# The use of deception in dementia-care robots: Should robots tell “white
lies” to limit emotional distress?

Samuel Rhys Cox<EMAIL_ADDRESS>0000-0002-4558-6610 National
University of Singapore ,  Grace Cheong<EMAIL_ADDRESS>0000-0002-6128-5834 Centre for Research on Successful Ageing (ROSA),
Singapore Management University and  Wei Tsang Ooi<EMAIL_ADDRESS>0000-0001-8994-1736 National University of Singapore

(2018)

###### Abstract.

With projections of ageing populations and increasing rates of dementia, there
is need for professional caregivers. Assistive robots have been proposed as a
solution to this, as they can assist people both physically and socially.
However, caregivers often need to use acts of deception (such as misdirection
or white lies) in order to ensure necessary care is provided while limiting
negative impacts on the cared-for such as emotional distress or loss of
dignity. We discuss such use of deception, and contextualise their use within
robotics.

Dementia-Care, Ageing, Robotic Assistants, Deception

††copyright: acmcopyright††journalyear: 2018††doi:
10.1145/1122445.1122456††conference: Woodstock ’18: ACM Symposium on Neural
Gaze Detection; June 03–05, 2018; Woodstock, NY††booktitle: Woodstock ’18: ACM
Symposium on Neural Gaze Detection, June 03–05, 2018, Woodstock, NY††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Applied computing Sociology

##  1\. Introduction

Population ageing coupled with lower fertility rates is an increasing concern
for many countries and with a growing population of older adults, there is an
increasing demand for professional caregivers to address the care needs of
older adults at a societal level. This is particularly so for persons with
dementia. Research estimates that the number of people living with dementia
will approximately triple from 57 million in 2019 to 153 million by 2050 (UK,
2022). Cognitive functioning deteriorates as the disease progresses,
inhibiting the independence and functioning of persons afflicted with the
disease. Depending on the severity of impairment, persons with dementia may
require extensive, consistent care in daily living.

To address this gap, research has pointed towards the use of robots to address
the care-giving needs of older persons (Bodenhagen et al., 2019; Feil-Seifer
and Matarić, 2011). Assistive robots have been deployed in a multiplicity of
care settings (Martinez-Martin and Costa, 2021; Asgharian et al., 2022;
Ghafurian et al., 2021; Khaksar et al., 2023). However, unlike human care-
givers, assistive robots at present lack the reflexivity to respond according
to the changing needs and behaviour of their cared-fors. This then also raises
the question of how and if robots should adapt their response according to the
changing cognitive state of their cared-fors.

On from this, advice for human carers would adapt depending on the severity of
the cognitive decline associated with the cared-for’s dementia. Strategies in
early stages of dementia emphasise orienting the recipient’s consciousness to
be more reality based (Carrion et al., 2013; Kanov, 2017) (such as via stating
the year, time of day and current weather, and the cared-for’s name, age and
significant relationships). However, in later stages of dementia such an
orientation towards reality may cause emotional distress to the cared-for
(James et al., 2003) and different approaches may be needed (Elvish et al.,
2010; Erdmann and Schnepp, 2016). For this reason, advice would dictate that
the care-giver could deploy acts that may be considered deceptive in order to
fulfil the care needs of cared-for while limiting emotional distress and loss
of dignity (Øye and Jacobsen, 2020; Cantone et al., 2019). In addition, while
it is in the interests of the care-giver to ensure that the cared-for is given
sufficient care (such as ensuring hygiene is maintained, and that cared-for
are safe) delusions from cognitive decline may conflict with these needs and
cause distress. These equally would lead to care-givers potentially using
techniques that are in some way acts of deception.

A robot that fully emulates a human carer would then, on occasion, deceive
those that it is caring for. Related to this, schools of thought for robotic
deception could be seen in two camps: those who oppose the use of deception as
(Sharkey and Sharkey, 2021) (perhaps due to leading to over-dependence, loss
of human relationships or lost sense of reality); and those who view deception
as a necessary and inevitable act (Carli and Najjar, 2023; Isaac and
Bridewell, 2017) (as robots continue to anthropomorphise and adopt human
characteristics, thereby increasing acceptability and effectiveness (Short et
al., 2010; Esposito et al., 2018)). At a philosophical and ethical level, the
context of robot lies have been discussed within healthcare. For example,
Matthias described that lies should always be in the patient’s best interests,
increase patient autonomy, not lead to harm, and be transparent in deception
(Matthias, 2015), yet these do not address precise strategies that could be
used in dementia-care by assistive robots, or people’s perception of these. On
from this, we discuss a number of strategies (involving deception) for
dementia-care that are either recommended practice, or have been reported as
used by care-givers. Afterwards, we discuss potential implications and
concerns that could be raised if these same strategies were used by assistive
robots. For example, when could a robot need to deceive people in order to
better complete its tasks, and why would it need to deceive someone rather
than revealing the truth?

##  2\. Human carers and uses of deception

A goal of human carers is to deliver necessary care, while balancing emotional
distress, dignity, and humanity in order to limit both physical and emotional
harm. For example, while allowing for autonomy and maintaining independent
activities improves well-being and cognitive functions, if such freedom
exists, so too does the possibility for resistiveness due to conflicting wants
between care-givers and the cared-fors (Watts et al., 2019), with such
resistiveness causing distress to both parties (Mortensen et al., 2022).
Additionally, in the case of persons with dementia in particular, cognitive
impairments may hinder the ability to make rational decisions and complete
tasks that are beneficial to one’s well-being.

As a consequence of this, human carers may use techniques to lessen
resistiveness to care that could be seen as deceptive, such as therapeutic
lying (Cantone et al., 2019; Elvish et al., 2010; James et al., 2003; James et
al., 2006), informal use of restraint (Øye and Jacobsen, 2020), or forms of
indirect coercion (Sander, 2005). While some of these techniques are widely
adopted (if not sometimes contentiously (Øye and Jacobsen, 2020; James and
Caiazza, 2017; Caiazza and James, 2016; Cantone et al., 2019)) among carer-
givers, additional debate is needed for their potential use by assistive
robots.

Specifically, therapeutic lying (lying that is used for the benefit of the
cared-for, rather than for the care-giver) can be used to limit emotional
distress while delivering care (Cantone et al., 2019; Elvish et al., 2010;
James et al., 2003; James et al., 2006). Therapeutic lies could be used for
“tricks” (Elvish et al., 2010) such as to simplify ingestion of medication
(Cantone et al., 2019), to avoid aggressive behaviour, to limit time spent
giving explanations, to go along with a care-for’s misconception, and to
alleviate stress (see (Cantone et al., 2019) for examples of therapeutic
lies). For example, a person with dementia may ask their care-giver where
their deceased parent is, to which the care-giver could reply “they’ll come
tomorrow” (Cantone et al., 2019). These therapeutic lies involve some form of
deception based in verbal communication, that while uncertain in their levels
of ethics and acceptability, would become more feasible as the abilities of
language models improves.

Similarly, Øye and Jacobsen surveyed nursing homes in Norway, and identified
five distinct types of “informal restraint” enacted by caregivers (Øye and
Jacobsen, 2020). While not acts of formal restraint (such as physical
restraint that would limit autonomy and cause physical and psychological
pain), informal restraint limits freedom of movement and personal preference
in order to provide care. Specifically, they identified diverting residents’
attention (such as showing photographs to distract a person with dementia
during washing that they may otherwise protest to); white lies (to affirm the
perceived realities of persons with dementia); persuasion and interpersonal
pressure; offers (to incentivise adherence to the care-giver’s requests); and
threats (such as sending them back to their room to enforce compliance) as
forms of informal restraint. They additionally identified “grey-zone”
constraints, such as seating a resident in a chair that is deep and low so
that they cannot get up on their own, and therefore reduce their chance of
wandering and hurting themselves.

##  3\. Discussion

We have provided an overview of some deception-based techniques that care-
givers may use when providing care for people with dementia. We will now
discuss issues related to their potential use and application by assistive
robots.

While the use of deception techniques such as therapeutic lying is well
investigated for both its prevalence and acceptability in care-giving
environments (Cantone et al., 2019), such beliefs are not well investigated
with regards to assistive robots, and there are potential ethical concerns
that a person with dementia may not be able to distinguish between a
relationship with a robot and a relationship with a person (Sætra, 2020).
However, previous literature has demonstrated that perhaps there is a
difference in attitude between the philosophical ideals and clinical needs and
practice when deploying forms of deception in dementia-care. For example, Koh
et al. (Koh et al., 2022) investigated the use of robot pets in nursing homes
for dementia, and found that, while people were apprehensive about the
potential for deception (i.e., people with dementia believing a robot pet to
be real), once stakeholder’s personally experienced real-world use of robots,
most were comfortable with its adoption. Although it is difficult to expand
such a discussion to the more direct forms of deception in Section 2, it could
be forseen that such acceptance would be more normalised and likely as
adoption and capabilities of robotics increases.

In addition, it is important to note that some of the uses of deception may be
related to lack of resources for care-givers (Øye and Jacobsen, 2020; Cantone
et al., 2019). For example, the use of grey-zone restraints (Øye and Jacobsen,
2020) could be avoided in a well-resourced home, or if a vision of a future
with more carer-givers due to assistive robots is met. Such forms of deception
are less ethically and morally defensible and (while ensuring the safety of
the cared-for) would perhaps be less desirable and less acceptable if used by
a robot.

