

Similarly, to get a global numerical lower bound for $\mathcal{A}_{1}(n)$, we
can use the inequalities (1.5) to show the following result.

######  Theorem 2.2.

For all natural numbers $n\geqslant 2$, we have

(2.2) $\alpha_{1}\leqslant\mathcal{A}_{1}(n)$

with the best possible constant $\alpha_{1}=\frac{8}{7}-\log\log 7$ and the
equality only for $n=7$.

The problem of obtaining a global numerical upper bound for
$\mathcal{A}_{1}(n)$ is quite different by the above ones. Although,
computations show that $\mathcal{A}_{1}(n)<\beta_{1}$ for any $n\geqslant 2$
with the best possible constant $\beta_{1}=M^{\prime}$, but the inequalities
(1.5) are not enough sharp to show this fact. To deal with this difficulty, we
made explicit all steps of the proof of (1.2) by following Saffari’s argument
in [12], and hence, we could to prove the following result.

######  Theorem 2.3.

For all natural numbers $n\geqslant\mathrm{e}^{14167}\approxeq 4.466\times
10^{6152}$, we have

(2.3) $\mathcal{A}_{1}(n)<\beta_{1}$

with the best possible constant $\beta_{1}=M^{\prime}$. Moreover, if we assume
that the Riemann hypothesis is true, then (2.3) holds for all natural numbers
$n\geqslant 1400387903260$.

To prove Theorem 2.3 we use explicit forms of the prime number theorem with
error term. Let $\pi(x)=\sum_{p\leqslant x}1$ be the prime counting function,
and $\mathrm{li}(x)=\int_{0}^{x}\frac{1}{\log t}\,\mathrm{d}t$ be the
logarithmic integral function, defined as the Cauchy principle value of the
integral. By $f=O^{\ast}(g)$ we mean $|f|\leqslant g$, providing an explicit
version of Landau’s notation. It is known [15, Theorem 2] that

$\pi(x)=\mathrm{li}(x)+O^{\ast}\left(0.2795\,x(\log
x)^{-\frac{3}{4}}\,\mathrm{e}^{-\sqrt{(\log x)/6.455}}\right)\qquad(x\geqslant
229).$

Modifying the above to the classical form, for any $x>1.2$ we have

(2.4) $\pi(x)=\mathrm{li}(x)+O^{\ast}(R(x)),\qquad
R(x)=x\,\mathrm{e}^{-\frac{1}{3}\sqrt{\log x}}.$

This is however a weaker approximation, but it is suitable for our arguments,
because of its global validity. We will use it to prove the following
unconditional results.

######  Theorem 2.4.

For any fixed integer $m\geqslant 1$ and for any $x\geqslant\mathrm{e}$ we
have

(2.5) $\sum_{n\leqslant x}\omega(n)=x\log\log
x+Mx+x\sum_{j=1}^{m}\frac{a_{j}}{\log^{j}x}+O^{\ast}\left(\mathcal{E}_{\omega}(x,m)\right),$

where

$\mathcal{E}_{\omega}(x,m)=2^{m+1}m!\,\frac{x}{\log^{m+1}x}+(2^{m+1}+1)\,\mathrm{e}m!\,\frac{\sqrt{x}}{\log
x}\\\ +x\,\mathrm{e}^{-\frac{\sqrt{2}}{6}\sqrt{\log x}}\left(\frac{1}{2}\log
x+3\sqrt{2}\sqrt{\log x}+21\right)+\sqrt{x}.$

######  Corollary 2.5.

For $x\geqslant\mathrm{e}^{14167}\approxeq 4.466\times 10^{6152}$ we have

(2.6) $\sum_{n\leqslant x}\omega(n)=x\log\log
x+Mx-\left(1-\gamma\right)\frac{x}{\log
x}+O^{\ast}\left(\frac{5x}{\log^{2}x}\right),$

and consequently $\frac{1}{x}\sum_{n\leqslant x}\omega(n)-\log\log x<M$.

In order to transfer an average result on the function $\omega$ to an average
result on the function $\Omega$, we may consider the average difference
$\mathcal{J}(x):=\sum_{n\leqslant x}\left(\Omega(n)-\omega(n)\right)$, for
which it is known [7, Theorem 1.1] that for each integer $n\geqslant 1$,

(2.7) $nM^{\prime\prime}-25\frac{\sqrt{n}}{\log
n}<\mathcal{J}(n)<nM^{\prime\prime}-\frac{\sqrt{n}}{\log
n}\Big{(}2-\frac{20}{\log n}\Big{)}.$

Modifying the above approximation, we will prove in Lemma 3.4 that
$\mathcal{J}(x)=M^{\prime\prime}x+O^{\ast}(\frac{33\sqrt{x}}{\log x})$ for any
$x\geqslant 2$. Thus, Theorem 2.4 and Corollary 2.5 transfer to the following
results.

######  Theorem 2.6.

For any fixed integer $m\geqslant 1$ and for any $x\geqslant\mathrm{e}$ we
have

(2.8) $\sum_{n\leqslant x}\Omega(n)=x\log\log
x+M^{\prime}x+x\sum_{j=1}^{m}\frac{a_{j}}{\log^{j}x}+O^{\ast}\left(\mathcal{E}_{\Omega}(x,m)\right),$

where

$\mathcal{E}_{\Omega}(x,m)=\mathcal{E}_{\omega}(x,m)+\frac{33\sqrt{x}}{\log
x}.$

######  Corollary 2.7.

For $x\geqslant\mathrm{e}^{14167}\approxeq 4.466\times 10^{6152}$ we have

(2.9) $\sum_{n\leqslant x}\Omega(n)=x\log\log
x+M^{\prime}x-\left(1-\gamma\right)\frac{x}{\log
x}+O^{\ast}\left(\frac{6x}{\log^{2}x}\right),$

and consequently $\frac{1}{x}\sum_{n\leqslant x}\Omega(n)-\log\log
x<M^{\prime}$.

