# Charge Transfer and Zhang-Rice Singlet Bands in the Nickelate Superconductor
$\mathrm{La_{3}Ni_{2}O_{7}}$ under Pressure

Wéi Wú<EMAIL_ADDRESS>Center for Neutron Science and Technology,
Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices,
School of Physics, Sun Yat-sen University, Guangzhou, Guangdong 510275, China
Zhihui Luo  Center for Neutron Science and Technology, Guangdong Provincial
Key Laboratory of Magnetoelectric Physics and Devices, School of Physics, Sun
Yat-sen University, Guangzhou, Guangdong 510275, China  Dao-Xin Yao  Center
for Neutron Science and Technology, Guangdong Provincial Key Laboratory of
Magnetoelectric Physics and Devices, School of Physics, Sun Yat-sen
University, Guangzhou, Guangdong 510275, China  Meng Wang  Center for Neutron
Science and Technology, Guangdong Provincial Key Laboratory of Magnetoelectric
Physics and Devices, School of Physics, Sun Yat-sen University, Guangzhou,
Guangdong 510275, China

###### Abstract

Recently, high-Tc superconductivity is reported in the bulk nickelate
$\mathrm{La_{3}Ni_{2}O_{7}}$ at pressures above 14 GPa. Here, we study an
eleven-band Hubbard model with hopping parameters derived from ab initio
calculations of $\mathrm{La_{3}Ni_{2}O_{7}}$, by employing large scale
determinant quantum Monte Carlo and cellular dynamical mean-field theory
methods. Our result suggests that the superexchange couplings in this system
are comparable to that of cuprates. The system is a charge transfer insulator
as hole concentration becomes four per site at large Hubbard $U$. Upon hole
doping, two low-energy spin-singlet bands emerge in the system exhibiting
distinct correlation properties: while the one composed of the out-of-plane
Ni-$d_{3z^{2}-r^{2}}$ and O-${p_{z}}$ orbitals demonstrates strong
antiferromagnetic correlations and narrow effective bandwidth, the in-plane
singlet band consisting of the Ni-$d_{x^{2}-y^{2}}$ and O-${p_{x}}/{p_{y}}$
orbitals is in general more itinerant. Over a broad range of hole doping, the
doped holes occupy primarily the $d_{x^{2}-y^{2}}$ and ${p_{x}}/{p_{y}}$
orbitals, as the $d_{3z^{2}-r^{2}}$ and ${p_{z}}$ orbitals retain underdoped.
We propose an effective $t-J$ model to capture the relevant physics and
discuss the implications of our result for comprehending the
$\mathrm{La_{3}Ni_{2}O_{7}}$ superconductivity.

Nickelates superconductivity, Pairing symmetry

††preprint: APS/123-QED

Introduction -  Since the discovery of cuprate superconductors Bednorz and
Müller (1986), understanding and searching for novel high transition
temperature (high-$T_{c}$) superconductors Sun  _et al._ (2023); Luo  _et al._
(2023); Zhang and Rice (1988a); Kotliar (1988); Lee  _et al._ (2006);
Scalapino (2007); Keimer  _et al._ (2015); Maeno  _et al._ (1994); Kamihara
_et al._ (2006); Maeno  _et al._ (2001); Chen  _et al._ (2008); Chubukov  _et
al._ (2008); Hu  _et al._ (2015); Kitatani  _et al._ (2023) has been one of
the major topics in the condensed matter physics. The discovery of infinite
layer nickelate superconductor Li  _et al._ (2019); Anisimov  _et al._ (1999);
Lee and Pickett (2004); Middey  _et al._ (2016); Haule and Pascut (2017); Peil
_et al._ (2019); Jiang  _et al._ (2020); Botana and Norman (2020); Zhang  _et
al._ (2020); Werner and Hoshino (2020); Wang  _et al._ (2020); Lechermann
(2020); Worm  _et al._ (2022); Gu  _et al._ (2020); Chen  _et al._ (2022)
marks a notable recent advancement, although in which the superconductivity
(SC) has been only observed in thin films on substrates Li  _et al._ (2019),
but not yet in bulk samples Ding  _et al._ (2023). The very recently
discovered bulk superconductor $\mathrm{La_{3}Ni_{2}O_{7}}$ under high
pressures Sun  _et al._ (2023) , which exhibits a remarkable $T_{c}$ of $\sim$
80 Kelvins, then represents a significant breakthrough in this field. As
revealed by the density-functional theory (DFT) calculations Luo  _et al._
(2023); Sun  _et al._ (2023); Pardo and Pickett (2011), a hallmark of the
nickelate bi-layer $\mathrm{La_{3}Ni_{2}O_{7}}$ Sun  _et al._ (2023); Liu  _et
al._ (2023a) is the activating of both 3${d_{x^{2}-y^{2}}}$ and
3${d_{3z^{2}-r^{2}}}$ orbitals in vicinity of Fermi level Luo  _et al._
(2023); Gao  _et al._ (2015). This distinct feature may lead to
superconductivity that differs significantly from the cuprates and infinite
layer nickelates. From a theoretical perspective, several crucial questions
arise. First, to understand the driving force behind the SC, it is necessary
to elucidate the magnetic exchange couplings Jiang  _et al._ (2020); Lu  _et
al._ (2021) among the four active $e_{g}$ orbitals in the $\mathrm{NiO_{2}}$
bi-layer. Furthermore, the $e_{g}$ orbitals of $\mathrm{La_{3}Ni_{2}O_{7}}$
under pressure possess a large hole concentration, as the nominal valence of
$\mathrm{Ni}$ is $\mathrm{Ni}^{2.5+}$ here, indicating an average of 1.25
holes per $e_{g}$ orbital. This high hole filling level is on the verge of
quenching SC by overdoping in the context of cuprates. Therefore, resolving
the distributions of the holes in the Ni-3${d_{x^{2}-y^{2}}}$,
Ni-3${d_{3z^{2}-r^{2}}}$, and the correlated O-2$p$ orbitals is crucial for
understanding the correlation effects in $\mathrm{La_{3}Ni_{2}O_{7}}$ under
pressure.

Figure 1:  The seven hopping terms in our 11-band Hubbard model Luo  _et al._
(2023). (a)-(d): Four hopping processes between Ni-$d$ and O-$p$ orbitals that
lead to major superexchanges between Ni-$d$ orbitals. Here
$t_{1}=-1.56,t_{2}=0.75,t_{3}=-1.63,t_{6}=1.37$. The site energy
$\epsilon_{{d_{x^{2}-y^{2}}}}=-1.06,\epsilon_{{d_{3z^{2}-r^{2}}}}=-1.16,\epsilon_{p_{x}/p_{y}}=-4.94,\epsilon_{p_{z}}=-4.30,\epsilon_{p^{\prime\prime}_{z}}=\epsilon_{p^{\prime}_{z}}=-3.77$.
Note that the hopping process (c) is from combining (a) and(b). The
superexchange between the on-site intra-layer ${d_{x^{2}-y^{2}}}$ and
${d_{3z^{2}-r^{2}}}$ orbitals (not shown here) vanishes due to symmetry. (e)
Hoppings between O-$p$ orbitals are shown on a cartoon depicting the structure
of the bilayer $\mathrm{NiO_{2}}$ planes. The 3$d$ orbitals in $\mathrm{Ni}$
(black dots) are not shown for clarity. Here
$t_{4}=0.58,t_{5}=0.49,t_{7}=0.43$. These seven hopping terms combining the
site-energies define our 11-band Hubbard model in the non-interacting limit.

To address above questions, here we study an 11-band Hubbard model that
includes four 3${d_{x^{2}-y^{2}}}$ / 3${d_{3z^{2}-r^{2}}}$ orbitals of nickel,
and seven most relevant 2$p$ orbitals of oxygen in the $\mathrm{NiO_{2}}$ bi-
layer per site. We carry out determinant quantum Monte Carlo simulations
(DQMC) Blankenbecler  _et al._ (1981); Assaad and Evertz (2008) and cellular
dynamical mean-field theory (CDMFT) Georges (1996); Maier  _et al._ (2005)
calculations in the normal state of the system. Our result suggests that the
superexchange couplings in this system are in general comparable to that in
cuprates Weber  _et al._ (2012), supporting a magnetic correlation origin of
the high $T_{c}$ superconductivity in $\mathrm{La_{3}Ni_{2}O_{7}}$ under
pressure. We show that at large Hubbard $U$, the system is a charge-transfer
insulator Karp  _et al._ (2020) in the Zaanen-Sawatzky-Allen (ZSA)
schemeZaanen  _et al._ (1985) at half-filling ( i.e., four holes per site).
Upon hole doping, the spin-singlet band associating with the vertical
Ni-${d_{3z^{2}-r^{2}}}$ \- O-${p_{z}}$ orbitals possesses strong
antiferromagnetic correlations, as where a small hole doping level is
retained. In contrast, the singlet band that consists of the in-plane
Ni-$d_{x^{2}-y^{2}}$ and $\mathrm{O}-{p_{x}}/{p_{y}}$ orbitals, drawing an
analogy to the Zhang-Rice singlet band (ZRSB) in cuprates, exhibits a higher
propensity for hole doping and greater itinerancy. We discuss the interplay
between the two spin-singlet bands and propose an effective $t-J$ model that
considers the leading-order exchange couplings in $\mathrm{La_{3}Ni_{2}O_{7}}$
under pressure.