The changing nature of interventions used by care-givers could also be
challenging for robotic assistants to overcome. For example, a therapeutic lie
may have limited effectiveness the more it is used, and additional alternative
therapeutic lies may need to be used in order to still deliver care while
limiting the emotional distress of the cared-for (Cantone et al., 2019). It is
unclear whether giving a robot the freedom and flexibility to devise such
deceptions would be seen as acceptable and ethical, and if this necessary
flexibility would have the potential for harm. This changing nature could also
lead to technical challenges due to differences in each person and their
perceived independent autonomy and application of care as such. For example,
how would such a robot be designed to differentiate and perceive the level of
cognitive abilities of individuals and thereby the potential application of
deception to aid in delivery of care? Such a robot would need to be able to
distinguish between situations where it can apply deception (and to whom) and
situations and persons where this is not appropriate, or where orientating in
reality (rather than using therapeutic lies for example) would be acceptable.

In addition, there are also potential legal issues caused by deception-led
robotic interventions with questions surrounding liability if such methods are
seen as as harmful, distressing and coercive (Loh, 2018; Gunkel et al., 2022).
Furthermore, if such interactions are (presumably) recorded and actionable to
one robotics corporation this centralised control could be more liable to
litigation, or in need of more strict oversight.

##  4\. Conclusion

In conclusion, we have discussed a number of techniques (that adopt forms of
deception) used by human care-givers when caring for people with dementia.
While we cannot provide a clear consensus on the acceptability, effectiveness
or ethics of adopting each technique, we would advise that assistive robots
behave so as to enhance emotional and physical well-being (Coeckelbergh,
2011). With robots and language models becoming more capable and prevalent in
the provision of care, it is hoped that our discussion will lead researchers
to reflect on the likely potential adoption of such deceptive techniques, as
well as draw attention to the need for additional studies of robotic ethics in
innovative contexts of use (Stahl and Coeckelbergh, 2016).

###### Acknowledgements.

This research is part of the programme DesCartes and is supported by the
National Research Foundation, Prime Minister’s Office, Singapore under its
Campus for Research Excellence and Technological Enterprise (CREATE)
programme.

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# An explicit lower bound for large gaps between some consecutive primes

Keiju Sono

###### Abstract

Let $p_{n}$ denote the $n$th prime and for any fixed positive integer $k$ and
$X\geq 2$, put

$G_{k}(X):=\max_{p_{n+k}\leq
X}\min\\{p_{n+1}-p_{n},\ldots,p_{n+k}-p_{n+k-1}\\}.$

Ford, Maynard and Tao [6] proved that there exists an effective absolute
constant $c_{LG}>0$ such that

$G_{k}(X)\geq\frac{c_{LG}}{k^{2}}\frac{\log X\log\log X\log\log\log\log
X}{\log\log\log X}$

holds for any sufficiently large $X$. The main purpose of this paper is to
clarify the numerical value of the constant $c_{LG}$ such that the above
inequality holds. We see that $c_{LG}$ is determined by several factors
related to analytic number theory, for example, the ratio of integrals of
functions in the multidimensional sieve of Maynard [14], the distribution of
primes in arithmetic progressions to large moduli, and the coefficient of
upper bound sieve of Selberg. We prove that the above inequality is valid at
least for $c_{LG}\approx 2.0\times 10^{-17}$.

0002020 Mathematics Subject Classification. 11N36000Key Words and Phrases.
large gaps between primes, explicit lower bound, probabilistic method

##  1 Introduction

For a positive integer $k$ and a real number $X\geq 3$, put

$G_{k}(X):=\max_{p_{n+k}\leq
X}\min\\{p_{n+1}-p_{n},\ldots,p_{n+k}-p_{n+k-1}\\}$

and in particular, put $G(X):=G_{1}(X)$, which denotes the largest prime gap
below $X$. The prime number theorem yields the number of primes below $x$ is
asymptotically $x/\log x$, so the average gap of consecutive primes below $x$
is $(1+o(1))\log x$. Hence we get the trivial lower bound

$G(X)\geq(1+o(1))\log X.$

Harald Cramér [1] constructed a simple probabilistic model of the set of prime
numbers, and following his model, he conjectured that $G(X)\sim(\log X)^{2}$.
(See also refinements of Cramér’s model by Granville [8], Firoozbakht [20] and
Wolf [24].) In 1931, Westzynthius [23] made the first quantitative improvement
on the trivial bound and proved

$G(X)\gg\frac{\log X\log_{3}X}{\log_{4}X},$

where $\log_{n}$ denotes the $n$ times composition of the logarithmic
function, i.e., $\log_{n}X:=(\underbrace{\log\circ\cdots\circ\log}_{n})X$. In
particular, it follows that $G(X)/\log X\to\infty$ as $X\to\infty$, so $G(X)$
can be arbitrarily large compared with the average gap. In 1934, Ricci [21]
slightly improved this and showed $G(X)\gg\log X\log_{3}X$. Erdős [2] improved
on Westzynthius’ result and obtained

$G(X)\gg\frac{\log X\log_{2}X}{(\log_{3}X)^{2}}.$

Rankin [18] made a further improvement and showed that

$G(X)\geq(c+o(1))\frac{\log X\log_{2}X\log_{4}X}{(\log_{3}X)^{2}}$ (1.1)

holds with $c=\frac{1}{3}$. Several mathematicians improved on the value of
the coefficient $c$ above (see Schönhage [22] with $c=\frac{1}{2}e^{\gamma}$,
Rankin [19] with $c=e^{\gamma}$, Maier and Pomerance [13] with
$c=1.31256e^{\gamma}$, and Pintz [16] with $c=2e^{\gamma}$). Erdős conjectured
that (1.1) holds with arbitrarily large $c$. This problem had been unsolved
for a long time, but Maynard [14] and a team of Ford, Green, Konyagin, Tao [5]
independently solved at almost the same time in August 2014. Several months
later, in a joint work [4], they obtained

$G(X)\gg\frac{\log X\log_{2}X\log_{4}X}{\log_{3}X}.$ (1.2)

This was the first quantitative improvement of Rankin’s bound (1.1) in almost
80 years.

Let $k$ be a fixed positive integer. The $G_{k}(X)$ above has also been
studied for a long time. Following the argument of Cramér, it is conjectured
that $G_{k}(X)\asymp\frac{1}{k}\log^{2}X$. Erdős [2] considered the case $k=2$
and proved

$G_{2}(X)/\log X\to\infty$

as $X\to\infty$. Maier [12] showed

$G_{k}(X_{n})\gg_{k}\frac{\log
X_{n}\log_{2}X_{n}\log_{4}X_{n}}{(\log_{3}X_{n})^{2}}$

by combining his famous Maier matrix method and Pintz’s ideas in [16], where
$(X_{n})$ is some monotonically increasing sequence such that $X_{n}\to\infty$
as $n\to\infty$. Pintz [17] improved on Maier’s result and proved

$G_{k}(X_{n})/\left(\frac{\log
X_{n}\log_{2}X_{n}\log_{4}X_{n}}{(\log_{3}X_{n})^{2}}\right)\to\infty.$

An issue of Maier’s argument is that one has to avoid (possible) Siegel zeros,
and this is the reason why the results of Maier and Pintz above are restricted
to a special sequence $(X_{n})$, rather than all sufficiently large $X$.

In [6], Ford, Maynard and Tao succeeded in handling this difficulty and gave a
lower bound for $G_{k}(X)$ for any sufficiently large $X$. Concretely, they
proved that

$G_{k}(X)\gg\frac{1}{k^{2}}\frac{\log X\log_{2}X\log_{4}X}{\log_{3}X}$

holds as $X\to\infty$, and the implied constant above is absolute and
effective. In other words, for any sufficiently large $X\geq 3$, there exists
an absolute computable constant $c_{LG}>0$ such that

$G_{k}(X)\geq\frac{c_{LG}}{k^{2}}\frac{\log X\log_{2}X\log_{4}X}{\log_{3}X}$
(1.3)

holds. In addition to the technique to handle the possibility of the existence
of exceptional zeros of $L$-functions, their result relies on hypergraph
covering theorem, the construction of multidimensional sieve weight, and very
clever and highly technical probabilistic arguments in [4].