###  2.2. Conditional results

As we observe in Corollary 2.5, approximation (2.5), even with its initial
parameter $m=1$, gives explicit bounds for $\sum_{n\leqslant x}\omega(n)$ for
large values of $x$. The reason is using approximation (2.4) with the
remainder term $R(x)$, and appearing the term
$x\,\mathrm{e}^{-\frac{\sqrt{2}}{6}\sqrt{\log x}}$ in
$\mathcal{E}_{\omega}(x,m)$. This term comes essentially from the classical
zero-free regions for the Riemann zeta function $\zeta(s)$. The situation
changes as well, when we use approximations for $\pi(x)$ under assuming the
Riemann hypothesis (RH), which asserts that $\Re(s)>\frac{1}{2}$ is a zero-
free region, and indeed it is the best possible zero-free region, for
$\zeta(s)$. Accordingly, it is known [13, Corollary 1] that if the Riemann
hypothesis is true, then

$\pi(x)=\mathrm{li}(x)+O^{\ast}\left(\frac{1}{8\pi}\sqrt{x}\log
x\right)\qquad(x\geqslant 2657).$

By computation, we observe that one may drop the coefficient $\frac{1}{8\pi}$
and get an easy to use bound for global range $x\geqslant 2$, as follows

(2.10)
$\pi(x)=\mathrm{li}(x)+O^{\ast}\left(\widehat{R}(x)\right),\qquad\widehat{R}(x)=\sqrt{x}\log
x.$

Note that the above approximations are close to optimal, because on one hand
von Koch [16] showed that the Riemann hypothesis is equivalent to
$\pi(x)=\mathrm{li}(x)+O(\sqrt{x}\log x)$, and on the other hand Littlewood
[11] proved that letting $b(x)=\frac{\log\log\log x}{\log x}$, there are
positive constants $c_{1}$ and $c_{2}$ such that there are arbitrarily large
values of $x$ for which $\pi(x)>\mathrm{li}(x)+c_{1}\sqrt{x}\,b(x)$ and that
there are also arbitrarily large values of $x$ for which
$\pi(x)<\mathrm{li}(x)-c_{2}\sqrt{x}\,b(x)$. By using conditional
approximation (2.10), we obtain the following analogues of Theorems 2.4, 2.6,
and Corollaries 2.5, 2.7.

######  Theorem 2.8.

Assume that the Riemann hypothesis is true. For any fixed integer $m\geqslant
1$ and for any $x\geqslant\mathrm{e}$ we have

(2.11) $\sum_{n\leqslant x}\omega(n)=x\log\log
x+Mx+x\sum_{j=1}^{m}\frac{a_{j}}{\log^{j}x}+O^{\ast}\left(\widehat{\mathcal{E}}_{\omega}(x,m)\right),$

and

(2.12) $\sum_{n\leqslant x}\Omega(n)=x\log\log
x+M^{\prime}x+x\sum_{j=1}^{m}\frac{a_{j}}{\log^{j}x}+O^{\ast}\left(\widehat{\mathcal{E}}_{\Omega}(x,m)\right),$

where

$\widehat{\mathcal{E}}_{\omega}(x,m)=\left(\frac{3}{2}\right)^{m+1}m!\,\frac{x}{\log^{m+1}x}+4x^{\frac{2}{3}}\log
x+9x^{\frac{2}{3}}\\\
+\left(\left(\frac{3}{2}\right)^{m+1}+1\right)\,\mathrm{e}m!\,\frac{x^{\frac{2}{3}}}{\log
x}+15\sqrt{x}\log x,$

and
$\widehat{\mathcal{E}}_{\Omega}(x,m)=\widehat{\mathcal{E}}_{\omega}(x,m)+\frac{33\sqrt{x}}{\log
x}$.

######  Corollary 2.9.

Assume that the Riemann hypothesis is true, and let $x_{0}=1400387903260$.
Then, for $x\geqslant x_{0}$ we have

(2.13) $\sum_{n\leqslant x}\omega(n)=x\log\log
x+Mx-\left(1-\gamma\right)\frac{x}{\log
x}+O^{\ast}\left(\frac{11x}{\log^{2}x}\right),$

and

(2.14) $\sum_{n\leqslant x}\Omega(n)=x\log\log
x+M^{\prime}x-\left(1-\gamma\right)\frac{x}{\log
x}+O^{\ast}\left(\frac{12x}{\log^{2}x}\right),$

and consequently $\frac{1}{x}\sum_{n\leqslant x}\omega(n)-\log\log x<M$ and
$\frac{1}{x}\sum_{n\leqslant x}\Omega(n)-\log\log x<M^{\prime}$.

######  Remark 2.10.

According to partial computations we could run, it seems that the inequality
$\mathcal{A}_{0}(n)<M$ holds for $n\geqslant 16$, however, it fails for
$n=15$. Also, as we mentioned above, the inequality
$\mathcal{A}_{1}(n)<M^{\prime}$ holds for any integer $n\geqslant 2$. A
computational challenge is to check validity of them up to $x_{0}$, hence we
will get a global conditional bound under RH. More generally, we ask about
finding bounds for the difference $\mathcal{A}_{s}(n)$ for any fixed real
$s>0$. A strategy to attack this problem is to make explicit the argument used
in [9] to approximate the average difference
$\mathcal{J}_{s}(n):=\sum_{k\leqslant n}\left(\Omega_{s}(k)-\omega(k)\right)$,
for which it is proved that

$2^{s}\frac{\sqrt{n}}{\log n}\ll
nM^{\prime}_{s}-\mathcal{J}_{s}(n)\ll(2+\varepsilon)^{s}\frac{\sqrt{n}}{\log
n},$

holds for each pair of fixed real numbers $s>0$ and $\varepsilon>0$, and for
$n$ sufficiently large.