Model and Methods -  To fully take into account the superexchange couplings in
$\mathrm{La_{3}Ni_{2}O_{7}}$ at pressure, here we consider an 11-band Hubbard
model Luo  _et al._ (2023) that can be written as,

$\displaystyle
H=\sum_{i,j,\alpha,\beta,\sigma}t_{i,j,\alpha,\beta}d_{i\alpha\sigma}^{\dagger}c_{j\beta\sigma}+\sum_{i,j,\alpha,\beta,\sigma}t_{i,j,\alpha,\beta}c_{i\alpha\sigma}^{\dagger}c_{j\beta\sigma}$
$\displaystyle+\sum_{i\alpha\sigma}(\epsilon_{\alpha}-\mu)n_{i\alpha\sigma}^{d}+\sum_{i\alpha\sigma}(\epsilon_{\alpha}-\mu)n_{i\alpha\sigma}^{c}$
(1)
$\displaystyle+\sum_{i\alpha}U_{dd}n_{i\alpha\uparrow}^{d}n_{i\alpha\downarrow}^{d}+\sum_{i,\alpha\neq\beta,\sigma,\sigma^{\prime}}U_{dd^{\prime}}n_{i\alpha\sigma}^{d}n_{i\beta\sigma^{\prime}}^{d}-\sum_{i\alpha\sigma}E_{dc}n_{i\alpha\sigma}^{d}$

where $t_{i,j,\alpha,\beta}$ denote hoppings between electrons on sites
($i,j$) and orbital ($\alpha,\beta$)( can be either Ni-$d$ or O-$p$ orbitals).
$d^{\dagger}_{\alpha,i,\sigma}$ ($c^{\dagger}_{\alpha,i,\sigma}$) is the
creation operator for electrons on $\alpha\in$ 3$d$ ($\in$ 2$p$) orbital.
$\epsilon_{\alpha}$ is the site-energy of $\alpha-$ orbital. $U_{dd}$ is the
Hubbard interaction between two electrons on the same $d$-orbital
(${d_{x^{2}-y^{2}}}$ or ${d_{3z^{2}-r^{2}}}$) and $U_{dd^{\prime}}$ is for
that on two different $d$-orbital. $E_{dc}$ is the double counting (DC) term
Anisimov  _et al._ (1991); Karolak  _et al._ (2010); Wang  _et al._ (2012) to
be subtracted in the DQMC or CDMFT. Here we use the Held’s formula Held
(2007): $E_{dc}=\frac{1}{3}(U_{dd}+2U_{dd^{\prime}})(n^{0}_{d}-0.5)$, with
$n^{0}_{d}$ being the occupation number of $d-$orbitals in the non-interacting
limit, $n^{0}_{d}\approx 2.16$. We have carefully checked that this DC term
does not shift the non-interacting Fermi surface (FS) significantly Karolak
_et al._ (2010). We adopt the hopping parameters and site-energies proposed in
Ref. Luo  _et al._ (2023), which is obtained by downfolding the DFT result in
the maximally localized Wannier orbitals. See Fig. 1 for all the hopping
parameters and site-energy values. In line with DFT result Luo  _et al._
(2023), we assume that the chemical potential $\mu=0$ in above Hamiltonian
corresponds to the single crystal $\mathrm{La_{3}Ni_{2}O_{7}}$ material at
pressures $>14$GPa, without considering other potential doping effects (oxygen
deficiency for example). We will also vary $\mu$ to explore regimes with
different hole concentrations that defined as
$n_{h}=(22-\sum_{\alpha,\sigma}n_{\alpha,\sigma}^{d})/4$, namely, the average
number of holes per $d-$ orbitals per site. $n_{h}=1$ corresponds to half-
filling in our study. In this work, we neglect the Hund’s coupling and keep
$U_{dd^{\prime}}=0.7U_{dd}$ when $U_{dd}$ is varied for the 11-band Hubbard
model. Here we use two typical values of $U_{dd}$, $U_{dd}=7$ eV and
$U_{dd}=9$ eV, where no qualitative difference is found between the two
results. Below we use electron volt ($\mathrm{eV}$) as the energy unit
throughout the paper. For the calculations on cuprate, we employ a canonical
set of parameters for the three-band Hubbard model Weber  _et al._ (2012) (one
${d_{x^{2}-y^{2}}}$ orbital and two ${p_{x}}$ / ${p_{y}}$ orbitals in the
$\mathrm{CuO_{2}}$ plane):
$t_{pd}=1.39,t_{pp}=0.64,t^{\prime}_{pp}=0.103,\Delta_{dp}\equiv\epsilon_{d}-\epsilon_{p}=2.6,U_{dd}=8.5,\mathrm{E_{DC}}=3.12$.
This set of parameters is assumed to be most relevant for the $\mathrm{LSCO}$
compound, which has been used in different studies of cuprates Weber  _et al._
(2012); Kung  _et al._ (2016); Kowalski  _et al._ (2021); Mai  _et al._
(2021).

Figure 2: Magnetic correlations of the 11-band Hubbard model (a): The spin-
spin correlation function $\langle S_{i,\alpha}\cdot S_{j,\beta}\rangle$ for
four neighboring d-orbitals are shown in numbers to profile the relative
strength of the antiferromagnetic exchange couplings in the system. Brown
(blue) symbols denote ${d_{x^{2}-y^{2}}}$ (${d_{3z^{2}-r^{2}}}$) orbitals.
Result is from DQMC calculations on $6\times 6\times 11$ lattice at half-
filling ($n_{h}=1$), $T=0.25$. (b) Magnetic correlation between a pair of
neighboring $d$ orbitals in Ni-11-band and Cu-3-band model as a function of
temperature $T$ at half-filling. For $\mathrm{Ni}$, we use $6\times 6\times
11$ orbitals and $6\times 6\times 3$ orbitals for Cu. Black curve shows result
for intra-layer (IR) ${d_{x^{2}-y^{2}}}$\- ${d_{x^{2}-y^{2}}}$ correlations,
while green curve denotes the correlation between the inter-layer (IT)
${d_{3z^{2}-r^{2}}}$ \- ${d_{3z^{2}-r^{2}}}$ orbitals. (c) The spin structure
factor $S(Q)=\frac{1}{N}\sum_{i,j}\langle S_{i\alpha}\cdot S_{j\beta}\rangle
e^{-iQ\cdot(R_{i}-Rj)}$ for Ni (intra-layer, $\alpha=\beta={d_{x^{2}-y^{2}}}$
component) as a function of hole concentration $n_{h}$ compared with that of
cuprate. Here $U_{dd}=7$ for nickelate and $U_{dd}=8.5$ for cuprate.

For DQMC simulation, we use a two dimensional $6\times 6\times 11=396$
orbitals lattice with periodic conditions for the 11-band Hubbard model, on
which we have verified that the finite size effects are negligible in the
parameter regime we study. For CDMFT study, we carry out computations in the
normal state, where the $2\times 2\times 11=44$ orbitals cluster effective
impurity model is used. The Hirsch-Fye quantum Monte Carlo (HFQMC) is used as
impurity solver. The time discretions are $\Delta\tau=0.0625$ (DQMC) and
$\Delta\tau=0.078$ (HFQMC).

Superexchanges- We first discuss the property of the magnetic exchange
couplings in the system. As shown in Fig. 1, there are a few hopping processes
can give rise to significant superexchanges. In the atomic limit of the
charge-transfer picture, the spin singlet state of two Ni-$d$ electrons
acquires an energy gain of
$J=\frac{-4t^{4}_{pd}}{(U_{dd}+\Delta_{pd})^{2}}\times(\frac{1}{U_{dd}}+\frac{1}{U_{dd}+\Delta_{pd}})$
over the spin triplet states, where $\Delta_{pd}=\epsilon_{d}-\epsilon_{p}$,
and $t_{pd}$ is the hopping between Ni-$d$ and O-$p$ orbitals. In principle,
inclusion of the hopping integrals between O-$p$ orbitals ($t_{pp}$) and
interaction effects may modify this superexchange coupling Weber  _et al._
(2012). The numbers shown in Fig. 2a are the magnitudes of spin correlation
function $\langle S_{i,\alpha}\cdot S_{j,\beta}\rangle$ for a few pairs of
neighboring $d-$ orbitals at $T=0.25$, where $S_{i,\alpha}$ is the spin
operator at site $i$ and orbital $\alpha$. This result profiles the fact that
there are three types of main exchange couplings in the system: the inter-
layer (IT) on-site ${d_{3z^{2}-r^{2}}}$-${d_{3z^{2}-r^{2}}}$ antiferromagnetic
exchange [ $\langle S\cdot S\rangle=-0.113(2)$] dominates the exchange
couplings in the system. Then it is the intra-layer (IR) nearest-neighboring
${d_{x^{2}-y^{2}}}$-${d_{x^{2}-y^{2}}}$ exchange [$\langle S\cdot
S\rangle=-0.054(1)$]. The intra-layer ${d_{3z^{2}-r^{2}}}$-${d_{x^{2}-y^{2}}}$
superexchange, in contrast, is significantly weaker than the aforementioned
two [$\langle S\cdot S\rangle=-0.017(1)$]. Finally, we note that the intra-
layer ${d_{3z^{2}-r^{2}}}$-${d_{3z^{2}-r^{2}}}$ exchanges are less than 1/20
of inter-layer ${d_{3z^{2}-r^{2}}}$-${d_{3z^{2}-r^{2}}}$ coupling, hence can
be neglected in further analysis. It is worthy noting that in CDMFT at lower
temperatures ($T\sim 0.1$), we observe similar relative strengths of the
magnetic correlations at half-filling(not shown). Fig. 2b and 2c show
respectively $\langle S_{i,\alpha}\cdot S_{j,\beta}\rangle$ as a function of
temperature at half-filling, and spin structure factor $S_{\alpha,\beta}(Q)$
as a function of $n_{h}$ at $T=0.3$, with comparison from the result of 3-band
Hubbard model of cuprates. As one can see that the intra-layer
${d_{x^{2}-y^{2}}}$-${d_{x^{2}-y^{2}}}$ antiferromagnetic correlations are
suggested to be in general comparable to its cuprate counterpart, while the
inter-layer ${d_{3z^{2}-r^{2}}}$-${d_{3z^{2}-r^{2}}}$ coupling seems to be
essentially stronger than the former. This result implies that the
antiferromagnetic correlations between inter-layer ${d_{3z^{2}-r^{2}}}$
orbitals could be at the origin of the observed superconductivity in
$\mathrm{La_{3}Ni_{2}O_{7}}$ under pressure. Fig. 3a and 3b show $\langle
S_{i,\alpha}\cdot S_{j,\beta}\rangle$ respectively as a function of $U_{dd}$
at $n_{h}=1.25$, and as a function of $n_{h}$ at $U_{dd}=7$ in CDMFT, from
which one sees that varying the value of $U_{dd}$ in the range of $(4\sim 9)$
eV does not change substantially the magnetic correlations between the inter-
layer ${d_{3z^{2}-r^{2}}}$ orbitals. More importantly, as shown in Fig. 3b, it
does not vanish until a huge hole doping $p=n_{h}-1>0.4$ is approached.