Though the lower bound in their theorem is effective, the explicit value of
$c_{LG}$ (resp. the implied constant of (1.2)) is not mentioned in [6] (resp.
[4]). (In the blog on the paper [4], Tao says “we manage to avoid the use of
the (ineffective) Siegel-Walfisz theorem by deleting an exceptional prime from
the multidimensional Selberg sieve, leading to an effective (but quite small)
value of $c$.” ) However, clarifying the value of $c_{LG}$ seems to be
significant because it has the merit of making clear which parameter affects
the coefficient $c_{LG}$ and how. With this reason, the main purpose of this
paper is to clarify what value is appropriate as the coefficient $c_{LG}$ in
(1.3) and how several factors in analytic number theory are related. It is
fully expected that this attempt might be a clue to establish a new
quantitative improvement on the lower bound of $G_{k}(X)$. We prove that one
can take

$c_{LG}=\frac{C_{PAP}^{2}\theta c_{I,J}e^{4\gamma}}{737280000\log
5\;C_{UB}M(1+D_{PAP}^{-1})^{4}(25C_{UB}+20e^{\gamma}M)}$

(see Theorem 3.6), where $\theta$ is a parameter which describes how primes
are equidistributed in arithmetic progressions (see [14], Hypothesis 1),
$c_{I,J}$ is the ratio of integrals of functions (see (2.3)) in the
multidimensional sieve of Maynard [14], $C_{PAP}$ and $D_{PAP}$ are constants
in the statement on distribution of primes in arithmetic progressions to large
moduli (see Assumption 3.2), and $C_{UB}$, $D_{UB}$ are the constants of upper
bound sieve of Selberg (see Assumption 3.4) and $M:=\max\\{D_{PAP},D_{UB}\\}$.
In Sections 2-3, we will see that unconditionally one can take
$\theta=\frac{1}{3}$, $c_{I,J}=\frac{1}{4}$, $C_{PAP}=1-e^{-2}$,
$D_{PAP}=160$, $C_{UB}=8e^{2\gamma}$ and $D_{UB}>0$ arbitrarily. Numerically,
one can take

$c_{LG}\approx 2.0\times 10^{-17}.$

Obviously, this value is not the best that can be obtained with current
techniques and could be improved to some extent with some effort, for example,
by applying recent results on zero density estimates and explicit zero free
regions of Dirichlet $L$-functions. In any case, however, as long as we rely
on current methods, the coefficient is likely to have to be fairly small.

##  2 Notation

Let $c>0$ be a fixed constant (to be determined later) and $x\geq 10$ a
sufficiently large real number. Put

$y:=c\frac{x\log x\log_{3}x}{\log_{2}x},\quad
z:=x^{\frac{\log_{3}x}{4\log_{2}x}}.$ (2.1)

Let $B_{0}$ be either $1$ or a prime number satisfying

$\log x\ll B_{0}\leq x.$ (2.2)

Define three disjoint sets of primes ${\cal S}$, ${\cal P}$ and ${\cal Q}$ by

${\cal S}:=\\{s:\mathrm{prime}\;|\;\log^{20}x<s\leq z,\;s\neq B_{0}\\},$
${\cal P}:=\\{p:\mathrm{prime}\;|\;\frac{x}{2}<p\leq x,\;p\neq B_{0}\\},$
${\cal Q}:=\\{q:\mathrm{prime}\;|\;x<q\leq y,\;q\neq B_{0}\\}.$

For vectors of residue classes $\vec{a}=(a_{s}\;\textrm{mod}\;s)_{s\in{\cal
S}}$, $\vec{n}=(n_{p}\;\textrm{mod}\;p)_{p\in{\cal P}}$, put

$S(\vec{a}):=\\{n\in\mathbb{Z}\;|\;n\not\equiv
a_{s}\;(\mathrm{mod}\;s),\;\forall s\in{\cal S}\\},$
$S(\vec{n}):=\\{n\in\mathbb{Z}\;|\;n\not\equiv
n_{p}\;(\mathrm{mod}\;p),\;\forall p\in{\cal P}\\}.$

Let $0<\theta<1$ be a parameter of Hypothesis 1 of [14]. In our situation,
unconditionally one can take

$\theta=\frac{1}{3}$

(see [4], Section 8).

Next, for $r\in\mathbb{Z}_{\geq 2}$, we denote by ${\cal F}_{r}$ the set of
square-integrable symmetric functions $F:\mathbb{R}^{r}\to\mathbb{R}$
supported in

${\cal
R}_{r}:=\\{(x_{1},\ldots,x_{r})\in\mathbb{R}^{r}\;|\;x_{1},\;\ldots,x_{r}\geq
0,x_{1}+\ldots+x_{r}\leq 1\\}.$

For $F\in{\cal F}_{r}$, put

$I_{r}(F):=\int_{0}^{\infty}\cdots\int_{0}^{\infty}F(t_{1},\ldots,t_{r})^{2}dt_{1}\cdots
dt_{r},$
$J_{r}(F):=\int_{0}^{\infty}\cdots\int_{0}^{\infty}\left(\int_{0}^{\infty}F(t_{1},\ldots,t_{r})dt_{r}\right)^{2}dt_{1}\cdots
dt_{r-1}.$

We suppose that for some positive constant $c_{I,J}$, the inequality

$\sup_{F\in{\cal F}_{r}}\frac{J_{r}(F)}{I_{r}(F)}\geq c_{I,J}\frac{\log
r}{r}\left(1+O\left(\frac{1}{\log r}\right)\right)$ (2.3)

holds as $r\to\infty$. Maynard showed that one can take $c_{I,J}=\frac{1}{4}$
(see (8.27) of [14]).

##  3 The sieve of intervals and the explicit lower bound for $G_{k}(X)$

######  Proposition 3.1.

Let $A\geq 1$ be an arbitrarily fixed constant and $x\geq 2$ sufficiently
large real number. Let $y$ be a parameter defined by (2.1) with

$c=\frac{\theta c_{I,J}}{12800\log 5}.$ (3.1)

Let $B_{0}$ be either $1$ or a prime number with $\log x\ll B_{0}\leq x$.
Then, for any prime number $p\leq x$ with $p\neq B_{0}$, there exists a
residue class $a_{p}\;(\textrm{mod}\;p)$ for which the set

${\cal T}:=\\{n\in[y]\backslash[x]\;|\;n\not\equiv
a_{p}\;(\mathrm{mod}\;p),\;\forall p\leq x,\;p\neq B_{0}\\}$

satisfies the following three conditions.
$\cdot$ (Upper bound)

$\\#{\cal T}\leq 5A(1+o(1))\frac{x}{\log x}.$ (3.2)

$\cdot$ (Lower bound)

$\\#{\cal T}\geq A(1+o(1))\frac{x}{\log x}.$ (3.3)

$\cdot$ (Upper bound in short intervals) For any fixed $0\leq\alpha<\beta\leq
1$, we have

$\\#({\cal T}\cap(\alpha y,\beta y])\leq
5A(2|\beta-\alpha|+\varepsilon)(1+o(1))\frac{x}{\log x}.$ (3.4)

We will prove this proposition in subsequent sections. We now give a lower
bound for $G_{k}(X)$ as a consequence of this proposition. Let $Q\geq 100$.
Then it is known that there exists a positive integer $B_{Q}$ which is either
$1$ or a prime number with

$B_{Q}\gg\log_{2}Q$ (3.5)

such that if a Dirichlet character $\chi$ with conductor less than $Q$ and
coprime to $B_{Q}$ satisfies

$L(\sigma+it,\chi)=0\quad(\sigma,t\in\mathbb{R}),$

then

$1-\sigma\geq\frac{c_{ZFR}}{\log(Q(1+|t|))}$ (3.6)

holds for some absolute constant $c_{ZFR}>0$. We introduce the following two
assumptions on distribution of primes in arithmetic progressions.

######  Assumption 3.2 (Assumption PAP).

Suppose that all $L$-functions associated to Dirichlet characters $\chi$
modulo $q$ do not have any zero in the region (3.6) (with $Q$ replaced by
$q$). Then, there exist absolute constants $0<C_{PAP}\leq 1$ and $D_{PAP}\geq
1$ for which for any positive integers $a,q$ with $(a,q)=1$ and $x\geq
q^{D_{PAP}}$, one has

$\\#\\{p:\mathrm{prime}\;|\;p\leq x,\;p\equiv a\;(\mathrm{mod}\;q)\\}\geq
C_{PAP}(1+o(1))\frac{x}{\varphi(q)\log x}.$ (3.7)

######  Remark 3.3.

Later we will show that the above assumption is valid unconditionally for
$D_{PAP}=160$, $C_{PAP}=1-e^{-2}$.

For $X>1$, we denote by $[X]$ the set of integers in the interval $[1,X]$.

######  Assumption 3.4 (Assumption UB).

There exist absolute constants $C_{UB}\geq 1$, $D_{UB}>0$ for which the
following holds. For any sufficiently large $x$ and a positive integer $B_{0}$
which is either $1$ or a prime number with

$\log x\ll B_{0}\leq x,$ (3.8)

put $P:=P(x)/B_{0}$, where $P(x):=\prod_{p\leq}p$ denotes the product of all
primes equal to or less than $x$. Then, for any $Z\geq P^{D_{UB}}$ and
$a,b\in[P]$ with $a\neq b$, one has

$\displaystyle\\#\\{z\in[Z]\;|\;Pz+a,\;Pz+b:\mathrm{prime}\\}\leq
C_{UB}(1+o(1))\left(\frac{\log x}{\log Z}\right)^{2}Z.$ (3.9)

######  Remark 3.5.

Later we will show that one can unconditionally take $C_{UB}=8e^{2\gamma}$ and
$D_{UB}>0$ arbitrarily.

With these notations, we have the following theorem.