######  Remark 2.11.

The Meissel–Mertens constant $M$ [4, pp. 94–98] is determined by

$M=\gamma+\sum_{p}\left(\log\Big{(}1-p^{-1}\Big{)}+p^{-1}\right),$

where $\gamma$ is the Euler–Mascheroni constant [4, pp. 24–40]. Also, see the
impressive survey [10] for more information about $\gamma$. Among several
properties of the constants $M$ and $M^{\prime}$ we have the following rapidly
converging series

$M=\gamma+\sum_{k=2}^{\infty}\frac{\mu(k)\log\zeta(k)}{k},\quad\text{and}\quad
M^{\prime}=\gamma+\sum_{k=2}^{\infty}\frac{\varphi(k)\log\zeta(k)}{k},$

where $\mu$ is the Möbus function and $\varphi$ is the Euler function.
Computations based on the above series representations yields that

$\displaystyle M$ $\displaystyle\approxeq
0.26149721284764278375542683860869585905156664826120,$ $\displaystyle
M^{\prime}$ $\displaystyle\approxeq
1.03465388189743791161979429846463825467030798434439.$

We have used these values in our numerical verifications of the results of the
present paper. All of computations have been done over Maple software111We
mention that the Maple command to compute $\Omega(n)$ is bigomega(n) and
accordingly, a Maple code to compute $\omega(n)$ is given by
with(numtheory):
rad:= n -> convert(numtheory:-factorset(n), ‘*‘):
smallomega:=n->bigomega(rad(n)); .

##  3\. Proof of unconditional approximations

###### Proof of Theorem 2.1.

Considering the left hand side of (1.4), we observe that the inequalities

$\mathcal{A}_{0}(n)>M-\frac{1.133}{\log n}>\alpha_{0}$

hold when $n>\mathrm{e}^{1.133/(M-\alpha_{0})}\approxeq 102841.56$. Thus, we
obtain the left hand side of (2.1) for any integer $n\geqslant 102842$. By
computation, it holds also for $2\leqslant n\leqslant 102841$ with equality
only for $n=32$. Also, considering the right hand side of (1.4), we observe
that the inequalities

$\mathcal{A}_{0}(n)<M+\frac{1}{2\log^{2}n}<\beta_{0}$

hold when $n>\mathrm{e}^{1/\sqrt{2(\beta_{0}-M)}}\approxeq 2.48$. This
completes the proof. ∎

###### Proof of Theorem 2.2.

Since $\mathrm{e}^{1.175/(M^{\prime}-\alpha_{1})}\approxeq 8.23$, for any
integer $n\geqslant 9$ we have $n>\mathrm{e}^{1.175/(M^{\prime}-\alpha_{1})}$,
or equivalently $M^{\prime}-1.175/\log n>\alpha_{1}$. By using this
inequality, and the left hand side of (1.5) we deduce that
$\mathcal{A}_{1}(n)>\alpha_{1}$ holds for $n\geqslant 24$. By computation, it
holds also for $2\leqslant n\leqslant 24$ with equality only for $n=7$. This
completes the proof. ∎

Proof of Theorems 2.4 and 2.6 and their corollaries based on a series of
lemmas. As in [12], we start by using Dirichlet’s hyperbola method [14,
Theorem 3.1] to get the following result.

######  Lemma 3.1.

For any $x$ and $y$ satisfying $1\leqslant y\leqslant x$, we have

(3.1) $\sum_{n\leqslant x}\omega(n)=\sum_{p\leqslant
y}\left[\frac{x}{p}\right]+\sum_{n\leqslant\frac{x}{y}}\pi\left(\frac{x}{n}\right)-\left[\frac{x}{y}\right]\pi(y).$

###### Proof.

Let $\mathbf{1}(n)=1$ be the unitary arithmetic function, and $\varpi(n)$ be
the characteristic function of primes; that is $\varpi(n)=1$ when $n$ is
prime, and $\varpi(n)=0$ otherwise. We consider Dirichlet convolution of these
two functions,

$\mathbf{1}\ast\varpi(n)=\varpi\ast\mathbf{1}(n)=\sum_{d|n}\varpi(d)\,\mathbf{1}\left(\frac{n}{d}\right)=\sum_{d|n}\varpi(d)=\sum_{p|n}1=\omega(n).$

Note that $[x]=\sum_{n\leq x}\mathbf{1}(n)$, and $\pi(x)=\sum_{n\leqslant
x}\varpi(n)$. Thus, by using Dirichlet’s hyperbola method, for any $y$
satisfying $1\leqslant y\leqslant x$ we deduce that

$\sum_{n\leqslant x}\omega(n)=\sum_{n\leqslant
x}\mathbf{1}\ast\varpi(n)=\sum_{n\leqslant
y}\left[\frac{x}{n}\right]\varpi(n)+\sum_{n\leqslant\frac{x}{y}}\pi\left(\frac{x}{n}\right)-\left[\frac{x}{y}\right]\pi(y).$

This gives (3.1). ∎

######  Lemma 3.2.

For any $x$ and $y$ satisfying $1.2<y\leqslant x$, we have

(3.2) $\sum_{p\leqslant y}\left[\frac{x}{p}\right]=x\log\log
y+Mx+O^{\ast}\left(h_{1}(x,y)\right),$

where

$h_{1}(x,y)=x\,\mathrm{e}^{-\frac{1}{3}\sqrt{\log y}}\left(6\sqrt{\log
y}+19\right)+y.$

###### Proof.

We have

$\sum_{p\leqslant y}\left[\frac{x}{p}\right]=\sum_{p\leqslant
y}\left(\frac{x}{p}-\left\\{\frac{x}{p}\right\\}\right)=x\sum_{p\leqslant
y}\frac{1}{p}+O^{\ast}(y).$

The Stieltjes integral and integration by parts gives

$\displaystyle\sum_{p\leqslant y}\frac{1}{p}$
$\displaystyle=\int_{2^{-}}^{y}\frac{\mathrm{d}\pi(t)}{t}=\frac{\pi(y)}{y}+\int_{2}^{y}\frac{\mathrm{li}(t)}{t^{2}}\,\mathrm{d}t+\int_{2}^{y}\frac{\pi(t)-\mathrm{li}(t)}{t^{2}}\,\mathrm{d}t$
$\displaystyle=\frac{\mathrm{li}(y)}{y}+O^{\ast}\left(\frac{R(y)}{y}\right)+\int_{2}^{y}\frac{\mathrm{li}(t)}{t^{2}}\,\mathrm{d}t+\int_{2}^{y}\frac{\pi(t)-\mathrm{li}(t)}{t^{2}}\,\mathrm{d}t.$