Figure 3: Magnetic correlations between two neighboring $d-$ orbitals from
CDMFT. (a): The spin-spin correlation function $\langle S_{i,\alpha}\cdot
S_{j,\beta}\rangle$ at $n_{h}=1.25\quad(\mu\sim 0),T=0.1$ as a function of
$U_{dd}$. Results are for two inter-layer (IT) ${d_{3z^{2}-r^{2}}}$ orbitals
and two intra-layer (IR) ${d_{x^{2}-y^{2}}}$ orbitals. (b) $\langle
S_{i,\alpha}\cdot S_{j,\beta}\rangle$ as a function of hole concentration
$n_{h}$ at $U_{dd}=7,T=0.1$. Dashed line indicates $\langle S_{i,\alpha}\cdot
S_{j,\beta}\rangle$ between the neighboring ${d_{x^{2}-y^{2}}}$ orbitals of
the 3-band Hubbard model for cuprate at fixed $n_{h}=1.05,T=0.1$.  Figure 4:
The hole filling per $d-$ orbital $n_{h}$ as a function of hole chemical
potential $\mu_{h}\equiv-\mu$. Both DQMC (dots) and CDMFT (diamonds) suggest
that a charge gap opens as hole chemical potential decreases to around
$\mu_{h}\sim-1.6$. Hollow symbols show that our CDMFT result are in excellent
agreement with that of DQMC at $T=0.3$. Green arrows indicate the $\mu_{h}$
range where charge gap opens in CDMFT. Inset: The change of hole concentration
$\Delta n^{\alpha}_{h}$ for different orbital-$\alpha$ as a function of hole
chemical potential in CDMFT at $T=0.1$. $\Delta n^{\alpha}_{h}$ is defined as
$\Delta n^{\alpha}_{h}\equiv
n^{\alpha}_{h}(\mu_{h})-n^{\alpha}_{h}(\mu_{h}=-1.6)$, where
$n^{\alpha}_{h}(\mu_{h}=-1.6)$ is the hole filling of orbital-$\alpha$ at
$\mu_{h}=-1.6$ ( where $n_{h}\approx 1.0$ ). Here $U_{dd}=9$. Similar charge-
transfer insulating behaviour is also observed at $U_{dd}=7$ in CDMFT (not
shown).

Charge transfer insulator- We now focus on the metal- insulator transition in
the 11-band Hubbard model. Fig. 4 displays the hole filling per Ni-$d$ orbital
$n_{h}$ as a function of hole chemical potential $\mu_{h}\equiv-\mu$. As one
can see that the DQMC result (dots) of $n_{h}$ exhibit an inflection point
around $n_{h}=1$ ( $\mu_{h}\sim-1.6$) with small values of compressibility
$\partial n_{h}/\partial\mu_{h}$, implying the forming of a charge gap. CDMFT
result are obtained at lower temperatures (diamonds), where one indeed see a
flat plateau in the $n_{h}$ curve between $\mu_{h}\in(-1.6\sim-2.4)$,
indicating the opening of a charge gap at half-filling. In Fig.4, DQMC and
CDMFT results at the same temperature $T=0.3$ are shown to be in good
agreement. To understand the nature of this insulating behavior, we further
study how the hole concentration changes in different orbitals as a function
$\mu_{h}$ in the Inset of Fig. 4. In other words, as chemical potential
$\mu_{h}$ is increased from the half-filling value $\mu_{h}\sim-1.6$, holes
can be added into different Ni-$d$ and O-$p$ orbitals of the system, which is
denoted by $\Delta n^{\alpha}_{h}$ in the Inset of Fig. 4. As one can see that
the doped holes go primarily to the oxygen orbitals, which unambiguously
points to the charge-transfer nature of the insulating state at half-filling
of this system. Similar to cuprates Kowalski  _et al._ (2021), here
${d_{x^{2}-y^{2}}}$ orbital also has a sizable portion of the doped holes. It
is remarkable that, however, the hole content of ${d_{3z^{2}-r^{2}}}$ orbital
almost do not change with $\mu_{h}$ for $\mu_{h}<0$ ( or for hole doping
$p\lesssim 22\%$ ). We note that in cuprates, a smaller portion of holes
residing on cations in general indicates a larger superexchange and higher
superconducting $T_{c}$ Ruan  _et al._ (2016); Kowalski  _et al._ (2021).

Zhang-Rice singlet band- In Fig. 5a, and Fig. 5b, we plot the local density of
states (DOS) for Ni-${d_{x^{2}-y^{2}}}$ and Ni-${d_{3z^{2}-r^{2}}}$ orbitals
respectively at $U_{dd}=9$. That of the in-plane O-${p_{x}}$/${p_{y}}$
orbitals, and of the out-of-plane O${p_{z}}$ orbitals are also plotted along
respectively. Here $n_{h}\approx 1.05,T=0.1$. Near the Fermi level, the local
DOS are shown to have mixed weights of Ni-$d$ and O-$p$ orbitals. To be more
specific, as depicted in Fig. 5a, the low energy DOS of ${d_{x^{2}-y^{2}}}$
and O-${p_{x}}$ /${p_{y}}$ orbitals share similar structure (peak-dip-peak)
near Fermi level, suggesting the presence of a Zhang-Rice singlet bandZhang
and Rice (1988b) (ZRSB-I) with correlated
${d_{x^{2}-y^{2}}}$-${p_{x}}$/${p_{y}}$ electrons. Likewise, in Fig. 5b, the
Ni-${d_{3z^{2}-r^{2}}}$ orbital and O-${p_{z}}$ orbitals also exhibit similar
structure (a single narrow peak) in the low-energy DOS, suggesting the
formation of another singlet band (ZRSB-II) along the c-axis. It is remarkable
that the vertical ZRSB-II singlet band are much narrower in bandwidth
comparing to ZRSB-I, which suggests a more localized nature of the
${d_{3z^{2}-r^{2}}}$ orbital.

There are a few points we would like to emphasize. First, the vertical
${d_{3z^{2}-r^{2}}}$ singlet band is quite different from a conventional ZRSB
in cuprates. In cuprates, a doped hole at oxygen sites hybridize with a
$\mathrm{Cu^{2+}}$ hole in terms of the superposition of four O$p$ hole states
adjacent to the $\mathrm{Cu^{2+}}$ iron, forming a spin singlet. Then the
singlet moves effectively in the antiferromagnetic background of
$\mathrm{Cu^{2+}}$ lattice with a bandwidth being $(2\sim 3)$eV Kowalski  _et
al._ (2021). Here the vertical singlet states of
Ni-${d_{3z^{2}-r^{2}}}$-O-${p_{z}}$ have a narrow bandwidth, and they barely
interact with each other directly. Instead, the doped holes in the in-plane
${p_{x}}/{p_{y}}$ orbitals can hybridize with the vertical singlets via
${d_{3z^{2}-r^{2}}}$ \- ${p_{x}}$/${p_{y}}$ hopping ($t_{2}=0.75$), or via the
${p_{x}}$ -${p_{z}}$ hopping ($t_{5}=0.49,t_{7}=0.43$ , see also Fig. 1). As a
result, the vertical ${d_{3z^{2}-r^{2}}}$ singlet band may behave more like
scattering centers with antiferromagntic characteristics in the system.
Finally, upon heavy hole doping, one can expect that the in-plane ZRSB will be
destroyed, and, in contrast, the vertical singlet band can maintain intact,
due to the imbalanced distributions of the doped holes in the two charge-
transfer bands.

Figure 5: Spin singlet bands and charge-transfer gaps in the local density of states (DOS). (a): DOS of the in-plane ${d_{x^{2}-y^{2}}}$ and ${p_{x}}$ (${p_{y}}$) orbitals. (b): DOS of the out-of-plane ${d_{3z^{2}-r^{2}}}$ and ${p_{z}}{{}^{\prime}}$ (${p_{z}}{{}^{\prime\prime}}$) orbitals. Dashed vertical lines indicate the Zhang-Rice singlet band (ZRSB) while arrows show the charge-transfer gap (CTG). UHB stands for upper Hubbard band, and LHB stands for lower Hubbard band. DOS are obtained by maximum entropy (MEM) analytic continuation Bergeron and Tremblay (2016) of the Matsubara Green’s functions that obtained by CDMFT at $T=0.1,n_{h}=1.05$.  |  $\mu_{h}$ | $n_{h}$ |  $n^{IP}_{h}\hskip 14.22636pt$ | $n^{OP}_{h}$
---|---|---|---|---
DQMC ($U_{dd}=7$) | 0.0 | 1.243 | 1.412 | 1.075
CDMFT ($U_{dd}=7$) | 0.0 | 1.255 | 1.446 | 1.064
CDMFT ($U_{dd}=9$) | 0.0 | 1.216 | 1.384 | 1.047
CDMFT ($U_{dd}=6^{*}$) | 0.0 | 1.262 | 1.428 | 1.095
Table 1:  Hole concentrations of ${d_{x^{2}-y^{2}}}-$, and
${d_{3z^{2}-r^{2}}}-$ correlated orbitals at $\mu_{h}=0$.
$n^{IP}_{h}=n^{h}_{\mathrm{Ni}{d_{x^{2}-y^{2}}}}+n^{h}_{O{p_{x}}}+n^{h}_{O{p_{y}}}$
counts the number of holes in the in-plane ${d_{x^{2}-y^{2}}}$ orbital and
${p_{x}}$ /${p_{y}}$ orbitals, while
$n^{OP}_{h}=n^{h}_{\mathrm{Ni}{d_{3z^{2}-r^{2}}}}+0.5(n^{h}_{O{p_{z}}}+n^{h}_{O{p_{z}}^{\prime}}+n^{h}_{O{p_{z}}^{\prime\prime})}$
counting the number of holes in the out-of-plane ${d_{3z^{2}-r^{2}}}$ orbital
and ${p_{z}}$ orbitals. $n_{h}=(n^{IP}_{h}+n^{OP}_{h})/2$. See also Fig. 1.
DQMC is done at $T=0.25$, while CDMFT results are obtained at $T=0.1$, except
the last one with $U_{dd}=6,U_{dd^{\prime}}=0$, where $T=0.05$.

In Table-I we inspect specifically the hole filling level of the system at
$\mu=0$, which may arguably correspond to that of the real material of
$\mathrm{La_{3}Ni_{2}O_{7}}$ under pressure. As one can see that, the 11-band
Hubbard model is about $21\sim 26\%$ hole doped at $\mu=0$, with small
variations up to the value of Hubbard $U$ and temperatures $T$. This result
roughly coincides with nominal average doping level $(=25\%)$ of the $e_{g}$
orbitals in $\mathrm{La_{3}Ni_{2}O_{7}}$. We find that the hole doping in the
out-of-plane ${d_{3z^{2}-r^{2}}}$ and ${p_{z}}$ orbitals combined,
$p=(n^{OP}_{h}-1)\approx(5\%\sim 8\%)$ , while that of the in-plane
${d_{x^{2}-y^{2}}}$ and ${p_{x}}/{p_{y}}$ orbitals, $p=(n^{IP}_{h}-1)$ is
about $\approx 40\%$ at $\mu=0$. Given this large value of hole doping of
$n^{IP}_{h}$ , one may expect that the in-plane ${d_{x^{2}-y^{2}}}$ and
${p_{x}}/{p_{y}}$ orbitals can be seen as itinerant orbitals in
$\mathrm{La_{3}Ni_{2}O_{7}}$. However, we find that in CDMFT at $\mu=0$, the
magnetic correlations between intra-layer
${d_{x^{2}-y^{2}}}-{d_{x^{2}-y^{2}}}$ orbital is weak but not vanishing ( see
Fig. 3b). It is notable that the strange metal (SM) phase can be very
sensitive to weak magnetic correlations Wú  _et al._ (2022). Hence, if the
observed strange metal state in $\mathrm{La_{3}Ni_{2}O_{7}}$ at
$P>18\mathrm{GPa}$ is related to also the magnetic correlations in
${d_{x^{2}-y^{2}}}$ orbitals, then the latter may cannot be simplified as pure
itinerant.