######  Theorem 3.6 (An explicit version of the theorem of Ford, Maynard and
Tao [6]).

Let $k$ be a fixed positive integer. Let $\theta$ be a parameter of Maynard’s
sieve in Hypothesis 1 of [14], $c_{I,J}$ a constant satisfying (2.3),
$C_{PAP}$ and $D_{PAP}$ constants in Assumption PAP and $C_{UB},D_{UB}$
constants in Assumption UB and put $M:=\max\\{D_{PAP},D_{UB}\\}$. Then we have

$\displaystyle G_{k}(X)\geq$ $\displaystyle\frac{C_{PAP}^{2}\theta
c_{I,J}e^{4\gamma}}{737280000\log
5\;C_{UB}M(1+D_{PAP}^{-1})^{4}(25C_{UB}+20e^{\gamma}M)}\frac{1}{k^{2}}\frac{\log
X\log_{2}X\log_{4}X}{\log_{3}X}$ (3.10)

fot any sufficiently large $X$.

######  Lemma 3.7.

Assume Assumptions PAP and UB. Let $B_{0}$ be a positive integer which is
either $1$ or a prime number satisfying (3.8). Put $P=P(x)/B_{0}$. Then, for
any $Z\geq P^{D_{PAP}}$ and $a\in[P]$ with $(a,P)=1$, we have

$\\#\\{z\in[Z]\;|\;Pz+a:\mathrm{prime}\\}\geq\frac{e^{\gamma}}{1+D_{PAP}^{-1}}(1+o(1))\frac{\log
x}{\log Z}Z.$ (3.11)

###### Proof.

We apply the Assumption PAP with a zero free region (3.6) with
$Q=P(x)=\prod_{p\leq x}p$. In the Assumption UB, we set $B_{0}=1$ if
$B_{P(x)}>x$ and otherwise put $B_{0}=B_{P(x)}$. Note that in the latter case,
due to the condition (3.5) and a consequence of the prime number theorem
($P(x)\sim e^{(1+o(1))x}$), it follows that

$B_{0}=B_{P(x)}\gg\log_{2}P(x)=\log_{2}\prod_{p\leq x}p\sim\log_{2}e^{x}=\log
x,$

so (3.8) is satisfied. Therefore, by the condition (3.7), for $Z\geq
P^{D_{PAP}}$, we have

$\displaystyle\\#\\{z\in[Z]\;|\;Pz+a:\mathrm{prime}\\}$ (3.12)
$\displaystyle=\\#\\{P+a\leq p\leq PZ+a\;|\;p\equiv
a\;(\mathrm{mod}\;P),\;p:\mathrm{prime}\\}$ $\displaystyle\geq
C_{PAP}(1+o(1))\frac{PZ+a}{\varphi(P)\log(PZ+a)}-\\#\\{p<P+a\;|\;p\equiv
a\;(\mathrm{mod}\;P),\;p:\mathrm{prime}\\}.$

By Mertens’ formula, we have

$\frac{P}{\varphi(P)}=\frac{P}{P\prod_{p\leq x,p\neq
B_{0}}\left(1-\frac{1}{p}\right)}=\frac{1}{\prod_{p\leq x,p\neq
B_{0}}\left(1-\frac{1}{p}\right)}\sim e^{\gamma}\log x.$ (3.13)

Hence the first term of the last line of (3.12) is asymptotically

$\frac{PZ}{\varphi(P)\log PZ}\geq e^{\gamma}\log x\frac{Z}{\log
Z^{1+\frac{1}{D_{PAP}}}}=\frac{e^{\gamma}}{1+D_{PAP}^{-1}}\frac{\log x}{\log
Z}Z.$

On the other hand, the second term of the last line of (3.12) is at most

$\\#\\{p<P\;|\;p:\mathrm{prime}\\}\leq\frac{2P}{\log P}\leq
2D_{PAP}\frac{Z^{\frac{1}{D_{PAP}}}}{\log Z}=o\left(\frac{\log x}{\log
Z}Z\right).$

Combining these estimates, we obtain (3.11). ∎

We write $\mathbb{P}$ for probability and $\mathbb{E}$ for expectation. Put
$Z=P^{M}$, where $M:=\max\\{D_{UB},D_{PAP}\\}$. Let $\mathbf{z}$ be a random
variable in $[Z]$ which is chosen uniformly and $y$, ${\cal T}$ and
$a_{p}\;(\textrm{mod}\;p)$ are those in Proposition 3.1. By Chinese remainder
theorem, there exists $m\in[Z]$ such that $m\equiv-a_{p}\;(\textrm{mod}\;p)$
holds for any $p\leq x,p\neq B_{0}$. Therefore, the interval
$\mathbf{z}P+m+{\cal T}$ exactly consists of elements of
$\mathbf{z}P+m+[y]\backslash[x]$ which is coprime to $P$. In particular, all
prime numbers in $\mathbf{z}P+m+[y]\backslash[x]$ are contained in
$\mathbf{z}P+m+{\cal T}$. By (3.11) and $Z=P^{M}=e^{(1+o(1))Mx}$, we have

$\displaystyle\mathbb{P}(\mathbf{z}P+m+a:\mathrm{prime})$
$\displaystyle\geq\frac{C_{PAP}e^{\gamma}}{1+D_{PAP}^{-1}}(1+o(1))\frac{\log
x}{\log Z}$ (3.14)
$\displaystyle=\frac{C_{PAP}e^{\gamma}}{(1+D_{PAP}^{-1})M}(1+o(1))\frac{\log
x}{x}$

for any $a\in{\cal T}$. On the other hand, by Brun-Titchmarsh type estimate

$\pi(x;q,a)\leq\frac{2x}{\varphi(q)\log\frac{x}{q}}\quad(\forall x>q)$

and Mertens type estimate (3.13), we have

$\displaystyle\mathbb{P}(\mathbf{z}P+m+a:\mathrm{prime})$
$\displaystyle=\frac{1}{Z}\\#\\{z\in[Z]\;|\;Pz+m+a:\mathrm{prime}\\}$ (3.15)
$\displaystyle\leq\frac{1}{Z}\\#\\{p\leq PZ+m+a\;|\;p\equiv
m+a\;(\mathrm{mod}\;P)\\}$ $\displaystyle\leq\frac{1}{Z}\\#\\{p\leq
2PZ\;|\;p\equiv m+a\;(\mathrm{mod}\;P)\\}$
$\displaystyle\leq\frac{1}{Z}\frac{2\cdot 2PZ}{\varphi(P)\log\frac{2PZ}{P}}$
$\displaystyle\leq\frac{4e^{\gamma}\log x}{\log(2Z)}$
$\displaystyle\leq\frac{4e^{\gamma}\log x}{\log
Z}\sim\frac{4e^{\gamma}}{M}\frac{\log x}{x}.$

Furthermore, by Assumption UB,

$\displaystyle\mathbb{P}(\mathbf{z}P+m+a,\;\mathbf{z}P+m+b:\mathrm{prime})$
$\displaystyle\leq C_{UB}(1+o(1))\left(\frac{\log x}{\log Z}\right)^{2}$
(3.16) $\displaystyle=\frac{C_{UB}}{M^{2}}(1+o(1))\left(\frac{\log
x}{x}\right)^{2}$

for any $a\neq b\in{\cal T}$. Let $\mathbf{N}$ denote the number of primes in
$\mathbf{z}P+m+{\cal T}$ (hence in $\mathbf{z}P+m+[y]\backslash[x]$). By
(3.14) and (3.3), we have

$\displaystyle\mathbb{E}\mathbf{N}$ $\displaystyle=\sum_{a\in{\cal
T}}\mathbb{P}(\mathbf{z}P+m+a:\mathrm{prime})\geq(1+o(1))\frac{C_{PAP}e^{\gamma}}{(1+D_{PAP}^{-1})M}A.$
(3.17)

On the other hand, by (3.15), (3.16) and (3.2),

$\displaystyle\mathbb{E}\mathbf{N}^{2}$ $\displaystyle=\sum_{a,b\in{\cal
T}}\mathbb{P}(\mathbf{z}P+m+a,\mathbf{z}P+m+b:\mathrm{prime})$ (3.18)
$\displaystyle\leq\frac{C_{UB}}{M^{2}}(1+o(1))\left(\frac{\log
x}{x}\right)^{2}(\\#{\cal T})^{2}+\frac{4e^{\gamma}}{M}(1+o(1))\frac{\log
x}{x}\\#{\cal T}$
$\displaystyle\leq\left(\frac{25A^{2}C_{UB}}{M^{2}}+\frac{20Ae^{\gamma}}{M}\right)(1+o(1))$
$\displaystyle\leq\frac{25C_{UB}+20e^{\gamma}M}{M^{2}}(1+o(1))A^{2}.$

(The first term of the second line is an upper bound for the contribution of
the terms with $a\neq b$, and the second term is that of the terms with $a=b$.
In the last line, we used $A\geq 1$.) Suppose that $0\leq\alpha<\beta\leq 1$
satisfy $\beta-\alpha\leq 2\varepsilon$ for $0<\varepsilon<1$. Then by (3.16)
and (3.4), the probability that the interval $\mathbf{z}P+m+[\alpha y,\beta
y]$ contains at least two primes is at most

$\left(6A(4\varepsilon+\varepsilon)\frac{x}{\log
x}\right)^{2}\frac{C_{UB}}{M^{2}}(1+o(1))\left(\frac{\log
x}{x}\right)^{2}\leq\frac{900\varepsilon^{2}A^{2}C_{UB}}{M^{2}}(1+o(1)).$