The last integral is dominated by
$\int_{2}^{\infty}\frac{R(t)}{t^{2}}\,\mathrm{d}t$, so it is convergent as
$y\to\infty$. Thus, we have

$\int_{2}^{y}\frac{\pi(t)-\mathrm{li}(t)}{t^{2}}\,\mathrm{d}t=\int_{2}^{\infty}\frac{\pi(t)-\mathrm{li}(t)}{t^{2}}\,\mathrm{d}t+O^{\ast}\left(\int_{y}^{\infty}\frac{R(t)}{t^{2}}\,\mathrm{d}t\right).$

Note that

$\int_{y}^{\infty}\frac{R(t)}{t^{2}}\,\mathrm{d}t=\mathrm{e}^{-\frac{1}{3}\sqrt{\log
y}}\left(6\sqrt{\log y}+18\right).$

Also, integration by parts implies

$\int_{2}^{y}\frac{\mathrm{li}(t)}{t^{2}}\,\mathrm{d}t=-\frac{\mathrm{li}(t)}{t}\Big{|}_{2}^{y}+\int_{2}^{y}\frac{\mathrm{d}t}{t\log
t}=\log\log y-\frac{\mathrm{li}(y)}{y}+\frac{\mathrm{li}(2)}{2}-\log\log 2.$

Combining the above approximations, we deduce that

$\sum_{p\leqslant y}\frac{1}{p}=\log\log
y+C+O^{\ast}\left(\mathrm{e}^{-\frac{1}{3}\sqrt{\log y}}\left(6\sqrt{\log
y}+19\right)\right),$

where

$C=\int_{2}^{\infty}\frac{\pi(t)-\mathrm{li}(t)}{t^{2}}\,\mathrm{d}t+\frac{\mathrm{li}(2)}{2}-\log\log
2.$

Mertens’ approximation concerning the sum of reciprocal of primes [14, Theorem
1.10] asserts that $\sum_{p\leqslant y}\frac{1}{p}-\log\log y\to M$ as
$y\to\infty$. This implies that $C=M$, and concludes the proof. Meanwhile, let
us mention that the equality $C=M$ also implies that

$\int_{2}^{\infty}\frac{\pi(t)-\mathrm{li}(t)}{t^{2}}\,\mathrm{d}t=M+\log\log
2-\frac{\mathrm{li}(2)}{2}\approxeq-0.62759759779276794.$

An additional output of the completed proof. ∎

######  Lemma 3.3.

Let $x$ and $y$ satisfy $x\geqslant\mathrm{e}$ and $1.2<x^{\delta}\leqslant
y\leqslant x^{\Delta}<x$ for some fixed $\delta,\Delta\in(0,1)$. Then, we have

(3.3)
$\sum_{n\leqslant\frac{x}{y}}\pi\left(\frac{x}{n}\right)=\left[\frac{x}{y}\right]\mathrm{li}(y)+x(\log\log
x-\log\log y)\\\
+x\sum_{j=1}^{m}\frac{a_{j}}{\log^{j}x}+O^{\ast}\left(h_{2}(x,y)\right),$

where

$h_{2}(x,y)=\frac{m!}{\delta^{m+1}}\,\frac{x}{\log^{m+1}x}\\\
+\left(1+\frac{1}{\delta^{m+1}}\right)\mathrm{e}m!\,\frac{x^{\Delta}}{\log
x}+x\,\mathrm{e}^{-\frac{1}{3}\sqrt{\log y}}\left(1+\log\frac{x}{y}\right).$

###### Proof.

For $n\leqslant\frac{x}{y}$ we have $\frac{x}{n}\geqslant y\geqslant
x^{\delta}>1.2$. Thus, we may use the approximation (2.4) to get

$\sum_{n\leqslant\frac{x}{y}}\pi\left(\frac{x}{n}\right)=\sum_{n\leqslant\frac{x}{y}}\mathrm{li}\left(\frac{x}{n}\right)+O^{\ast}\left(\sum_{n\leqslant\frac{x}{y}}R\left(\frac{x}{n}\right)\right).$

Since
$\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{li}\left(\frac{x}{t}\right)=-\frac{x}{t^{2}(\log
x-\log t)}$, the Stieltjes integral and integration by parts gives

$\sum_{n\leqslant\frac{x}{y}}\mathrm{li}\left(\frac{x}{n}\right)=\int_{1^{-}}^{\frac{x}{y}}\mathrm{li}\left(\frac{x}{t}\right)\mathrm{d}[t]=\left[\frac{x}{y}\right]\mathrm{li}(y)+x\int_{1}^{\frac{x}{y}}\frac{[t]}{t^{2}(\log
x-\log t)}\,\mathrm{d}t.$

We write $[t]=t-\\{t\\}$ to get

(3.4)
$\sum_{n\leqslant\frac{x}{y}}\mathrm{li}\left(\frac{x}{n}\right)=\left[\frac{x}{y}\right]\mathrm{li}(y)+x(\log\log
x-\log\log y)-\mathcal{E}(x,y),$

with the remainder $\mathcal{E}(x,y)$ given by

$\mathcal{E}(x,y)=x\int_{1}^{\frac{x}{y}}\frac{\\{t\\}}{t^{2}(\log x-\log
t)}\,\mathrm{d}t.$