Effective t-J model- Based on our study above, we propose a four-band $t-J$
model to describe the low-energy physics of the $\mathrm{La_{3}Ni_{2}O_{7}}$
under pressure. The proposed Hamiltonian can be written as
$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{J}$ with $\mathcal{H}_{0}=\sum_{{\rm
k}\sigma}\Psi_{{\rm k}\sigma}^{\dagger}H({\rm k})\Psi_{{\rm k}\sigma}$, which
reads,

$\displaystyle H(k)_{1,1}=H(k)_{3,3}=-2t^{x}_{1}[\rm cos(k_{x})+cos(k_{y})]$
$\displaystyle-4t^{x}_{2}\rm cos(k_{x})cos(k_{y})+\epsilon_{x}$ $\displaystyle
H(k)_{2,2}=H(k)_{4,4}=-2t^{z}_{1}[\rm cos(k_{x})+cos(k_{y})]$
$\displaystyle-4t^{z}_{2}\rm cos(k_{x})cos(k_{y})+\epsilon_{z}$ $\displaystyle
H(k)_{1,2}=H(k)_{3,4}=-2t^{xz}[\rm cos(k_{x})-cos(k_{y})]$ $\displaystyle
H(k)_{2,4}=-2V_{\perp}[\rm cos(k_{x})-cos(k_{y})]\vskip 12.0pt plus 4.0pt
minus 4.0pt$ (2)

where
$\Psi_{\sigma}=\left(d_{x_{1}\sigma},d_{z_{1}\sigma},d_{x_{1}\sigma},d_{z_{2}\sigma}\right)^{T}$,
denoting the annihilation operators of ${d_{x^{2}-y^{2}}}$ and
${d_{3z^{2}-r^{2}}}$ orbitals in the two $\mathrm{NiO_{2}}$ layer. The
$\mathcal{H}_{0}$ part is taken from the down-folded tight-binding model from
Luo et al’s work in Ref. Luo  _et al._ (2023), namely, $t^{x}_{1}\approx
0.5,t^{x}_{2}\approx 0.07,t^{z}_{1}\approx 0.11,t^{x}_{2}\approx
0.02,t^{xz}=-0.24,V_{\perp}=0.64$ . Note that the renormalization factors
$\mathrm{g_{t}}$ Zhang  _et al._ (1988) associating the corresponding hopping
amplitudes in Eq. Charge Transfer and Zhang-Rice Singlet Bands in the
Nickelate Superconductor $\mathrm{La_{3}Ni_{2}O_{7}}$ under Pressure are not
explicitly written down.

For the interacting part $\mathcal{H}_{J}$, we consider three main magnetic
exchanges terms,

$\displaystyle\mathcal{H}_{J}=J_{1}\sum_{i}{(S_{i,z_{1}}S_{i,z_{2}}-\frac{1}{4}n_{i,z_{1}}n_{i,z_{2}})}$
$\displaystyle+J_{2}\sum_{\langle
i,j\rangle,\alpha=x_{1},x_{2}}(S_{i,\alpha}S_{j,\alpha}-\frac{1}{4}n_{i,\alpha}n_{j,\alpha})$
$\displaystyle+J_{3}\sum_{\begin{subarray}{c}{\langle i,j\rangle}\\\
{\alpha,\beta=(x_{1},z_{1})/(x_{2},z_{2})}\end{subarray}}(S_{i,\alpha}S_{j,\beta}-\frac{1}{4}n_{i,\alpha}n_{j,\beta})$
(3)

where $J_{1}$ captures the exchange couplings between the on-site inter-layer
${d_{3z^{2}-r^{2}}}$ orbitals, $J_{2}$ the exchanges between the intra-layer
${d_{x^{2}-y^{2}}}$ orbitals on nearest neighboring (NN) sites, and finally
$J_{3}$ for the intra-layer ${d_{x^{2}-y^{2}}}$ \- ${d_{3z^{2}-r^{2}}}$
exchanges on NN sites. Considering our magnetic correlation results compared
with LSCO cuprate Wakimoto  _et al._ (2007); Wang  _et al._ (2022), typical
vales of the antiferromagnetic exchange couplings can be set around $J_{1}\sim
0.18\mathrm{eV}$, $J_{2}\sim 0.09\mathrm{eV}$ , and $J_{3}\sim
0.03\mathrm{eV}$. Note that if the derived effects from Hund’s coupling are
considered De’Medici  _et al._ (2011), then $J_{3}$ may be enhanced to be
comparable to $J_{2}$.

Discussion and Conclusion -  After the discovery of high-$T_{c}$
superconductivity in $\mathrm{La_{3}Ni_{2}O_{7}}$ under pressure Sun  _et al._
(2023), a number of effective interacting models have been proposed to study
the pairing symmetry Yang  _et al._ (2023); Shen  _et al._ (2023); Gu  _et
al._ (2023); Sakakibara  _et al._ (2023), as inspired by the ab initio
calculations Luo  _et al._ (2023); Zhang  _et al._ (2023); Lechermann  _et
al._ (2023); Christiansson  _et al._ (2023); Shilenko and Leonov (2023) or
phenomenological insights. These effective models ignore the oxygen degrees of
freedom, incorporating the magnetic correlations between Ni orbitals in terms
of direct exchanges. Hence, the charge transfer property, which has shown to
be one of the key ingredients in determining $T_{c}$ of cuprate
superconductors Weber  _et al._ (2012); Ruan  _et al._ (2016); Rybicki  _et
al._ (2016); Kowalski  _et al._ (2021); O’Mahony  _et al._ (2022), as well as
the superexchange couplings, are beyond the scope of those effective models.
In this work, we have studied an 11-band Hubbard model including both Ni-$3d$
orbitals and relevant O-$2p$ orbitals. We reveal the relative strengths of the
superexchange couplings between different Ni-$3d$ orbitals. We find two spin-
singlet bands in the system with strikingly different hole concentrations and
correlation strengths. These results suggest that the strong antiferromagnetic
correlations within the inter-layer ${d_{3z^{2}-r^{2}}}$ orbitals might be the
driving force of the SC in $\mathrm{La_{3}Ni_{2}O_{7}}$. The role of the more
itinerant ${d_{x^{2}-y^{2}}}$ orbital in SM and SC states, owing to the
remnant magnetic correlations within and the specific geometry of the
$\alpha,\beta-$ sheets of the Fermi surface Luo  _et al._ (2023), might be
complicated though. Finally, we propose a four band $t-J$ model with three
different exchange coupling $J_{1},J_{2},J_{3}$ . Whether this $t-J$ model can
be further simplified, for example, by neglecting $J_{2}$ or $J_{3}$ terms
Shen  _et al._ (2023), can be scrutinized in further experimental or
theoretical studies on the strange metal state Sun  _et al._ (2023); Wu  _et
al._ (2001); Wú  _et al._ (2022); Phillips  _et al._ (2022) or the
superconducting phase in $\mathrm{La_{3}Ni_{2}O_{7}}$ under pressure.

Note added - During the preparation of this work, we become aware of an
experimental work Liu  _et al._ (2023b) probing the optical response in
$\mathrm{La_{3}Ni_{2}O_{7}}$, which suggests the proximity to Mottness of the
electron correlations in the material. This finding is in accordance with our
doped charge-transfer insulator description of the
$\mathrm{La_{3}Ni_{2}O_{7}}$.

Acknowledgment-  We thank Hualei Sun, Mi Jiang, K. Le Hur, and Xunwu Hu for
useful discussions. W.W is indebted to A. -M. Tremblay for useful discussions
and insightful suggestions. W.W. acknowledge help from Dong Meng in preparing
the illustrations in Figure 1. Work at Sun Yat-Sen University was supported by
the National Natural Science Foundation of China (Grants No.12274472, No.
92165204, No.12174454, No.11974432), the National Key Research and Development
Program of China (Grants No. 2022YFA1402802, 2018YFA0306001), the Guangdong
Basic and Applied Basic Research Foundation (Grants No. 2022A1515011618, No.
2021B1515120015), Guangdong Provincial Key Laboratory of Magnetoelectric
Physics and Devices (Grant No. 2022B1212010008), Shenzhen International
Quantum Academy (Grant No. SIQA202102), and Leading Talent Program of
Guangdong Special Projects (201626003). We acknowledge the support from
GuangZhou National Supercomputing Center (Tianhe-II).

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# Karatsuba’s divisor problem and related questions

Mikhail R. Gabdullin, Vitalii V. Iudelevich, Sergei V. Konyagin  Steklov
Mathematical Institute, Gubkina str., 8, Moscow, Russia, 119991
<EMAIL_ADDRESS>Moscow State University, Leninskie Gory str., 1,
Moscow, Russia, 119991<EMAIL_ADDRESS>Steklov Mathematical
Institute, Gubkina str., 8, Moscow, Russia, 119991<EMAIL_ADDRESS>
###### Abstract.

We prove that

$\sum_{p\leqslant x}\frac{1}{\tau(p-1)}\asymp\frac{x}{(\log
x)^{3/2}}\quad\mbox{\text{and}}\quad\sum_{n\leqslant
x}\frac{1}{\tau(n^{2}+1)}\asymp\frac{x}{(\log x)^{1/2}},$

where $\tau(n)=\sum_{d|n}1$ is the number of divisors of $n$, and the
summation in the first sum is over primes.

##  1\. Introduction

In 2004, A. A. Karatsuba in his seminar “Analytic number theory and
applications” posed the following problem: find the asymptotics for the sum

$\Phi(x)=\sum_{p\leqslant x}\frac{1}{\tau(p-1)}$

as $x\to\infty$, where $\tau(n)=\sum_{d|n}1$ is the divisor function and the
summation is over primes not exceeding $x$. This is a natural “hybrid” of the
following two classical problems from analytic number theory.