We cover the interval $[0,1]$ by intervals

$I_{1}=[0,2\varepsilon],\;I_{2}=[\varepsilon,3\varepsilon],\;\ldots,\;I_{n_{\varepsilon}}=[(n_{\varepsilon}-1)\varepsilon,1],$

where $n_{\varepsilon}=\lceil\frac{1}{\varepsilon}\rceil$. Then two elements
$a,b\in[0,1]$ with $|a-b|\leq\varepsilon$ are contained in the same interval
$I_{j}$ for some $1\leq j\leq n_{\varepsilon}$. Therefore, the probability
that “The interval $\mathbf{z}P+m+[y]\backslash[x]$ contains two primes with
gap at most $\varepsilon y$ ” is equal to or less than the probability that
“some of the $n_{\varepsilon}$ intervals

$\mathbf{z}P+m+[0,2\varepsilon y],\;\mathbf{z}P+m+[\varepsilon y,3\varepsilon
y],\;\ldots,\;\mathbf{z}P+m+[(n_{\varepsilon}-1)\varepsilon y,y]$

contains two primes”, and by the above consideration, this probability is at
most

$n_{\varepsilon}\frac{900\varepsilon^{2}A^{2}C_{UB}}{M^{2}}(1+o(1))<\frac{1800\varepsilon
A^{2}C_{UB}}{M^{2}},$

since
$n_{\varepsilon}=\left\lceil\frac{1}{\varepsilon}\right\rceil<\frac{2}{\varepsilon}$.
We take

$\varepsilon=\frac{M^{2}}{1800A^{2}C_{UB}}\frac{e^{2\gamma}}{4(1+D_{PAP}^{-1})^{2}(25C_{UB}+20e^{\gamma}M)}.$
(3.19)

(Later we will see that $\varepsilon<1$ with an appropriate choice of $A$.)
Then the above probability is less than

$\frac{e^{2\gamma}}{4(1+D_{PAP}^{-1})^{2}(25C_{UB}+20e^{\gamma}M)}.$

In other words, all primes in $\mathbf{z}P+m+[y]\backslash[x]$ are separated
more than $\varepsilon y$ to each other with probability greater than

$1-\frac{e^{2\gamma}}{4(1+D_{PAP}^{-1})^{2}(25C_{UB}+20e^{\gamma}M)}.$ (3.20)

On the other hand, for any fixed $c_{1}>0$, by Cauchy-Schwarz inequality, we
have

$\displaystyle\mathbb{E}\mathbf{N}$ $\displaystyle\leq
c_{1}A\mathbb{P}(\mathbf{N}\leq
c_{1}A)+\mathbb{E}\mathbf{N}1_{\mathbf{N}>c_{1}A}\leq
c_{1}A+(\mathbb{E}\mathbf{N}^{2})^{\frac{1}{2}}\mathbb{P}(\mathbf{N}>c_{1}A)^{\frac{1}{2}}.$

Therefore, by (3.17) and (3.18), we have

$\displaystyle\mathbb{P}(\mathbf{N}>c_{1}A)$
$\displaystyle\geq\frac{(\mathbb{E}\mathbf{N}-c_{1}A)^{2}}{\mathbb{E}\mathbf{N}^{2}}\geq\frac{\left(\frac{C_{PAP}e^{\gamma}A}{(1+D_{PAP}^{-1})M}-c_{1}A\right)^{2}}{\frac{(25C_{UB}+20e^{\gamma}M)A^{2}}{M^{2}}}=\frac{\left(\frac{C_{PAP}e^{\gamma}}{1+D_{PAP}^{-1}}-c_{1}M\right)^{2}}{25C_{UB}+20e^{\gamma}M},$
(3.21)

provided that $c_{1}<\frac{C_{PAP}e^{\gamma}}{(1+D_{PAP}^{-1})M}$. We take

$c_{1}=\frac{C_{PAP}e^{\gamma}}{2(1+D_{PAP}^{-1})M}.$

Then,

$\mathbb{P}(\mathbf{N}>c_{1}A)\geq\frac{e^{2\gamma}}{4(1+D_{PAP}^{-1})^{2}(25C_{UB}+20e^{\gamma}M)}.$
(3.22)

Since the sum of (3.20) and the right hand side of (3.22) is equal to $1$, it
follows that there exists some integer $z\in[Z]$ such that the interval
$zP+m+[y]\backslash[x]$ contains at least

$c_{1}A=\frac{e^{\gamma}A}{2M(1+D_{PAP}^{-1})}$

primes and all of them are separated at least

$\varepsilon
y=\frac{yM^{2}}{1800A^{2}C_{UB}}\frac{e^{2\gamma}}{4(1+D_{PAP}^{-1})^{2}(25C_{UB}+20e^{\gamma}M)}$

to each other, where

$y=c\frac{x\log x\log_{3}x}{\log_{2}x},\quad c=\frac{\theta c_{I,J}}{12800\log
5}.$

We take

$A=2e^{-\gamma}C_{PAP}^{-1}M(1+D_{PAP}^{-1})k.$ (3.23)

Then, the interval $zP+m+[y]\backslash[x]$ contains $k$ consecutive primes
which are separated at least $\varepsilon y$ to each other. Therefore,

$\displaystyle G_{k}(ZP+m+y)$ $\displaystyle\geq\varepsilon y$
$\displaystyle=\frac{\theta c_{I,J}}{12800\log
5}\frac{M^{2}}{1800A^{2}C_{UB}}\frac{e^{2\gamma}}{4(1+D_{PAP}^{-1})^{2}(25C_{UB}+20e^{\gamma}M)}\frac{x\log
x\log_{3}x}{\log_{2}x}.$

Since

$ZP+m+y\leq 2ZP=2P^{M+1}=2e^{(1+o(1))(M+1)x}\leq e^{1.5Mx},$

by putting $X=e^{1.5Mx}$, we have

$\displaystyle G_{k}(X)$ $\displaystyle\geq\frac{\theta c_{I,J}}{12800\log
5}\frac{M^{2}}{1800A^{2}C_{UB}}\frac{e^{2\gamma}(1+o(1))}{4(1+D_{PAP}^{-1})^{2}(25C_{UB}+20e^{\gamma}M)}\frac{2}{3M}\frac{\log
X\log_{2}X\log_{4}X}{\log_{3}X}$ $\displaystyle\geq\frac{\theta
c_{I,J}}{12800\log
5}\frac{M^{2}}{1800A^{2}C_{UB}}\frac{e^{2\gamma}}{4(1+D_{PAP}^{-1})^{2}(25C_{UB}+20e^{\gamma}M)}\frac{1}{2M}\frac{\log
X\log_{2}X\log_{4}X}{\log_{3}X}.$

Finally, by substituting (3.23), we obtain (3.10). $\Box$

##  4 Possible values of $C_{UB}$ and $D_{UB}$ in Assumption UB

To establish the upper bound (3.9) in Assumption UB, we use the following
theorem.

######  Theorem 4.1 ([9], Theorem 5.7).

For $g\in\mathbb{N}$ and integers $a_{i},b_{i}$ $(i=1,\ldots,g)$, suppose

$E:=\prod_{i=1}^{g}a_{i}\prod_{1\leq r<s\leq g}(a_{r}b_{s}-a_{s}b_{r})\neq 0.$

For any prime number $p$, let $\rho(p)$ denote the number of solutions
$n\;(\textrm{mod}\;p)$ of the equation

$\prod_{i=1}^{g}(a_{i}n+b_{i})\equiv 0\quad(\mathrm{mod}\;p),$

and suppose that $\rho(p)<p$ holds for any primes $p$. Then, for any $1<y\leq
x$, we have

$\displaystyle\\#\\{n\;|\;x-y<n\leq x,a_{i}n+b_{i}:\mathrm{prime}\\}$
$\displaystyle\quad\leq
2^{g}g!\prod_{p}\left(1-\frac{\rho(p)-1}{p-1}\right)\left(1-\frac{1}{p}\right)^{-g+1}\frac{y}{\log^{g}y}\left(1+O((\log
y)^{-1}(\log_{2}3y+\log_{2}3|E|))\right).$

We apply this theorem with $x=y=Z$, $a_{1}=a_{2}=P$, and $b_{1}=a$, $b_{2}=b$.
Then, $E=P^{3}(a-b)\neq 0$ and

$|E|\ll P^{3}Z\ll e^{3(1+o(1))x}Z.$

Note that since $a,b\in{\cal T}\subset[y]\backslash[x]$, we have $|a-b|\leq
y=o(x\log x)$. We will use this later (see (4.5) below).