Letting $g_{x}(t)=(1-\frac{\log t}{\log x})^{-1}$, we have

$\mathcal{E}(x,y)=\frac{x}{\log
x}\int_{1}^{\frac{x}{y}}\frac{\\{t\\}}{t^{2}}\,g_{x}(t)\,\mathrm{d}t=\mathcal{E}_{1}(x,y)-\mathcal{E}_{2}(x,y),$

with

$\mathcal{E}_{1}(x,y)=\frac{x}{\log
x}\int_{1}^{\infty}\frac{\\{t\\}}{t^{2}}\,g_{x}(t)\,\mathrm{d}t,\qquad\mathcal{E}_{2}(x,y)=\frac{x}{\log
x}\int_{\frac{x}{y}}^{\infty}\frac{\\{t\\}}{t^{2}}\,g_{x}(t)\,\mathrm{d}t.$

Since $y\geqslant x^{\delta}$, we have $1\leqslant
t\leqslant\frac{x}{y}\leqslant x^{1-\delta}$, and consequently
$0\leqslant\frac{\log t}{\log x}\leqslant 1-\delta<1$. We use Taylor’s formula
with remainder [1, Theorem 5.19] for the function $u\mapsto(1-u)^{-1}$, which
asserts that if $0\leqslant u\leqslant 1-\delta$ for some fixed
$\delta\in(0,1)$, as in our case, then for any given integer $m\geqslant 1$,

(3.5)
$(1-u)^{-1}=\sum_{r=0}^{m-1}u^{r}+O^{\ast}\left(\frac{1}{\delta^{m+1}}\,u^{m}\right).$

Taking $u=\frac{\log t}{\log x}$ in (3.5), we get

(3.6) $g_{x}(t)=\sum_{r=0}^{m-1}\left(\frac{\log t}{\log
x}\right)^{r}+O^{\ast}\left(\frac{1}{\delta^{m+1}}\left(\frac{\log t}{\log
x}\right)^{m}\right).$

Thus,

$\mathcal{E}_{1}(x,y)=\frac{x}{\log
x}\int_{1}^{\infty}\frac{\\{t\\}}{t^{2}}\sum_{r=0}^{m-1}\left(\frac{\log
t}{\log x}\right)^{r}\,\mathrm{d}t+h_{\delta}(x),$

where

$\displaystyle|h_{\delta}(x)|$ $\displaystyle\leqslant\frac{x}{\log
x}\int_{1}^{\infty}\frac{\\{t\\}}{t^{2}}\frac{1}{\delta^{m+1}}\left(\frac{\log
t}{\log x}\right)^{m}\,\mathrm{d}t$
$\displaystyle\leqslant\frac{1}{\delta^{m+1}}\,\frac{x}{\log^{m+1}x}\int_{1}^{\infty}\frac{\log^{m}t}{t^{2}}\,\mathrm{d}t=\frac{m!}{\delta^{m+1}}\,\frac{x}{\log^{m+1}x}.$

Also, we have

$\frac{x}{\log
x}\int_{1}^{\infty}\frac{\\{t\\}}{t^{2}}\sum_{r=0}^{m-1}\left(\frac{\log
t}{\log x}\right)^{r}\,\mathrm{d}t\\\
=\sum_{j=1}^{m}\frac{x}{\log^{j}x}\int_{1}^{\infty}\frac{\\{t\\}}{t^{2}}\log^{j-1}t\,\mathrm{d}t=-x\sum_{j=1}^{m}\frac{a_{j}}{\log^{j}x}.$

Hence, the following approximation holds for any fixed integer $m\geqslant 1$,
with the coefficients $a_{j}$ given by (1.3),

(3.7)
$\mathcal{E}_{1}(x,y)=-x\sum_{j=1}^{m}\frac{a_{j}}{\log^{j}x}+O^{\ast}\left(\frac{m!}{\delta^{m+1}}\,\frac{x}{\log^{m+1}x}\right).$

To deal with $\mathcal{E}_{2}(x,y)$ we note that by induction on $n\geqslant
0$, we obtain the following anti-derivative formula with the coefficients
$P(n,j)={n\choose j}j!$,

(3.8)
$\int\frac{\log^{n}t}{t^{2}}\,\mathrm{d}t=-\frac{1}{t}\sum_{j=0}^{n}P(n,j)\log^{n-j}t.$

Since $y\leqslant x^{\Delta}$ we get $\frac{x}{y}\geqslant x^{1-\Delta}$.
Thus, for any integer $n\geqslant 0$ we have

$\int_{\frac{x}{y}}^{\infty}\frac{\\{t\\}}{t^{2}}\,\log^{n}t\,\mathrm{d}t\leqslant\int_{x^{1-\Delta}}^{\infty}\frac{\\{t\\}}{t^{2}}\,\log^{n}t\,\mathrm{d}t<\int_{x^{1-\Delta}}^{\infty}\frac{\log^{n}t}{t^{2}}\,\mathrm{d}t$

By using (3.8), and assuming that $x\geqslant\mathrm{e}$, we get

$\displaystyle\int_{x^{1-\Delta}}^{\infty}\frac{\log^{n}t}{t^{2}}\,\mathrm{d}t$
$\displaystyle=\frac{\log^{n}x}{x^{1-\Delta}}\sum_{j=0}^{n}P(n,j)(1-\Delta)^{n-j}\frac{1}{\log^{j}x}$
$\displaystyle<\frac{\log^{n}x}{x^{1-\Delta}}\sum_{j=0}^{n}P(n,j)=\frac{\log^{n}x}{x^{1-\Delta}}\sum_{j=0}^{n}\frac{n!}{j!}<\mathrm{e}n!\,\frac{\log^{n}x}{x^{1-\Delta}}.$

Thus, for any integer $n\geqslant 0$ we obtain

(3.9)
$\mathcal{I}_{n}(x,y):=\int_{\frac{x}{y}}^{\infty}\frac{\\{t\\}}{t^{2}}\,\log^{n}t\,\mathrm{d}t<\mathrm{e}n!\,\frac{\log^{n}x}{x^{1-\Delta}}.$