The first one (so-called Titchmarsh divisor problem) is asking for the
asymptotic behaviour of the sum

$D(x)=\sum_{p\leqslant x}\tau(p-1).$

It is well-known (see [13], [1], [6]) that

$D(x)\sim\frac{\zeta(2)\zeta(3)}{\zeta(6)}\,x,\quad\quad x\to\infty,$

where $\zeta(s)$ denotes the Riemann zeta-function. The other problem is to
find the asymptotics for the sum

$T(x)=\sum_{n\leqslant x}\frac{1}{\tau(n)}.$

This problem was solved by S. Ramanujan [10], who showed that

(1.1) $T(x)=c_{0}\frac{x}{\sqrt{\log x}}\left(1+O\left(\frac{1}{\sqrt{\log
x}}\right)\right),$

where

$c_{0}=\frac{1}{\sqrt{\pi}}\prod_{p}\sqrt{p^{2}-p}\log\frac{p}{p-1}=0.5486\ldots$

The sum $\Phi(x)$ has been studied before. In the recent work [4] it was shown
that

$\Phi(x)\leqslant 4K\dfrac{x}{(\log x)^{3/2}}+O\left(\frac{x\log\log x}{(\log
x)^{5/2}}\right),$

where

$K=\frac{1}{\sqrt{\pi}}\prod_{p}\sqrt{\dfrac{p}{p-1}}\left(p\log\dfrac{p}{p-1}-\dfrac{1}{p-1}\right)=0.2532\ldots$

Note that the bound $\Phi(x)\ll\frac{x}{(\log x)^{3/2}}$ follows from
Corollary 1.2 of [7], and can also be derived by arguing similarly to the
proof of upper bound in Theorem 1.2 below. We make the conjecture that

$\Phi(x)\sim K\frac{x}{(\log x)^{3/2}},\quad\quad x\to\infty,$

but this is probably hard to show.

In this paper, we prove that the aforementioned upper bound for $\Phi(x)$ is
sharp up to a constant.

######  Theorem 1.1.

We have

$\Phi(x)\gg\frac{x}{(\log x)^{3/2}}.$

Thus, we find the correct order of magnitude of the sum $\Phi(x)$:

$\Phi(x)\asymp\frac{x}{(\log x)^{3/2}},$

which answers the question of Karatsuba in the first approximation.

In addition to $\Phi(x)$, we consider the sum

$F(x)=\sum_{n\leqslant x}\frac{1}{\tau(n^{2}+1)},$

and establish the following.

######  Theorem 1.2.

We have

$F(x)\asymp\frac{x}{(\log x)^{1/2}}.$

Now we discuss the main ideas of the proofs. In the sum $\Phi(x)$, for each
prime $p\leqslant x$, we write $p-1$ in the form $ab$, where $a$ consists of
prime factors not exceeding $z$, and $b$ has only prime factors greater than
$z$; here $z=x^{\varepsilon}$ for some small fixed $\varepsilon>0$. Now
$\Phi(x)$ can be rewritten as

(1.2) $\Phi(x)=\sum_{\begin{subarray}{c}a\leqslant x\\\ p|a\Rightarrow
p\leqslant
z\end{subarray}}\frac{1}{\tau(a)}\sum_{\begin{subarray}{c}b\leqslant\frac{x-1}{a}\\\
p|b\Rightarrow p>z\\\ ab+1\text{ prime}\end{subarray}}\frac{1}{\tau(b)},$

and, since $\tau(b)=O_{\varepsilon}(1)$, we have

$\Phi(x)\gg_{\varepsilon}\sum_{a\leqslant
x^{\varepsilon}}\frac{1}{\tau(a)}\sum_{\begin{subarray}{c}b\leqslant\frac{x-1}{a}\\\
p|b\Rightarrow p>x^{\varepsilon}\\\ ab+1\text{ prime}\end{subarray}}1.$

Using the Brun-Hooley sieve, one can estimate the inner sum from below by a
quantity of order

$\frac{x}{a(\log x)^{2}}-R(x;a),$

where the contribution of $R(x;a)$ to the outer sum will be negligible.
Combining these estimates, we see that

$\Phi(x)\gg_{\varepsilon}\frac{x}{(\log x)^{2}}\sum_{a\leqslant
x^{\varepsilon}}\frac{1}{a\tau(a)}\gg_{\varepsilon}\frac{x}{(\log x)^{3/2}},$

which gives the desired lower bound. Let us note that this argument does not
yield any upper bound for $\Phi(x)$, since estimation of the contribution of
numbers $a>x^{1-\varepsilon}$ to (1.2) is actually equivalent to the initial
problem.

The upper bound for $F(x)$ follows from Theorem 1 of [8]. We also note that
the same upper bound can be derived from the inequality $\tau(n)\geqslant
2^{\omega(n)}$ (here $\omega(n)$ denotes the number of distinct prime divisors
of $n$) and a bound for the number of $n\leqslant x$ with a given value of
$\omega(n^{2}+1)$. For completeness, we provide this argument.

We observe that the methods of this work can also be applied for other
functions similar to $\tau(n)$. For instance, let $\tau_{k}(n)$ be the
generalized divisor function, $\tau_{k}(n)=\sum_{n=d_{1}d_{2}\ldots d_{k}}1$;
then one can show that

$\sum_{p\leqslant x}\frac{1}{\tau_{k}(p-1)}\asymp_{k}x(\log
x)^{\frac{1}{k}-2}.$

Acknowledgements. The work of Mikhail Gabdullin was supported by the Russian
Science Foundation under grant no.19-11-00001,
https://rscf.ru/en/project/19-11-00001/. The work of Vitalii Iudelevich was
supported by the Theoretical Physics and Mathematics Advancement Foundation
“BASIS”.

##  2\. Notation

By $\varphi(n)=\\#\\{k\leqslant n:(k,n)=1\\}$ we denote the Euler totient
function, and we use $P^{+}(n)$ and $P^{-}(n)$ for the least and the greatest
prime divisors of a number $n>1$ respectively; by convention, $P^{+}(1)=0$ and
$P^{-}(1)=\infty$. By $\pi(x)$ we denote the number of primes up to $x$, and
$\pi(x;q,a)$ stands for the number of primes up to $x$ belonging to the
arithmetic progression $a\allowbreak\mkern 7.0mu({\operator@font mod}\,\,q)$;
we also set $R(x;q,a)=\pi(x;q,a)-\frac{\pi(x)}{\varphi(q)}$. The notation
$f(x)\ll g(x)$, as well as $f(x)=O(g(x))$, means that $|f(x)|\leqslant Cg(x)$
for some absolute constant $C>0$ and all possible values of $x$. We write
$f(x)\asymp g(x)$, if $f(x)\ll g(x)\ll f(x)$, and write $f(x)\ll_{k}g(x)$ if
we want to stress that the implied constant depends on $k$.

Now we recall some notation from the sieve methods. Let $\mathcal{A}$ be a
finite set of positive integers, and $\mathcal{P}$ be a finite subset of
primes. Define $P=\prod\limits_{p\in\mathcal{P}}p$ and
$S(\mathcal{A},\mathcal{P})=\\#\\{a\in\mathcal{A}:(a,P)=1\\}$. Let also
$\mathcal{A}_{d}=\\#\\{a\in\mathcal{A}:a\equiv 0\allowbreak\mkern
7.0mu({\operator@font mod}\,\,d)\\}$. We assume that, for all $d|P$,

(2.1) $\mathcal{A}_{d}=Xg(d)+r_{d},$

where $g(d)$ is a multiplicative function such that $0<g(p)<1$ for
$p\in\mathcal{P}$ and $g(p)=0$ for $p\not\in\mathcal{P}$.

Further, let the set $\mathcal{P}$ be divided into disjoint subsets
$\mathcal{P}_{1},\mathcal{P}_{2},\ldots,\mathcal{P}_{t}$, and let
$P_{j}=\prod\limits_{p\in\mathcal{P}_{j}}p$. Finally, let
$\\{k_{j}\\}_{r=1}^{t}$ be a sequence of even numbers. We set
$V_{j}=\prod\limits_{p\in\mathcal{P}_{j}}(1-g(p))$, $L_{j}=\log V_{j}^{-1}$,
$E=\sum\limits_{j=1}^{t}\frac{e^{L_{j}}(L_{j})^{k_{j}+1}}{(k_{j}+1)!}$,
$R=\sum\limits_{\begin{subarray}{c}d_{j}|P_{j}\\\ \omega(d_{j})\leqslant
k_{j}\end{subarray}}|r_{d_{1}\ldots d_{t}}|$, and
$R^{\prime}=\sum\limits_{l=1}^{t}\sum\limits_{\begin{subarray}{c}d_{j}|P_{j}\\\
\omega(d_{j})\leqslant k_{j},j\neq l\\\
\omega(d_{l})=k_{l}+1\end{subarray}}|r_{d_{1}\ldots d_{t}}|$.

##  3\. Auxiliary results

We need the following version of the Brun-Hooley sieve, which is due to Ford
and Halberstam [2].

######  Theorem 3.1.

Let (2.1) be met. Then

$S(\mathcal{A},\mathcal{P})\leqslant X\prod_{p\in\mathcal{P}}(1-g(p))e^{E}+R$

and

$S(\mathcal{A},\mathcal{P})\geqslant
X\prod_{p\in\mathcal{P}}(1-g(p))(1-E)-R-R^{\prime}.$

###### Proof.

See [2]. ∎

######  Lemma 3.2.

Let $a\leqslant x^{1/40}$ be an even positive integer and

$F_{a}(x)=\\#\left\\{n\leqslant(x-1)/a:an+1\text{ prime}\textit{ and
}P^{-}(n)>x^{{1}/{40}}\right\\}.$

Then, for $x\geqslant x_{0}$,

$F_{a}(x)\geqslant\frac{c_{1}\pi(x)}{\varphi(a)\log x}-R_{1},$

where $c_{1}>0$ is an absolute constant and

$0\leqslant R_{1}\leqslant\sum_{d\leqslant x^{13/40}}|R(x;ad,1)|$

######  Remark.

Analogous upper bounds for $F_{a}(x)$ without the error term $R_{1}$ can be
obtained by sifting the set $\\{(an+1)(n+P):n\leqslant(x-1)/a\\}$, where
$P=\prod_{p\leqslant z}p$.