We need to compute $\rho(p)$. First, for $p\leq x$ with $p\neq B_{0}$, if
either $p|a$ or $p|b$ holds, then

$Pn+a\equiv 0\quad(\mathrm{mod}\;p)\quad\mathrm{or}\quad Pn+b\equiv
0\quad(\mathrm{mod}\;p)$

holds for any $n\in\mathbb{N}$, so

$\\#\\{z\in[Z]\;|\;Pz+a,Pz+b:\mathrm{prime}\\}=0.$

Therefore, it suffices to assume that $p\nmid ab$ ($\forall p\leq x$) or
$p=B_{0}$ (in case of $B_{0}$: prime).
I) If $p\leq x$, $p\neq B_{0}$, then the condition

$(Pn+a)(Pn+b)\equiv 0\quad(\mathrm{mod}\>p)$

is equivalent to

$ab\equiv 0\quad(\mathrm{mod}\>p)$

which does not hold due to our assumption. Hence $\rho(p)=0$.
II) If $B_{0}$ is a prime number and $p=B_{0}$, then the equation

$(Pn+a)(Pn+b)\equiv 0\quad(\mathrm{mod}\;p)$

has two solutions. Hence $\rho(B_{0})=2$.
III) If $p>x$, then the number of the solutions $n\;(\textrm{mod}\;p)$ of the
equation

$(Pn+a)(Pn+b)\equiv 0\quad(\mathrm{mod}\;p)$

is $1$ if $a\equiv b\;(\textrm{mod}\;p)$, and 2 if $a\not\equiv
b\;(\textrm{mod}\;p)$.
Consequently,

$\displaystyle\\#\\{z\in[Z]\;|\;Pz+a,Pz+b:\mathrm{prime}\\}$ (4.1)
$\displaystyle\quad\leq 2^{2}\cdot 2!\underset{p\neq B_{0}}{\prod_{p\leq
x}}\left(1+\frac{1}{p-1}\right)\left(1-\frac{1}{p}\right)^{-1}\left[\left(1-\frac{1}{B_{0}-1}\right)\left(1-\frac{1}{B_{0}}\right)^{-1}\right]$
$\displaystyle\quad\quad\quad\times\underset{p\nmid(a-b)}{\prod_{p>x}}\left(1-\frac{1}{p-1}\right)\left(1-\frac{1}{p}\right)^{-1}\underset{p|(a-b)}{\prod_{p>x}}\left(1-\frac{1}{p}\right)^{-1}$
$\displaystyle\quad\quad\times\frac{Z}{\log^{2}Z}\left(1+O\left(\frac{\log_{2}Z+\log
x}{\log Z}\right)\right).$

Here, the factor $[\cdots]$ in the second line appears only when $B_{0}$ is
prime. Now,

$\underset{p\neq B_{0}}{\prod_{p\leq
x}}\left(1+\frac{1}{p-1}\right)\left(1-\frac{1}{p}\right)^{-1}\leq\prod_{p\leq
x}\left(1-\frac{1}{p}\right)^{-2}\sim e^{2\gamma}\log^{2}x,$ (4.2)
$\left(1-\frac{1}{B_{0}-1}\right)\left(1-\frac{1}{B_{0}}\right)^{-1}\sim 1,$
(4.3)
$\underset{p\nmid(a-b)}{\prod_{p>x}}\left(1-\frac{1}{p-1}\right)\left(1-\frac{1}{p}\right)^{-1}\leq
1,$ (4.4)

and

$\underset{p|(a-b)}{\prod_{p>x}}\left(1-\frac{1}{p}\right)^{-1}=\exp\left(-\underset{p|(a-b)}{\sum_{p>x}}\log\left(1-\frac{1}{p}\right)\right)\leq\exp\left(\underset{p|(a-b)}{\sum_{p>x}}\frac{2}{p}\right)\leq\exp\left(\frac{2\omega(|a-b|)}{x}\right)\sim
1,$ (4.5)

where $\omega(n)$ denotes the number of distinct prime factors of $n$.
Substituting (4.2)-(4.5) into (4.1), we have

$\displaystyle\\#\\{z\in[Z]\;|\;Pz+a,Pz+b:\mathrm{prime}\\}$
$\displaystyle\leq 8e^{2\gamma}\log^{2}x\frac{Z}{\log^{2}Z}(1+o(1))$ (4.6)
$\displaystyle=8e^{2\gamma}(1+o(1))\left(\frac{\log x}{\log Z}\right)^{2}Z,$

provided that $\log x=o(\log Z)$. Since $Z\geq
P^{D_{UB}}=e^{(1+o(1))D_{UB}x}$, this condition is valid for any $D_{UB}>0$.
Consequently, the Assumption UB is valid for

$C_{UB}=8e^{2\gamma}$ (4.7)

and arbitrary $D_{UB}>0$.

##  5 Possible values of $C_{PAP}$ and $D_{PAP}$ in Assumption PAP

To establish an inequality in Assumption PAP, we need some results on zero-
free regions of Dirichlet $L$-functions. We use the following results by
McCurley [11]. For a positive integer $q$, let ${\cal L}_{q}(s)$ be the
product of $\varphi(q)$ $L$-functions associated to Dirichlet characters
modulo $q$. Write $s=\sigma+it$, where $\sigma,t\in\mathbb{R}$.

######  Proposition 5.1 ([11], Theorem 1).

Let $M:=\max\\{q,q|t|,10\\}$, $R=9.6459\ldots$. Then ${\cal L}_{q}(s)$ has at
most one zero in

$\left\\{s=\sigma+it\;|\;\sigma\geq 1-\frac{1}{R\log M}\right\\}.$

The only possible zero is a real zero of the $L$-function associated to a non-
principal real character modulo $q$.

######  Proposition 5.2 ([11], Theorem 2).

For $i=1,2$, let $\chi_{i}$ be a real character modulo $q_{i}$ such that
$\chi_{1}\neq\chi_{2}$ and $\beta_{i}$ be a real zero of $L(s,\chi_{i})$. Put

$M_{1}=\max\left\\{\frac{q_{1}q_{2}}{17},13\right\\},\quad
R_{1}=\frac{5-\sqrt{5}}{15-10\sqrt{2}}.$

Then we have

$\min\\{\beta_{1},\beta_{2}\\}<1-\frac{1}{R_{1}\log M_{1}}.$

For $Q\geq 100$ and $q_{1},q_{2}\leq Q$, let $\chi_{1}$, $\chi_{2}$ be
distinct primitive characters modulo $q_{1}$, $q_{2}$ respectively. Suppose
that for $j=1,2$, $L(s,\chi_{j})$ has a zero $\rho_{j}=\beta_{j}+i\gamma_{j}$
in the domain

${\cal R}:\left\\{s=\sigma+it\;|\;\sigma\geq
1-\frac{1}{4R_{1}\log(Q(1+|t|))}\right\\}.$

Then for $M_{(j)}:=\max\\{q_{j},q_{j}|t|,10\\}$ ($j=1,2$),

$1-\frac{1}{R\log M_{(j)}}\leq 1-\frac{1}{4R_{1}\log(Q(1+|t|))}$

holds. Hence by Proposition 5.1, $\rho_{j}$ is a real zero of $L(s,\chi_{j})$
for $j=1,2$. By Proposition 5.2, we have

$\min\\{\beta_{1},\beta_{2}\\}\leq 1-\frac{1}{R_{1}\log M_{1}},$ (5.1)

where $M_{1}=\max\\{\frac{q_{1}q_{2}}{17},13\\}$. However, since

$1-\frac{1}{R_{1}\log M_{1}}\leq 1-\frac{1}{4R_{1}\log Q},$

the inequality (5.1) contradicts our assumption that both $\rho_{1}=\beta_{1}$
and $\rho_{2}=\beta_{2}$ lie in ${\cal R}$. Combining this and an argument in
the proof of Corollary 1 of [6], we have the following conclusion.

######  Proposition 5.3.

Put

$c_{ZFR}:=\frac{1}{24}\left(<\frac{1}{4R_{1}}=\frac{15-10\sqrt{2}}{4(5-\sqrt{5})}\right).$

Let $Q\geq 100$ be a fixed positive integer. Suppose that $L(s,\chi)=0$ holds
for some primitive Dirichlet character modulo at most $Q$. Then either one of

$1-\sigma\geq\frac{c_{ZFR}}{\log(Q(1+|t|))}\quad(s=\sigma+it,\sigma,t\in\mathbb{R})$

or “$t=0$ and $\chi=\chi_{Q}$ is a real character which is determined uniquely
by $Q$” holds.

Gallagher [7] proved that there exist some absolute constants $a,b>0$ for
which

$\sum_{q\leq
Q}\sum_{\chi(\mathrm{mod}\;q)}^{\quad\quad*}\left|\sum_{p=x}^{x+h}\chi(p)\log
p\right|\ll h\exp\left(-a\frac{\log x}{\log Q}\right)$ (5.2)

holds uniformly for $x/Q\leq h\leq x$, $\exp(x^{\frac{1}{2}}\log x)\leq Q\leq
x^{b}$. We need to make clear the value of $a$. Due to the above
consideration, the $L$-functions associated to primitive characters modulo at
most $T$ do not have zero in

$\sigma>1-\frac{3c_{ZFR}}{\log T},\quad|t|\leq T$

with at most one exception, and if the exceptional zero exists, then it is a
simple real zero. Hence one can take $c_{1}=3c_{ZFR}$ in [7], (26). The
estimate of [7], (30) yields

$\sum_{q\leq
Q}\sum_{\chi(\mathrm{mod}\;q)}^{\quad\quad*}\left|\sum_{p=x}^{x+h}\chi(p)\log
p\right|\ll h\left(\sum_{q\leq
Q}\sum_{\chi(\mathrm{mod}\;q)}^{\quad\quad*}\sum_{\beta}x^{\beta-1}+\frac{Q^{4}}{T}\right),$
(5.3)

where $\beta$ runs over zeros of non trivial zeros of $L(s,\chi)$ in
$\sigma\leq 1-3c_{ZFR}/\log T$, $|t|\leq T$. Due to the argument of [7], we
have

$\sum_{q\leq
Q}\sum_{\chi(\mathrm{mod}\;q)}^{\quad\quad*}\sum_{\beta}x^{\beta-1}\ll
x^{-\frac{1}{2}\frac{3c_{ZFR}}{\log T}}.$

Therefore, by putting $T=Q^{5}$, the right hand side of (5.3) is at most

$\exp\left(-\frac{3}{10}c_{ZFR}\frac{\log x}{\log Q}\right).$

Therefore, one can take

$a=\frac{3}{10}c_{ZFR}$

in (5.2).