Applying (3.6) we get

$\displaystyle\frac{\log x}{x}\,\mathcal{E}_{2}(x,y)$
$\displaystyle=\int_{\frac{x}{y}}^{\infty}\frac{\\{t\\}}{t^{2}}\,g_{x}(t)\,\mathrm{d}t$
$\displaystyle=\sum_{r=0}^{m-1}\frac{1}{\log^{r}x}\,\mathcal{I}_{r}(x,y)+O^{\ast}\left(\frac{1}{\delta^{m+1}\log^{m}x}\,\mathcal{I}_{m}(x,y)\right).$

Hence, by using (3.9) we deduce that

$\mathcal{E}_{2}(x,y)<\left(\frac{\mathrm{e}m!}{\delta^{m+1}}+\mathrm{e}\sum_{r=0}^{m-1}r!\right)\frac{x^{\Delta}}{\log
x}.$

Since $\sum_{r=0}^{m-1}r!\leqslant m!$, we obtain

(3.10)
$\mathcal{E}_{2}(x,y)=O^{\ast}\left(\left(1+\frac{1}{\delta^{m+1}}\right)\mathrm{e}m!\,\frac{x^{\Delta}}{\log
x}\right).$

Combining (3.4) with approximations (3.7) and (3.10) we obtain

$\sum_{n\leqslant\frac{x}{y}}\mathrm{li}\left(\frac{x}{n}\right)=\left[\frac{x}{y}\right]\mathrm{li}(y)+x(\log\log
x-\log\log y)+x\sum_{j=1}^{m}\frac{a_{j}}{\log^{j}x}\\\
+O^{\ast}\left(\frac{m!}{\delta^{m+1}}\,\frac{x}{\log^{m+1}x}+\left(1+\frac{1}{\delta^{m+1}}\right)\mathrm{e}m!\,\frac{x^{\Delta}}{\log
x}\right).$

Now, to conclude the proof of (3.3) we need just to approximate the sum
$\sum_{n\leqslant\frac{x}{y}}R\left(\frac{x}{n}\right)$. Since
$n\leqslant\frac{x}{y}$ we have $\frac{x}{n}\geqslant y$. Thus,

$\sum_{n\leqslant\frac{x}{y}}R\left(\frac{x}{n}\right)\leqslant
x\,\mathrm{e}^{-\frac{1}{3}\sqrt{\log
y}}\sum_{n\leqslant\frac{x}{y}}\frac{1}{n}\leqslant
x\,\mathrm{e}^{-\frac{1}{3}\sqrt{\log y}}\left(1+\log\frac{x}{y}\right).$

This completes the proof. ∎

###### Proof of Theorem 2.4.

Considering the hyperbolic identity (3.1) and approximations (3.2) and (3.3)
we get

(3.11) $\sum_{n\leqslant x}\omega(n)=x\log\log
x+Mx+x\sum_{j=1}^{m}\frac{a_{j}}{\log^{j}x}+O^{\ast}\left(h_{3}(x,y)\right),$

where

$h_{3}(x,y)=h_{1}(x,y)+h_{2}(x,y)+\left[\frac{x}{y}\right]\left(\mathrm{li}(y)-\pi(y)\right).$

By using (2.4) we deduce that

$\displaystyle\left[\frac{x}{y}\right]\left(\mathrm{li}(y)-\pi(y)\right)$
$\displaystyle=\left[\frac{x}{y}\right]O^{\ast}\left(R(y)\right)$
$\displaystyle=O^{\ast}\left(x\,\frac{R(y)}{y}\right)=O^{\ast}\left(x\,\mathrm{e}^{-\frac{1}{3}\sqrt{\log
y}}\right).$

Thus, (3.11) holds with
$h_{3}(x,y)=h_{1}(x,y)+h_{2}(x,y)+x\,\mathrm{e}^{-\frac{1}{3}\sqrt{\log y}}$,
or with

$h_{3}(x,y)=\frac{m!}{\delta^{m+1}}\,\frac{x}{\log^{m+1}x}+\left(1+\frac{1}{\delta^{m+1}}\right)\mathrm{e}m!\,\frac{x^{\Delta}}{\log
x}\\\ +x\,\mathrm{e}^{-\frac{1}{3}\sqrt{\log
y}}\left(\log\frac{x}{y}+6\sqrt{\log y}+21\right)+y.$

Now, we take $\delta=\Delta=\frac{1}{2}$, and hence $y=\sqrt{x}$. Note that
the assumption $x\geqslant\mathrm{e}$ covers $x^{\delta}=\sqrt{x}>1.2$. Thus,
we obtain (2.5), and the proof is complete. ∎

###### Proof of Corollary 2.5.

We use (2.5) with $m=1$. Letting

$h(z)=z^{4}\mathrm{e}^{-\frac{\sqrt{2}}{6}z}\left(\frac{z^{2}}{2}+3\sqrt{2}z+21\right)+z^{2}\mathrm{e}^{-\frac{z^{2}}{2}}\left(z^{2}+5\mathrm{e}\right),$

we have

$h(\sqrt{\log
x})=\frac{\log^{2}x}{x}\left(\mathcal{E}_{\omega}(x,1)-\frac{4x}{\log^{2}x}\right).$

By computation, we observe that $h(z)$ is decreasing for $z>23.97$, and
$h(119.02511)<1<h(119.02510)$. When $x\geqslant\mathrm{e}^{14167}$ we have
$\sqrt{\log x}\geqslant 119.02511$ and consequently, $h(\sqrt{\log x})<1$.
Also, we note that $\left(1-\gamma\right)\frac{x}{\log
x}>\frac{5x}{\log^{2}x}$ provided $x>\mathrm{e}^{5/(1-\gamma)}$, and this
holds for the values of $x$ we work here. Hence, we conclude the proof. ∎

By using the following key result, Theorem 2.4 and Corollary 2.5 imply Theorem
2.6 and Corollary 2.7, respectively.