###### Proof.

We apply Theorem 3.1 to the sets
$\mathcal{A}=\\{n\leqslant(x-1)/a:an+1\,\text{ prime}\\}$ and
$\mathcal{P}=\\{p\leqslant z\\}$, where $z=x^{1/40}$ (the other parameters
will be chosen later). Then $F_{a}(x)=S(\mathcal{A},\mathcal{P})$, and, for
any $d|P=\prod_{p\leqslant z}p$, we have

$\mathcal{A}_{d}=\\#\\{k\leqslant(x-1)/(ad):adk+1\,\text{
prime}\\}=\pi(x;ad,1)=\frac{\pi(x)}{\varphi(ad)}+R(x;ad,1).$

Now we write a number $d|P$ in the form $d=d_{1}d_{2}$, where $(d_{1},a)=1$
and $d_{2}$ consists only of primes which divide $a$. Then, since
$\varphi(ad_{2})=ad_{2}\prod_{p|a}(1-1/p)=d_{2}\varphi(a)$, we find that

$\frac{1}{\varphi(ad)}=\frac{1}{\varphi(d_{1})\varphi(ad_{2})}=\frac{1}{\varphi(d_{1})d_{2}\varphi(a)},$

and, hence, the equality (2.1) holds with $X=\frac{\pi(x)}{\varphi(a)}$, the
multiplicative function $g$ defined by

$g(p)=\begin{cases}\frac{1}{p-1},\ \quad\text{for}\ (p,a)=1,\\\
\frac{1}{p},\,\quad\quad\text{otherwise},\end{cases}$

and

$r_{d}=R(x;ad,1).$

Besides, we have

$\prod_{p\in\mathcal{P}}(1-g(p))\asymp\frac{1}{\log x},$

with an absolute implied constant.

Now we choose the partition of $\mathcal{P}$ and define numbers
$\\{k_{j}\\}_{j=1}^{t}$. Set $z_{j}=z^{2^{1-j}}$ and

$\mathcal{P}_{j}=\mathcal{P}\cap(z_{j+1},z_{j}],$

where $t$ is the unique positive integer with $z_{t+1}<2\leqslant z_{t}$. Let
$k_{j}=b+2(j-1)$, where $b\geqslant 2$ is even and will be chosen later. We
also set

$C=\prod_{p>2}\left(1+\frac{1}{p^{2}-2p}\right)\leqslant 1.52.$

By [11], Theorem 7 (see (3.26), (3.27)), for any $x>1$,

(3.1) $\frac{e^{-\gamma}}{\log
x}\left(1-\frac{1}{\log^{2}x}\right)<\prod_{p\leqslant
x}\left(1-1/p\right)<\frac{e^{-\gamma}}{\log
x}\left(1+\frac{1}{2\log^{2}x}\right),$

where $\gamma$ is Euler’s constant. Hence, for any $z\geqslant\sqrt{2}$,

(3.2) $\prod_{z<p\leqslant z^{2}}(1-1/p)^{-1}\leqslant 3;$

indeed, it follows from (3.1) for $z\geqslant 4$, and can be checked manually
for $\sqrt{2}\leqslant z<4$. Thus

$V_{j}^{-1}=\prod_{z_{j+1}<p\leqslant z_{j}}(1-g(p))^{-1}\leqslant
C\prod_{z_{j+1}<p\leqslant z_{j}}(1-1/p)^{-1}\leqslant 3C\leqslant 5.$

Hence $L_{j}=\log V_{j}^{-1}\leqslant L=\log 5$ and

$E=\sum_{j=1}^{t}\frac{e^{L_{j}}L_{j}^{k_{j}+1}}{(k_{j}+1)!}\leqslant
e^{L}\sum_{j=1}^{t}\frac{L^{b+2j-1}}{(b+2j-1)!}.$

Now we estimate $R+R^{\prime}$. If a number $d$ corresponds to a summand from
$R$, we have $d=d_{1}\ldots d_{t}$ with $d_{j}|{P}_{j}$ and
$\omega(d_{j})\leqslant k_{j}=b+2(j-1)$. Therefore,

$d\leqslant z_{1}^{k_{1}}\ldots z_{t}^{k_{t}}\leqslant
z^{b+(b+2)/2+(b+4)/4+...}=z^{2b+4}.$

If a number $d$ corresponds to a summand from $R^{\prime}$, we similarly find
that $d\leqslant z^{2b+4}z=z^{2b+5}$. Since the numbers $d$ corresponding to
the sums $R$ and $R^{\prime}$ are distinct and do not exceed $z^{2b+5}$, we
see that

$R+R^{\prime}\leqslant\sum_{d\leqslant z^{2b+5}}|r_{d}|.$

Now we take $b=4$. Then $2b+5=13$,

$E\leqslant 5\sum_{j=1}^{\infty}\frac{(\log 5)^{2j+3}}{(2j+3)!}\leqslant
0.48,$

and

$R+R^{\prime}\leqslant\sum_{d\leqslant x^{13/40}}|R(x;ad,1)|.$

The claim follows. ∎

Define

$\mathcal{M}=\left\\{a\geqslant 1:a\mid n^{2}+1\text{ for some }n\geqslant
1\right\\}.$

It is well-known that $a\in\mathcal{M}$ if and only if $4\nmid a$ and $p\nmid
a$ for any $p\equiv 3\,(\text{mod}\,4)$.

######  Lemma 3.3.

Let $2\leqslant a,z\leqslant x^{1/30}$, $a\in\mathcal{M}$ and
$P^{+}(a)\leqslant z$. Let

$W_{a}(x,z)=\\#\left\\{n\leqslant x:a|(n^{2}+1)\text{ and
}P^{-}\left(\frac{n^{2}+1}{a}\right)>z\right\\}.$

Then

$W_{a}(x,z)\asymp\frac{2^{\omega(a)}}{\varphi(a)}\frac{x}{\log z}.$

###### Proof.

Setting $\mathcal{A}=\\{k:ak=n^{2}+1\mbox{ for some }n\leqslant x\\}$ and

$\mathcal{P}=\begin{cases}\\{p\leqslant z:p\equiv 1\allowbreak\mkern
7.0mu({\operator@font mod}\,\,4)\\},&\text{ if $a$ is even};\\\
\\{2\\}\cup\\{p\leqslant z:p\equiv 1\allowbreak\mkern 7.0mu({\operator@font
mod}\,\,4)\\},&\text{ if $a$ is odd,}\end{cases}$

we have $W_{a}(x,z)=S(\mathcal{A},\mathcal{P})$. As before, we write each
$d|P=\prod_{p\in\mathcal{P}}p$ as $d=d_{1}d_{2}$, where $(d_{1},a)=1$ and
$d_{2}$ consists only of primes dividing $a$. Suppose that one of the numbers
$d$ and $a$ is even. Then by the Chinese remainder theorem and the fact that
the congruence $x^{2}+1\equiv 0\allowbreak\mkern 7.0mu({\operator@font
mod}\,\,p)$ has two solutions $p\equiv 1\allowbreak\mkern
7.0mu({\operator@font mod}\,\,4)$ and one solution for $p=2$, we get

$\mathcal{A}_{d}=\\#\\{n\leqslant x:n^{2}+1\equiv 0\\!\\!\allowbreak\mkern
7.0mu({\operator@font
mod}\,\,ad)\\}=\frac{x2^{\omega(ad)-1}}{ad}+O(2^{\omega(ad)})\\\
=\frac{x2^{\omega(a)+\omega(d_{1})-1}}{ad_{1}d_{2}}+O(2^{\omega(ad)}).$

If both $a$ and $d$ are odd, a similar argument shows that

$\mathcal{A}_{d}=\frac{x2^{\omega(a)+\omega(d_{1})}}{ad_{1}d_{2}}+O(2^{\omega(ad)}).$

In both cases we have

$\mathcal{A}_{d}=\frac{x2^{\omega(a)+\omega(d_{1})-{\mathbb{I}}(2|ad)}}{ad_{1}d_{2}}+O(2^{\omega(ad)})=\frac{x2^{\omega(a)-{\mathbb{I}}(2|a)}}{a}\dfrac{2^{\omega(d_{1})-{\mathbb{I}}(2|d_{1})}}{d_{1}d_{2}}+O\left(\tau(ad)\right),$

where ${\mathbb{I}}(2|l)$ is equal to one if $l$ is even and equals to zero
otherwise. Thus we see that the condition (2.1) holds with
$X=\frac{x2^{\omega(a)-{\mathbb{I}}(2|a)}}{a}$, the multiplicative function
$g$ defined on the primes from $\mathcal{P}$ by

$g(p)=\begin{cases}\frac{2}{p},\quad p\nmid a,\\\ \frac{1}{p},\quad
p|a,\end{cases}$

(and also $g(2)=1/2$ in the case of odd $a$), and

$r_{d}=O(\tau(ad)).$

It is well-known that

(3.3) $\prod\limits_{\begin{subarray}{c}p\leqslant x\\\
p\equiv\,1(\text{mod}\,4)\end{subarray}}\left(1-\dfrac{1}{p}\right)\asymp\dfrac{1}{\sqrt{\log
x}},$

(see [14]). Thus, in both cases we have

$\prod_{p\in\mathcal{P}}(1-g(p))\asymp\prod_{p\in\mathcal{P},\,p>2}(1-2/p)\prod_{p|a,\,p>2}\frac{1-1/p}{1-2/p}\asymp\frac{a}{\varphi(a)\log
z}$

with an absolute implied constant.

Now we choose the partition of $\mathcal{P}$ and define the numbers
$\\{k_{j}\\}_{j=1}^{t}$. We again set $z_{j}=z^{2^{1-j}}$ and

$\mathcal{P}_{j}=\mathcal{P}\cap(z_{j+1},z_{j}],$

where $t$ is a unique positive integer such that $z_{t+1}<2\leqslant z_{t}$
and $k_{j}=b+2(j-1)$ with some even $b\geqslant 2$. Now (3.2) implies that

$V_{j}^{-1}=\prod_{z_{j+1}<p\leqslant z_{j}}(1-g(p))^{-1}\leqslant
2\prod_{z_{j+1}<p\leqslant z_{j},\,p\neq 2}(1-2/p)^{-1}\\\
=2\prod_{z_{j+1}<p\leqslant z_{j},\,p\neq
2}(1-1/p)^{-2}\frac{(1-1/p)^{2}}{(1-2/p)}\leqslant 18C\leqslant 28.$

Therefore, $L_{j}=\log V_{j}^{-1}\leqslant L=\log 28$ and

$E=\sum_{j=1}^{t}\frac{e^{L_{j}}L_{j}^{k_{j}+1}}{(k_{j}+1)!}\leqslant
e^{L}\sum_{j=1}^{t}\frac{L^{b+2j-1}}{(b+2j-1)!}.$

Now we take $b=10$; then $E\leqslant 0.43$ and $2b+5=25$. As in the proof of
Lemma 3.2, we get

$R+R^{\prime}\leqslant\sum_{d\leqslant
z^{2b+5}}|r_{d}|\ll\tau(a)\sum_{d\leqslant z^{25}}\tau(d)\ll x^{1/30}z^{25}\ll
x^{26/30},$

and the main term is at least

$X\prod_{p\in\mathcal{P}}(1-g(p))\asymp\frac{x2^{\omega(a)}}{\varphi(a)\log
z}\gg\frac{x^{1-1/30}}{\log x}\,.$

Applying Theorem 3.1 concludes the proof. ∎

The next lemma is a special case of the result of J. Tenenbaum [12]. For
completeness, we provide the proof.