Next, by Theorem 6 of [7], for some constant $c_{ZD}>0$,

$\sum_{q\leq T}\sum_{\chi\;(\mathrm{mod}\;q)}^{\quad\quad*}N(\alpha,T,\chi)\ll
T^{c_{ZD}(1-\alpha)}$ (5.4)

holds uniformly for $0\leq\alpha\leq 1$. Here, $N(\alpha,T,\chi)$ denotes the
number of zeros of $L(s,\chi)$ in the domain $\alpha\leq\Re(s)\leq 1$,
$|\Im(s)|\leq T$. With this $c_{ZD}$, Theorem 7 of [7] is valid for
$T^{c_{ZD}}\leq x^{\frac{1}{2}}$, $T=Q^{5}$. Therefore, for
$\exp(\log^{\frac{1}{2}}x)\leq Q\leq x^{\frac{1}{10c_{ZD}}}$, (5.2) holds for
$x/Q\leq h\leq x$. Therefore, the proof og Lemma 2 of Maier [12] is valid for
$x\geq Q^{10c_{ZD}}$.

In [12], the contribution of the terms with $\chi=\chi_{0}$ to $\psi(x;q,a)$
is $(1+o(1))x/\varphi(q)$, whereas that of the terms with $\chi\neq\chi_{0}$
is $O_{\leq}(xe^{-aD}/\varphi(q))$. Here, $f(x)=O_{\leq}(g(x))$ means that
$|f(x)|\leq g(x)$ holds for any sufficiently large $x$. Therefore, we have

$\psi(x;q,a)\geq(1-e^{-aD})(1+o(1))\frac{x}{\varphi(q)}$

provided that $D>0$. Hence one can take $D_{PAP}=10c_{ZD}$,
$C_{PAP}=1-e^{-aD_{PAP}}$.

Next, we evaluate $c_{ZD}$ above. Put

$N^{*}(\alpha,T,Q):=\sum_{q\leq
Q}\sum_{\chi(\mathrm{mod}\;q)}^{\quad\quad*}N(\alpha,T,\chi).$

By Theorem 1 of Jutila [10], for any fixed $\varepsilon>0$,

$N^{*}(\alpha,T,Q)\ll_{\varepsilon}(Q^{2}T)^{(2+\varepsilon)(1-\alpha)}$

holds uniformly for $\frac{4}{5}\leq\alpha\leq 1$, $T\geq 1$. Put $Q=T$. Then
it follows that

$\sum_{q\leq
T}\sum_{\chi\;(\mathrm{mod}\;q)}^{\quad\quad*}N(\alpha,T,\chi)\ll_{\varepsilon}T^{(6+\varepsilon)(1-\alpha)}.$
(5.5)

On the other hand, if $0\leq\alpha\leq\frac{4}{5}$, by using the trivial
estimate

$N(\alpha,T,\chi)\ll T\log T,$

we have

$\displaystyle\sum_{q\leq
T}\sum_{\chi\;(\mathrm{mod}\;q)}^{\quad\quad*}N(\alpha,T,\chi)$
$\displaystyle\ll\sum_{q\leq T}qT\log T\ll T^{3}\log T\ll
T^{(15+\varepsilon)(1-\alpha)}.$ (5.6)

By (5.5) and (5.6),

$\sum_{q\leq
T}\sum_{\chi\;(\mathrm{mod}\;q)}^{\quad\quad*}N(\alpha,T,\chi)\ll_{\varepsilon}T^{(15+\varepsilon)(1-\alpha)}$

holds uniformly for $0\leq\alpha\leq 1$. Therefore, one can take $c_{ZD}=16$.
Consequently, one can take

$C_{PAP}=1-e^{-2},\quad D_{PAP}=160$ (5.7)

as the values of $C_{PAP}$ and $D_{PAP}$.

##  6 Proof of Proposition 3.1

Proposition 3.1 can be derived from the following proposition.

######  Proposition 6.1 (An explicit version of Theorem 3 of [6]).

Let $A\geq 1$ be an arbitrary real number and $x$ be a sufficiently large real
number and $B_{0}$ be a positive integer. Put

$y=c\frac{x\log x\log_{3}x}{\log_{2}x},\quad c=\frac{\theta c_{I,J}}{12800\log
5}$

and let $S(\vec{a})$, $S(\vec{n})$ be the subsets of $\mathbb{Z}$ defined in
Section $2$. Then, there exist $A^{\prime}=A^{\prime}(x)$ and
$c^{\prime}=c^{\prime}(x)$ satisfying

$A^{\prime}\sim c^{\prime}A,\quad 1\leq c^{\prime}\leq 5$

and choices of vectors
$\vec{\mathbf{a}}=(\mathbf{a}_{s}\;\mathrm{mod}\;s)_{s\in{\cal S}}$,
$\vec{\mathbf{n}}=(\mathbf{n}_{p}\;\mathrm{mod}\;p)_{p\in{\cal P}}$ such that
for arbitrarily fixed $0\leq\alpha<\beta\leq 1$,

$\\#({\cal Q}\cap S(\vec{\mathbf{a}})\cap S(\vec{\mathbf{n}})\cap(\alpha
y,\beta y])\sim A^{\prime}|\beta-\alpha|\frac{x}{\log x}$ (6.1)

holds with probability $1-o(1)$. The implied constant of $o(1)$ in the
probability may depend on $A,\alpha,\beta$, but that of (6.1) doesn’t.

(Proof of Proposition 6.1 $\Rightarrow$ Proposition 3.1) Let
$0<\varepsilon<1$. We decompose the interval $(0,1]$ into
$O(\varepsilon^{-1})$ disjoint intervals $(\alpha_{i},\beta_{i}]_{i\in I}$ of
lengths between $\varepsilon/2$ and $\varepsilon$. We apply Proposition 6.1
with $(\alpha,\beta)=(\alpha_{i},\beta_{i})$ ($i\in I$) and
$(\alpha,\beta)=(0,1)$. Then there exist $A^{\prime}$, $c^{\prime}$ with

$A^{\prime}\sim c^{\prime}A,\quad 1\leq c^{\prime}\leq 5$

and some vectors of residue classes
$\vec{a}=(a_{s}\;\textrm{mod}\;s)_{s\in{\cal S}}$,
$\vec{n}=(n_{p}\;\textrm{mod}\;p)_{p\in{\cal P}}$ for which

$\\#({\cal Q}\cap S(\vec{a})\cap S(\vec{n}))\sim A^{\prime}\frac{x}{\log x}$
(6.2)

and

$\\#({\cal Q}\cap S(\vec{a})\cap S(\vec{n})\cap(\alpha_{i}y,\beta_{i}y])\leq
A^{\prime}\varepsilon(1+o(1))\frac{x}{\log x}$ (6.3)

hold. For any real numbers $0\leq\alpha<\beta\leq 1$, the interval $(\alpha
y,\beta y]$ can be covered by at most
$\lfloor\frac{2}{\varepsilon}\rfloor|\beta-\alpha|+1$ intervals
$(\alpha_{i}y,\beta_{i}y]$. Hence by (6.3) we have

$\displaystyle\\#({\cal Q}\cap S(\vec{a})\cap S(\vec{n})\cap(\alpha y,\beta
y])$ $\displaystyle\leq
A^{\prime}\varepsilon\left(\left\lfloor\frac{2}{\varepsilon}\right\rfloor|\beta-\alpha|+1\right)(1+o(1))\frac{x}{\log
x}$ (6.4) $\displaystyle\leq
A^{\prime}(2|\beta-\alpha|+\varepsilon)(1+o(1))\frac{x}{\log x}.$

We extend $\vec{a}$ to $(a_{p})_{p\leq x}$ by

$a_{p}:=\begin{cases}n_{p}\quad&(p\in{\cal P})\\\ 0\quad&(p\not\in{\cal
S}\cup{\cal P}),\end{cases}$

and set

${\cal T}:=\\{n\in[y]\backslash[x]\;|\;n\not\equiv
a_{p}\;(\mathrm{mod}\;p),\forall p\leq x,p\neq B_{0}\\}.$

Following the argument of Section 4 of [6], we see that the set ${\cal T}$
only differs from ${\cal Q}\cap S(\vec{a})\cap S(\vec{n})$ by a set ${\cal R}$
consisting of $z$-smooth numbers in $[y]$ multiplied by powers of $B_{0}$, and
that ${\cal R}=o(x/\log x)$. Consequently we obtain Proposition 3.1. $\Box$
Hence to prove the main theorem, it suffices to prove Proposition 6.1. This
proposition can be derived from the following two propositions.