######  Lemma 3.4.

For any $x\geqslant 2$ we have

(3.12) $\mathcal{J}(x):=\sum_{n\leqslant
x}\left(\Omega(n)-\omega(n)\right)=M^{\prime\prime}x+O^{\ast}\left(\frac{33\sqrt{x}}{\log
x}\right).$

###### Proof.

Let $\kappa(x)=\frac{25\sqrt{[x]}}{\log[x]}$. By using the double sided
inequality (2.7), we deduce that

$\displaystyle\mathcal{J}(x)$
$\displaystyle=\sum_{k=1}^{[x]}\left(\Omega(k)-\omega(k)\right)$
$\displaystyle=M^{\prime\prime}[x]+O^{\ast}\left(\kappa(x)\right)=M^{\prime\prime}x+O^{\ast}\left(\kappa(x)+M^{\prime\prime}\right).$

By computation, we observe that
$\kappa(x)+M^{\prime\prime}<\frac{33\sqrt{x}}{\log x}$ for $x\geqslant 2$. ∎

###### Proof of Corollary 2.7.

Approximations (2.6) and (3.12) imply

$\sum_{n\leqslant x}\Omega(n)=x\log\log
x+M^{\prime}x-\left(1-\gamma\right)\frac{x}{\log
x}+O^{\ast}\left(\frac{5x}{\log^{2}x}+\frac{33\sqrt{x}}{\log x}\right).$

We note that

(3.13) $\frac{33\sqrt{x}}{\log x}<\frac{x}{\log^{2}x},\qquad(x\geqslant
155652).$

This completes the proof. ∎

##  4\. Proof of conditional approximations

To prove conditional results, under assuming the Riemann hypothesis, we
reconstruct Lemma 3.2 and Lemma 3.3, replacing $R(x)$ by $\widehat{R}(x)$.

######  Lemma 4.1.

Assume that the Riemann hypothesis is true. Then, for any $x$ and $y$
satisfying $2\leqslant y\leqslant x$, we have

(4.1) $\sum_{p\leqslant y}\left[\frac{x}{p}\right]=x\log\log
y+Mx+O^{\ast}\left(\frac{x}{\sqrt{y}}\,(3\log y+4)+y\right).$

###### Proof.

Note that

$\int_{y}^{\infty}\frac{\widehat{R}(t)}{t^{2}}\,\mathrm{d}t=\frac{2\log
y+4}{\sqrt{y}}.$

Thus, following similar argument as the proof of Lemma 3.2 and by using
(2.10), we deduce that assuming RH, for any $y\geqslant 2$ we have

$\sum_{p\leqslant y}\frac{1}{p}=\log\log y+M+O^{\ast}\left(\frac{3\log
y+4}{\sqrt{y}}\right).$

This completes the proof. ∎

######  Lemma 4.2.

Assume that the Riemann hypothesis is true. Let $x$ and $y$ satisfy
$x\geqslant\mathrm{e}$ and $1.2<x^{\delta}\leqslant y\leqslant x^{\Delta}<x$
for some fixed $\delta,\Delta\in(0,1)$. Then, we have

(4.2)
$\sum_{n\leqslant\frac{x}{y}}\pi\left(\frac{x}{n}\right)=\left[\frac{x}{y}\right]\mathrm{li}(y)+x(\log\log
x-\log\log y)\\\
+x\sum_{j=1}^{m}\frac{a_{j}}{\log^{j}x}+O^{\ast}\left(\widehat{h}_{2}(x,y)\right),$

where

$\widehat{h}_{2}(x,y)=\frac{m!}{\delta^{m+1}}\,\frac{x}{\log^{m+1}x}+\left(1+\frac{1}{\delta^{m+1}}\right)\mathrm{e}m!\,\frac{x^{\Delta}}{\log
x}\\\ +\frac{2x}{\sqrt{y}}\left(\log y+2\right)+15\sqrt{x}\log x.$

###### Proof.

Following similar argument as the proof of Lemma 3.3, we should approximate
the sum $\sum_{n\leqslant\frac{x}{y}}\widehat{R}\left(\frac{x}{n}\right)$, for
which, we have

(4.3)
$\sum_{n\leqslant\frac{x}{y}}\widehat{R}\left(\frac{x}{n}\right)=\sqrt{x}\log
x\,\sum_{n\leqslant\frac{x}{y}}\frac{1}{\sqrt{n}}-\sqrt{x}\sum_{n\leqslant\frac{x}{y}}\frac{\log
n}{\sqrt{n}}.$

Letting $f_{0}(t)=\frac{1}{\sqrt{t}}$ and $f_{1}(t)=\frac{\log t}{t}$, we
observe that $f_{0}(t)$ is decreasing for $t\geqslant 1$, and with
$t_{0}=\mathrm{e}^{2}\approxeq 7.39$, the function $f_{1}(t)$ is increasing
for $1\leqslant t\leqslant t_{0}$ and decreasing for $t\geqslant t_{0}$.
Moreover,

$\max_{t\geqslant 1}f_{1}(t)=f_{1}(\mathrm{e}^{2})=\frac{2}{\mathrm{e}}<1.$

Thus, comparison of a sum and an integral of a monotonic function [14, Theorem
0.4] implies that there exists $\theta_{0}\in[0,1]$ such that

$\sum_{n\leqslant\frac{x}{y}}\frac{1}{\sqrt{n}}=1+\int_{1}^{[\frac{x}{y}]}f_{0}(t)\,\mathrm{d}t+\theta_{0}\left(f_{0}\left(\left[\frac{x}{y}\right]\right)-1\right).$

Since $\max_{t\geqslant 1}f_{0}(t)=f_{0}(1)=1$, we get

(4.4)
$\sum_{n\leqslant\frac{x}{y}}\frac{1}{\sqrt{n}}=\int_{1}^{[\frac{x}{y}]}f_{0}(t)\,\mathrm{d}t+O^{\ast}(3)=\int_{1}^{\frac{x}{y}}f_{0}(t)\,\mathrm{d}t+O^{\ast}(4).$

Also, we write

$\sum_{n\leqslant\frac{x}{y}}\frac{\log n}{\sqrt{n}}=\sum_{1<n\leqslant
7}\frac{\log n}{\sqrt{n}}+\frac{\log
8}{\sqrt{8}}+\sum_{8<n\leqslant\frac{x}{y}}\frac{\log n}{\sqrt{n}}.$