######  Lemma 3.4.

Let $R>0$ be fixed. There exist positive constants $A=A(R)$ and $B=B(R)$ such
that, for all $1\leqslant k\leqslant R\log\log x$,

$\\#\\{n\leqslant x:\omega(n^{2}+1)=k\\}\leqslant\frac{Ax(\log\log
x+B)^{k-1}}{(k-1)!\log x}.$

###### Proof.

For $m\geqslant 2$ and $y\geqslant 2$, we define “the $y$-smooth part” of $m$:

$d(m,y)=\max\\{d|m:P^{+}(d)\leqslant y\\}.$

For each $n\leqslant x$, the number $n^{2}+1$ can be written as the product
$ab$, where

$a=a(n)=\max\\{d(n^{2}+1,y):d(n^{2}+1,y)\leqslant x^{1/30}\\}$

and $b=b(n)=(n^{2}+1)/a$; note that $(a,b)=1$. Now we set

$A_{1}=\\{n\leqslant x:a\leqslant x^{1/60},P^{-}(b)>x^{1/60}\mbox{ and
}b>1\\},$ $A_{2}=\\{n\leqslant x:a\leqslant x^{1/60},P^{-}(b)\leqslant
x^{1/60}\mbox{ and }b>1\\},$

and

$A_{3}=\\{n\leqslant x:x^{1/60}<a\leqslant x^{1/30}\mbox{ and }b>1\\}.$

Since the number of $n\leqslant x$ for which $b(n)=1$ does not exceed
$x^{1/60}$, we conclude that

(3.4) $\\#\\{n\leqslant x:\omega(n^{2}+1)=k\\}=N_{1}+N_{2}+N_{3}+O(x^{1/60}),$

where

$N_{i}=\\#\\{n\in A_{i}:\omega(n^{2}+1)=k\\},\quad i=1,2,3.$

First, we estimate $N_{1}$. In this case $1\leqslant\omega(b)\leqslant 120$;
by Lemma 3.3, we have

(3.5) $N_{1}\leqslant\sum_{l=k-120}^{k-1}\sum_{\begin{subarray}{c}a\leqslant
x^{1/60},\,a\in{\mathcal{M}}\\\
\omega(a)=l\end{subarray}}W_{a}(x,x^{1/60})\ll\sum_{l=k-120}^{k-1}\sum_{\begin{subarray}{c}a\leqslant
x^{1/60},\,a\in{\mathcal{M}}\\\
\omega(a)=l\end{subarray}}\frac{x2^{\omega(a)}}{\varphi(a)\log x}\\\
\leqslant\frac{x}{\log
x}\sum_{l=k-120}^{k-1}2^{l}\sum_{\begin{subarray}{c}a\leqslant
x^{1/60},\,a\in{\mathcal{M}}\\\
\omega(a)=l\end{subarray}}\frac{1}{\varphi(a)}.$

Taking the logarithms of both sides of (3.3), we get

$\sum_{\begin{subarray}{c}p\leqslant x\\\ p\equiv
1(\text{mod}\,4)\end{subarray}}\dfrac{1}{p}=\frac{1}{2}\log\log x+O(1).$

Hence, for each $l\geqslant 0$, the inner sum in (3.5) is at most

$\frac{1}{l!}\left(\sum_{p\leqslant x^{1/60},\,p\nequiv
3\\!\\!\allowbreak\mkern 7.0mu({\operator@font
mod}\,\,4)}\frac{1}{\varphi(p)}+\frac{1}{\varphi(p^{2})}+...\right)^{l}\leqslant\frac{1}{l!}\left(0.5\log\log
x+B_{1}\right)^{l}$

for some absolute constant $B_{1}>0$. Thus

(3.6) $N_{1}\ll\frac{x(\log\log x+B_{1})^{k-1}}{(k-1)!\log
x}\left(1+R+...+R^{120}\right).$

Now we estimate $N_{2}$. Let $p=P^{-}(b)$, and let $r$ be the largest integer
such that $p^{r}|b$. Then, by definition of $a$, we have $ap^{r}>x^{1/30}$.
Thus, for any $n\in A_{2}$ we have $p^{r}>x^{1/60}$, and since $p\leqslant
x^{1/60}$, we see that $r\geqslant 2$. Let $\nu=\min\\{u\geqslant
1:p^{u}>x^{1/60}\\}$. Then $2\leqslant\nu\leqslant r$ and $p^{\nu-1}\leqslant
x^{1/60}$. Hence, $p^{\nu}\leqslant x^{1/60}p\leqslant x^{1/30}$. We conclude
that, for each $n\in A_{2}$, the number $n^{2}+1$ is divided by $p^{\nu}$,
where $p$ is a prime such that $x^{1/60}<p^{\nu}\leqslant x^{1/30}$ and
$\nu\geqslant 2$. Therefore,

$N_{2}\leqslant\sum_{x^{1/60}<p^{\nu}\leqslant x^{1/30},\,\nu\geqslant
2}\left(\frac{2x}{p^{\nu}}+O(1)\right)\ll\sum_{p^{\nu}>x^{1/60},\,\nu\geqslant
2}\frac{x}{p^{\nu}}.$

Note that $p^{\nu}\geqslant\max\\{p^{2},x^{1/60}\\}$, and therefore,
$p^{\nu}\geqslant px^{1/120}$. It follows that

(3.7) $N_{2}\ll\sum_{p\leqslant x^{1/30}}\frac{x}{px^{1/120}}\ll
x^{1-1/120}\log\log x.$

Finally, let us consider $N_{3}$. Setting $q=P^{+}(a)$, we have
$P^{-}(b)\geqslant q+1$ and

(3.8) $\omega(b)\leqslant\frac{\log(x^{2}+1)}{\log(q+1)}\leqslant\frac{2\log
x}{\log q}=:\eta.$

Lemma 3.3 implies that

(3.9) $N_{3}\leqslant\sum_{\begin{subarray}{c}q\leqslant x^{1/30}\\\
q\not{\equiv}3\\!\\!\allowbreak\mkern 7.0mu({\operator@font
mod}\,\,4)\end{subarray}}\sum_{k-\eta\leqslant l\leqslant
k-1}\sum_{\begin{subarray}{c}x^{1/60}<a\leqslant x^{1/30}\\\
P^{+}(a)=q,\,\omega(a)=l\end{subarray}}W_{a}(x,q)\\\ \ll
x\sum_{\begin{subarray}{c}q\leqslant x^{1/30}\\\
q\not{\equiv}3\\!\\!\allowbreak\mkern 7.0mu({\operator@font
mod}\,\,4)\end{subarray}}\frac{1}{\log q}\sum_{k-\eta\leqslant l\leqslant
k-1}2^{l}\sum_{\begin{subarray}{c}x^{1/60}<a\leqslant x^{1/30}\\\
P^{+}(a)=q,\,\omega(a)=l\end{subarray}}\frac{1}{\varphi(a)}.$

Let

(3.10) ${\delta}=\frac{C}{\log q},$

where $C=120\log(R+2)+60$. Let us denote by $N_{3}^{(1)}$ and $N_{3}^{(2)}$
the contribution of $q\leqslant e^{2C}$ and of $q>e^{2C}$, respectively, to
the sum in (3.9). Since $\varphi(a)\gg\frac{a}{\log\log
a}\gg\frac{x^{1/60}}{\log\log x}$, we have

(3.11) $N_{3}^{(1)}\ll x^{59/60}\log\log x\sum_{q\leqslant
e^{2C}}\sum_{1\leqslant
l\leqslant\pi(q)}2^{l}\sum_{\begin{subarray}{c}a\leqslant x^{1/30}\\\
P^{+}(a)=q\end{subarray}}1\ll x^{59/60}(\log x)^{c_{R}},$

where $c_{R}>0$ depends only on $R$. Now we turn to $N_{3}^{(2)}$. For fixed
$q$ and $l$, we see that

$S_{q,l}:=\sum_{\begin{subarray}{c}x^{1/60}<a\leqslant x^{1/30}\\\
P^{+}(a)=q,\,\omega(a)=l\end{subarray}}\frac{1}{\varphi(a)}\leqslant\sum_{\begin{subarray}{c}a\leqslant
x^{1/30}\\\
P^{+}(a)=q,\,\omega(a)=l\end{subarray}}\left(\frac{a}{x^{1/60}}\right)^{\delta}\frac{1}{\varphi(a)}\\\
\leqslant
x^{-{\delta}/60}\frac{1}{(l-1)!}\left(\sum_{\begin{subarray}{c}p<q\\\
p\not{\equiv}3\\!\\!\\!\\!\allowbreak\mkern 7.0mu({\operator@font
mod}\,\,4)\end{subarray}}\frac{p^{{\delta}}}{\varphi(p)}+\frac{p^{2{\delta}}}{\varphi(p^{2})}+...\right)^{l-1}\left(\frac{q^{{\delta}}}{\varphi(q)}+\frac{q^{2{\delta}}}{\varphi(q^{2})}+...\right).$

Note that
$\frac{q^{{\delta}}}{\varphi(q)}+\frac{q^{2{\delta}}}{\varphi(q^{2})}+...\ll_{R}1/q$
for $q>e^{2C}$, and the sum over primes $p$ is equal to (since $e^{u}\leqslant
1+O_{K}(u)$ for $0\leqslant u\leqslant K$)