######  Proposition 6.2 (An explicit version of Theorem 5 of [6]).

Let $x$ be a sufficiently large real number and put

$y=c\frac{x\log x\log_{3}x}{\log_{2}x},\quad c=\frac{\theta c_{I,J}}{12800\log
5}.$

Then, there exist a quantity $C=C(x)$ satisfying

$\frac{\theta c_{I,J}}{9600c}(1+o(1))\leq C\leq\frac{\theta
c_{I,J}}{4800c}(1+o(1))$ (6.5)

and choices of random vectors
$\vec{\mathbf{a}}=(\mathbf{a}_{s}\;\mathrm{mod}\;s)_{s\in{\cal S}}$,
$\vec{\mathbf{n}}=(\mathbf{n}_{p}\;\mathrm{mod}\;p)_{p\in{\cal P}}$, for which
the following three conditions hold.
$\cdot$ For any vector $\vec{a}$ in the essential range of $\vec{\mathbf{a}}$,

$\mathbb{P}(q\equiv\mathbf{n}_{p}\;(\mathrm{mod}\;p)\;|\;\vec{\mathbf{a}}=\vec{a})\leq
x^{-\frac{1}{2}-\frac{1}{10}}$ (6.6)

holds uniformly for all $p\in{\cal P}$.
$\cdot$ For any fixed $0\leq\alpha<\beta\leq 1$,

$\\#({\cal Q}\cap S(\vec{\mathbf{a}})\cap(\alpha y,\beta y])\sim
80c|\beta-\alpha|\frac{x}{\log x}\log_{2}x$ (6.7)

holds with probability $1-o(1)$.
$\cdot$ We say that a vector $\vec{a}$ in the essential range of
$\vec{\mathbf{a}}$ is “good” if

$\sum_{p\in{\cal
P}}\mathbb{P}(q\equiv\mathbf{n}_{p}\;(\mathrm{mod}\;p)\;|\;\vec{\mathbf{a}}=\vec{a})=C+O_{\leq}\left(\frac{1}{(\log_{2}x)^{2}}\right)$
(6.8)

holds for any $q\in{\cal Q}\cap S(\vec{\mathbf{a}})$ with at most $x/\log
x\log_{2}x$ exceptionals. Then $\vec{\mathbf{a}}$ is “good” with probability
$1-o(1)$.

######  Proposition 6.3 ([6], Theorem 4).

Let $x$ be a sufficiently large real number, and ${\cal P}^{\prime},{\cal
Q}^{\prime}$ sets of primes in $(\frac{x}{2},x]$, $(x,\log x]$ with $\\#{\cal
Q}^{\prime}>(\log_{2}x)^{3}$, respectively. For each $p\in{\cal P}^{\prime}$,
let $\mathbf{e}_{p}$ be a random subset of ${\cal Q}^{\prime}$ satisfying

$\\#\mathbf{e}_{p}\leq r=O\left(\frac{\log
x\log_{3}x}{\log_{2}^{2}x}\right)\quad(p\in{\cal P}^{\prime}).$ (6.9)

Assume the following two conditions.
$\cdot$ For any $p\in{\cal P}^{\prime}$, $q\in{\cal Q}^{\prime}$,

$\mathbb{P}(q\in\mathbf{e}_{p})\leq x^{-\frac{1}{2}-\frac{1}{10}}.$ (6.10)

$\cdot$ For any $q\in{\cal Q}^{\prime}$ with at most
$\frac{1}{(\log_{2}x)^{2}}\\#{\cal Q}^{\prime}$ exceptions, there exists some
quantity $C$ (which is independent of $q$) with

$\frac{5}{4}\log 5\leq C\ll 1$ (6.11)

such that

$\sum_{p\in{\cal
P}^{\prime}}\mathbb{P}(q\in\mathbf{e}_{p})=C+O_{\leq}\left(\frac{1}{(\log_{2}x)^{2}}\right)$
(6.12)

holds.
Then, for any positive integer $m$ with

$m\leq\frac{\log_{3}x}{\log 5},$ (6.13)

there exist random subsets $\mathbf{e}_{p}^{\prime}\subset{\cal Q}^{\prime}$
for each $p\in{\cal P}$ such that

$\\#\\{q\in{\cal Q}^{\prime}\;|\;q\not\in\mathbf{e}_{p}^{\prime},\forall
p\in{\cal P}^{\prime}\\}\sim 5^{-m}\\#{\cal Q}^{\prime}$

holds with probability $1-o(1)$. More generally, for any subset ${\cal
Q}^{\prime\prime}\subset{\cal Q}^{\prime}$ with $\\#{\cal
Q}^{\prime\prime}\geq(\\#{\cal Q}^{\prime})/\sqrt{\log_{2}x}$,

$\\#\\{q\in{\cal Q}^{\prime\prime}\;|\;q\not\in\mathbf{e}_{p}^{\prime},\forall
p\in{\cal P}^{\prime}\\}\sim 5^{-m}\\#{\cal Q}^{\prime\prime}$

holds with probability $1-o(1)$.

(Proof of Proposition 6.2 $\land$ Proposition 6.3 $\Rightarrow$ Proposition
6.1 ) Let $m$ be a positive integer with

$m\leq\frac{\log_{3}x}{\log 5},$ (6.14)

and put $A^{\prime}=5^{-m}80c\log_{2}x$. If

$m=\left\lfloor\frac{\log\left(\frac{80c\log_{2}x}{A}\right)}{\log
5}\right\rfloor,$

then $A\leq A^{\prime}\leq 5A$ holds. For this $m$ satisfy (6.14), we need

$1\ll\frac{80c}{A}\leq 1.$

Write

$c=\frac{A}{80k}\varepsilon_{0},\quad\varepsilon_{0}>0.$

Then by (6.5), it follows that

$\frac{k\theta c_{I,J}}{120A\varepsilon_{0}}(1+o(1))\leq C\leq\frac{k\theta
c_{I,J}}{60A\varepsilon_{0}}(1+o(1)).$

This $C$ must satisfy the condition (6.11) in Proposition 6.3. Hence the
$\varepsilon_{0}$ must satisfy

$1\ll\varepsilon_{0}\leq\frac{k\theta c_{I,J}}{150A\log 5}(1+o(1)).$

We take $\varepsilon_{0}=\frac{k\theta c_{I,J}}{160A\log 5}$. Then

$c=\frac{k\theta c_{I,J}}{12800\log 5}.$

Following the argument in Section 5 of [6], we see that the statement of
Proposition 6.1 is valid with this $c$. $\Box$

With the above discussion, all that remains is to prove Proposition 6.2. In
practice, however, we only need to check how the remaining coefficients are
determined following the arguments in [4] and [6].

(Proof of Proposition 6.2) Proposition 6.2 can be obtained by applying the
sieve of Maynard (Theorem 6 of [6]. For the proof, see Section 6 of [4]). In
particular, the constant $c_{0}$ in (6.1) of [6] can be chosen as
$c_{0}=\frac{1}{5}$, following the argument of Maynard [14]. The quantity $C$
in Theorem 5 of [6] is computed in the following way. Put

$\sigma=\prod_{s\in{\cal S}}\left(1-\frac{1}{s}\right),\quad y=c\frac{x\log
x\log_{3}x}{\log_{2}x}.$

By Mertens formula, we have

$\sigma y=\left(1+O\left(\frac{1}{\log_{2}^{10}x}\right)\right)80cx\log_{2}x.$

Following (8.3) of [4], we take

$u=\frac{\varphi(B)}{B}\frac{\log R}{\log x}\frac{rJ_{r}}{2I_{r}},$

where $B$ is either $1$ or prime, so

$\frac{1}{2}\leq\frac{\varphi(B)}{B}\leq 1.$

Following Theorem 6 of [4], we take $R=x^{\frac{\theta}{3}}$, where $\theta>0$
is a constant which satisfies the condition in Definition 2 of [4]. (As
mentioned above, one can take $\theta=\frac{1}{3}$.) Suppose

$J_{r}=c_{I,J}(1+o(1))\frac{\log r}{r}I_{r}$

holds as $r\to\infty$. Then

$\frac{\theta}{12}c_{I,J}(1+o(1))\log r\leq
u\leq\frac{\theta}{6}c_{I,J}(1+o(1))\log r.$

The parameter $r$ above can be chosen as

$r=\left\lfloor\log^{c_{0}}x\right\rfloor=\left\lfloor\log^{\frac{1}{5}}x\right\rfloor.$

Hence

$\frac{\theta}{60}c_{I,J}(1+o(1))\log_{2}x\leq
u\leq\frac{\theta}{30}c_{I,J}(1+o(1))\log_{2}x.$

Finally, following [6], one can take

$C=\frac{u}{\sigma}\frac{x}{2y}.$

Hence due to the above estimate of $u$, the statement of Proposition 6.1 holds
if the parameter satisfies the condition (6.5). $\Box$

##  7 Acknowledgements

This work is partially supported by the JSPS, KAKENHI Grant Number 24K06697.
The author would like to thank Professor Terence Tao for telling him an idea
to obtain a lower bound (3.21) in his blog.

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Kanto Gakuin University,
Kanazawa, Yokohama,
Kanagawa, Japan
E-mail address<EMAIL_ADDRESS>