There exists $\theta_{1},\theta_{2}\in[0,1]$ such that

$\sum_{1<n\leqslant 7}\frac{\log
n}{\sqrt{n}}=\int_{1}^{7}f_{1}(t)\,\mathrm{d}t+\theta_{1}f_{1}(7)=\int_{1}^{7}f_{1}(t)\,\mathrm{d}t+O^{\ast}\left(\frac{2}{\mathrm{e}}\right),$

and

$\displaystyle\sum_{8<n\leqslant\frac{x}{y}}\frac{\log n}{\sqrt{n}}$
$\displaystyle=\int_{8}^{[\frac{x}{y}]}f_{1}(t)\,\mathrm{d}t+\theta_{2}\left(f_{1}\left(\left[\frac{x}{y}\right]\right)-f_{1}(8)\right)$
$\displaystyle=\int_{8}^{[\frac{x}{y}]}f_{1}(t)\,\mathrm{d}t+O^{\ast}\left(\frac{4}{\mathrm{e}}\right).$

Thus,

$\sum_{n\leqslant\frac{x}{y}}\frac{\log
n}{\sqrt{n}}=\int_{1}^{[\frac{x}{y}]}f_{1}(t)\,\mathrm{d}t+O^{\ast}\left(\eta\right)=\int_{1}^{\frac{x}{y}}f_{1}(t)\,\mathrm{d}t+O^{\ast}\left(\eta+\frac{2}{\mathrm{e}}\right),$

where
$\eta=\frac{6}{\mathrm{e}}+f_{1}(8)+\int_{7}^{8}f_{1}(t)\,\mathrm{d}t\approxeq
3.68$. Since $\eta+\frac{2}{\mathrm{e}}<5$, we get

(4.5) $\sum_{n\leqslant\frac{x}{y}}\frac{\log
n}{\sqrt{n}}=\int_{1}^{\frac{x}{y}}f_{1}(t)\,\mathrm{d}t+O^{\ast}(5).$

By computation, we have

$\sqrt{x}\log
x\int_{1}^{\frac{x}{y}}f_{0}(t)\,\mathrm{d}t-\sqrt{x}\int_{1}^{\frac{x}{y}}f_{1}(t)\,\mathrm{d}t\\\
=\frac{2x}{\sqrt{y}}\left(\log y+2\right)-2\sqrt{x}\left(\log x+2\right).$

Thus, considering the identity (4.3) and the approximations (4.4) and (4.5),
we deduce that

$\sum_{n\leqslant\frac{x}{y}}\widehat{R}\left(\frac{x}{n}\right)=\frac{2x}{\sqrt{y}}\left(\log
y+2\right)+O^{\ast}\left(15\sqrt{x}\log x\right).$

This completes the proof. ∎

###### Proof of Theorem 2.8.

Considering the hyperbolic identity (3.1) and approximations (4.1) and (4.2)
we get

(4.6) $\sum_{n\leqslant x}\omega(n)=x\log\log
x+Mx+x\sum_{j=1}^{m}\frac{a_{j}}{\log^{j}x}+O^{\ast}\left(\widehat{h}_{3}(x,y)\right),$

where

$\widehat{h}_{3}(x,y)=\widehat{h}_{1}(x,y)+\widehat{h}_{2}(x,y)+\left[\frac{x}{y}\right]\left(\mathrm{li}(y)-\pi(y)\right),$

with $\widehat{h}_{1}(x,y)=\frac{x}{\sqrt{y}}\,(3\log y+4)+y$. By using (2.10)
we deduce that

$\displaystyle\left[\frac{x}{y}\right]\left(\mathrm{li}(y)-\pi(y)\right)$
$\displaystyle=\left[\frac{x}{y}\right]O^{\ast}\left(\widehat{R}(y)\right)$
$\displaystyle=O^{\ast}\left(\frac{x}{y}\,\widehat{R}(y)\right)=O^{\ast}\left(\frac{x\log
y}{\sqrt{y}}\right).$

Thus, (4.6) holds with
$\widehat{h}_{3}(x,y)=\widehat{h}_{1}(x,y)+\widehat{h}_{2}(x,y)+\frac{x\log
y}{\sqrt{y}}$, or with

$\widehat{h}_{3}(x,y)=\frac{m!}{\delta^{m+1}}\,\frac{x}{\log^{m+1}x}+\left(1+\frac{1}{\delta^{m+1}}\right)\mathrm{e}m!\,\frac{x^{\Delta}}{\log
x}\\\ +\frac{6x\log y}{\sqrt{y}}+\frac{8x}{\sqrt{y}}+15\sqrt{x}\log x+y.$

Now, we take $\delta=\Delta=\frac{2}{3}$, and hence $y=x^{\frac{2}{3}}$. Note
that the assumption $x\geqslant\mathrm{e}$ covers $x^{\delta}>1.2$. Thus, we
obtain (2.11), and consequently we get (2.12) by using (3.12). The proof is
complete. ∎

###### Proof of Corollary 2.9.

We use (2.11) with $m=1$. By computation, we observe that
$\widehat{\mathcal{E}}_{\omega}(x,1)<\frac{11x}{\log^{2}x}$ for $x\geqslant
x_{0}$. Thus, we get (2.13), and consequently (2.14), by using the
approximation (3.12) and the inequality (3.13). Also, we note that

$\left(1-\gamma\right)\frac{x}{\log
x}>\frac{12x}{\log^{2}x}>\frac{11x}{\log^{2}x},$

provided $x>\mathrm{e}^{12/(1-\gamma)}$. Since
$x_{0}>\mathrm{e}^{12/(1-\gamma)}$, we conclude the proof. ∎

### Acknowledgement

The author is greatly indebted to Prof. Horst Alzer for suggesting the problem
of finding global numerical bounds for the number-theoretic omega functions
and for many stimulating conversations.

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