$\sum_{\begin{subarray}{c}p<q\\\ p\not{\equiv}3\\!\\!\allowbreak\mkern
7.0mu({\operator@font
mod}\,\,4)\end{subarray}}\frac{p^{\delta}}{p}+O_{R}(1)\leqslant\sum_{\begin{subarray}{c}p<q\\\
p\not{\equiv}3\\!\\!\allowbreak\mkern 7.0mu({\operator@font
mod}\,\,4)\end{subarray}}\frac{1}{p}+O_{R}\left({\delta}\sum_{\begin{subarray}{c}p<q\\\
p\not{\equiv}3\\!\\!\allowbreak\mkern 7.0mu({\operator@font
mod}\,\,4)\end{subarray}}\frac{\log p}{p}+1\right)\leqslant 0.5\log\log x+B,$

where $B=B(R)>0$. Thus,

$S_{q,l}\ll_{R}\frac{x^{-{\delta}/60}}{q(l-1)!}\left(0.5\log\log
x+B\right)^{l-1}$

and

(3.12) $N_{3}^{(2)}\ll_{R}x\sum_{\begin{subarray}{c}e^{2C}<q\leqslant
x^{1/30}\\\
q\not{\equiv}3(\text{mod}\,4)\end{subarray}}\dfrac{x^{-\delta/60}}{q\log
q}\sum_{k-\eta\leqslant l\leqslant k-1}\dfrac{2^{l}(0.5\log\log
x+B)^{l-1}}{(l-1)!}\\\ \ll\frac{x(\log\log
x+2B)^{k-1}}{(k-1)!}\sum_{q\leqslant x^{1/30}}\frac{x^{-{\delta}/60}}{q\log
q}\left(1+R+...+R^{[\eta]}\right).$

Using the inequality

$1+R+\ldots+R^{[\eta]}\leqslant(R+2)^{\eta+1}$

and the fact that

$(R+2)^{\eta}x^{-\delta/60}=x^{-1/\log q},$

(which follows from the definitions (3.8) and (3.10) of $\eta$ and $\delta$
respectively), we see that the sum over primes $q$ in (3.12) is at most

$(R+2)\sum_{q\leqslant x^{1/30}}\frac{(R+2)^{\eta}x^{-{\delta}/60}}{q\log
q}\leqslant(R+2)\sum_{q\leqslant x^{1/2}}\frac{x^{-1/\log q}}{q\log q}\\\
\leqslant\frac{R+2}{\log x}\sum_{j\geqslant
1}\sum_{q\in\left(x^{2^{-(j+1)}},\,x^{-2^{j}}\right]}\frac{1}{q}\,2^{j+1}\exp(-2^{j})\ll_{R}\frac{1}{\log
x}\,.$

Substituting this into (3.12) and taking into account (3.11), we find that

(3.13) $N_{3}\ll_{R}\frac{x(\log\log x+2B)^{k-1}}{(k-1)!\log x}.$

Combining (3.4), (3.6), (3.7), and (3.13) we complete the proof. ∎

##  4\. Proof of Theorem 1.1

Writing each $p\leqslant x$ in the form $p-1=ab$, where $P^{+}(a)\leqslant
x^{1/40}$ and $P^{-}(b)>x^{1/40}$, we can rewrite the sum $\Phi(x)$ as

$\Phi(x)=\sum_{\begin{subarray}{c}a\leqslant x\\\ P^{+}(a)\leqslant
x^{1/40}\end{subarray}}\frac{1}{\tau(a)}\sum_{\begin{subarray}{c}b\leqslant\frac{x-1}{a}\\\
P^{-}(b)>x^{{1}/{40}}\\\ ab+1\text{ prime}\end{subarray}}\frac{1}{\tau(b)}.$

Let $b=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\ldots p_{s}^{\alpha_{s}}$ be the
prime factorization of $b$. Then

$x^{(\alpha_{1}+\alpha_{2}+\ldots+\alpha_{s})/40}<b\leqslant x,$

and, hence, $\alpha_{1}+\ldots+\alpha_{s}<40$ and

$\tau(b)=(\alpha_{1}+1)\ldots(\alpha_{s}+1)\leqslant
2^{\alpha_{1}+\ldots+\alpha_{s}}<2^{40}.$

Therefore,

$\Phi(x)\geqslant 2^{-40}\sum_{\begin{subarray}{c}a\leqslant x\\\
P^{+}(a)\leqslant
x^{1/40}\end{subarray}}\frac{1}{\tau(a)}\sum_{\begin{subarray}{c}b\leqslant\frac{x-1}{a}\\\
P^{-}(b)>x^{1/40}\\\ ab+1\text{ prime}\end{subarray}}1\geqslant
2^{-40}\sum_{\begin{subarray}{c}a\leqslant x^{1/40}\\\ a\text{
even}\end{subarray}}\frac{1}{\tau(a)}F_{a}(x).$

Lemma 3.2 implies that

(4.1) $\Phi(x)\geqslant\frac{c_{2}\pi(x)}{\log
x}\sum_{\begin{subarray}{c}a\leqslant x^{1/40}\\\ a\text{
even}\end{subarray}}\frac{1}{\tau(a)\varphi(a)}-R_{2},$

where $c_{2}>0$ and

$0\leqslant R_{2}\leqslant\sum_{a\leqslant x^{1/40}}\sum_{d\leqslant
x^{13/40}}|R(x;ad,1)|\leqslant\sum_{q\leqslant x^{7/20}}\tau(q)|R(x;q,1)|.$

Using the trivial bound $|R(x;q,1)|\ll x/q$ and the Cauchy-Schwarz inequality,
we get

$R_{2}\ll x^{1/2}\sum_{q\leqslant
x^{0.35}}\frac{\tau(q)}{q^{1/2}}(|R(x;q,1)|)^{1/2}\leqslant
x^{1/2}\left(\sum_{q\leqslant
x^{0.35}}\frac{\tau^{2}(q)}{q}\right)^{1/2}\left(\sum_{q\leqslant
x^{0.35}}|R(x;q,1)|\right)^{1/2}.$

Applying the bound

$\sum_{n\leqslant x}\frac{\tau^{2}(n)}{n}\asymp(\log x)^{4},$

which follows from $\sum_{n\leqslant x}\tau^{2}(n)\asymp x(\log x)^{3}$ (see
[9], Chapter III, Exercise 7) by partial summation and the Bombieri–Vinogradov
theorem, we find that, for any fixed $A>0$,

$R_{2}\ll_{A}x^{1/2}(\log x)^{2}\frac{x^{1/2}}{(\log x)^{A/2}}=\frac{x}{(\log
x)^{A/2-2}}.$

By taking $A=24$ (say), we obtain

(4.2) $R_{2}\ll\frac{x}{(\log x)^{10}}.$

Now we turn to the sum

$T_{1}=\sum_{\begin{subarray}{c}a\leqslant x^{1/40}\\\ a\text{
even}\end{subarray}}\frac{1}{\tau(a)\varphi(a)}=\sum_{l\leqslant
0.5x^{1/40}}\frac{1}{\tau(2l)\varphi(2l)}.$

Since $\tau(mn)\leqslant\tau(m)\tau(n)$ and $\varphi(n)\leqslant n$, we see
that

$T_{1}\gg\sum_{l\leqslant 0.5x^{1/40}}\frac{1}{\tau(l)l}.$

From (1.1) and partial summation we have

$T_{1}\gg(\log x)^{1/2}.$

Combining this with (4.1) and (4.2), we obtain

$\Phi(x)\gg\frac{\pi(x)}{(\log x)^{1/2}}-O\left(\frac{x}{(\log
x)^{10}}\right)\gg\frac{x}{(\log x)^{3/2}}.$

Now the claim follows.

##  5\. Proof of Theorem 1.2

We first prove the lower bound. For any $z\geqslant 2$,

$F(x)=\sum_{n\leqslant x}\dfrac{1}{\tau(n^{2}+1)}=\underset{P^{+}(a)\leqslant
z,\,P^{-}(b)>z}{\underset{ab=n^{2}+1,\,n\leqslant
x}{\sum\sum}}\frac{1}{\tau(ab)}.$

Setting $z=x^{1/30}$, we have $\tau(b)\leqslant 2^{60}$, and now by Lemma 3.3,

$F(x)\geqslant 2^{-60}\sum_{\begin{subarray}{c}a\leqslant x^{1/30}\\\
a\in\mathcal{M}\end{subarray}}\frac{1}{\tau(a)}W_{a}\left(x,x^{1/30}\right)\gg\frac{x}{\log
x}\sum_{\begin{subarray}{c}a\leqslant x^{1/30}\\\
a\in\mathcal{M}\end{subarray}}\frac{2^{\omega(a)}}{\tau(a)\varphi(a)}\\\
\geqslant\frac{x}{\log x}\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}a\leqslant
x^{1/30}\\\ a\in\mathcal{M}\\\ a\text{ square-
free}\end{subarray}}\\!\\!\\!\\!\\!\frac{1}{a}=\frac{x}{\log
x}\sum_{\begin{subarray}{c}a\leqslant x^{1/30}\\\
a\in\mathcal{M}\end{subarray}}\frac{1}{a}\,\sum_{\delta^{2}|a}\mu({\delta}).$

Changing the order of summation and using the fact that
$\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{\pi^{2}}{6}$, we have

(5.1) $F(x)\gg\frac{x}{\log x}\sum_{\begin{subarray}{c}{\delta}\leqslant
x^{1/60}\\\
{\delta}\in\mathcal{M}\end{subarray}}\frac{\mu({\delta})}{{\delta}^{2}}\sum_{\begin{subarray}{c}a\leqslant
x^{1/30}{\delta}^{-2}\\\
a\in\mathcal{M}\end{subarray}}\frac{1}{a}\geqslant\left(2-\frac{\pi^{2}}{6}\right)\frac{x}{\log
x}\sum_{\begin{subarray}{c}a\leqslant x^{1/30}\\\
a\in\mathcal{M}\end{subarray}}\frac{1}{a}.$

It is well-known (see [5], §183) that for $y\geqslant 2$ one has

$\\#\\{a\leqslant y:p|a\Rightarrow p\equiv
1\,(\text{mod\,}4)\\}\asymp\frac{y}{(\log y)^{1/2}}.$

By partial summation,

$\sum_{\begin{subarray}{c}a\leqslant y\\\ p|a\Rightarrow p\equiv
1\,(\text{mod\,}4)\end{subarray}}\frac{1}{a}\asymp(\log y)^{1/2}.$

Now the claim follows from (5.1).

Now we prove the upper bound. By Lemma 3.4,

$F(x)\leqslant\sum_{n\leqslant
x}\frac{1}{2^{\omega(n^{2}+1)}}=\sum_{k\leqslant 2\log\log
x}\frac{\\#\\{n\leqslant
x:\omega(n^{2}+1)=k\\}}{2^{k}}+O\left(\sum_{k>2\log\log
x}\frac{x}{2^{k}}\right)\\\ \ll\frac{x}{\log
x}\sum_{k=1}^{+\infty}\frac{(0.5\log\log
x+0.5B)^{k-1}}{(k-1)!}+O\left(\frac{x}{\log x}\right)\ll\frac{x}{(\log
x)^{1/2}}.$

This concludes the proof.

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