Luis A. Urtubeylogo_ttlM. Antonia Huertas, João Marcos, María Manzano, Sophie
Pinchinat,
François Schwarzentruber54th International Conference on Tools for Teaching
Logic11243TTL2015

# To Teach Modal Logic: An Opinionated Survey 111This work was partially
supported by the project ”Intentional Hybrid Logic” (MICINN FFI2013-47126-P);
Department of Philosophy, Logic and Aesthetics; University of Salamanca
(Spain). It has received also partial support from SeCyT (UNC-Argentina).

Luis A. Urtubey

###### Abstract

I aim to promote an alternative agenda for teaching modal logic chiefly
inspired by the relationships between modal logic and philosophy. The guiding
idea for this proposal is a reappraisal of the interest of modal logic in
philosophy, which do not stem mainly from mathematical issues, but which is
motivated by central problems of philosophy and language. I will point out
some themes to start elaborating a guide for a more comprehensive approach to
teach modal logic, and consider the contributions of dual-process theories in
cognitive science, in order to explore a pedagogical framework for the
proposed point of view.

###### keywords:

Modal Logic – Philosophy – Teaching Modal Reasoning –

##  1 Introduction

In his recent book Logic and how it gets that way, Dale Jaquette [5] quotes a
few lines of Nicholas Rescher’s Autobiography, where he talks about his
conception of logic. Rescher’s words read as follows:

> I never looked a logic as an accomplished fact –a field fully formed and
> merely in need of systematization and exposition, but saw it as an
> unfinished and malleable discipline to be reworked and structured for the
> sake of its applications. I always gravitated towards those logical issues
> and areas that were of potential utility for the elucidation and treatment
> of philosophical issues, and viewed logic not just as a self-contained
> discipline, but as a body of machinery for the accomplishment of extra-
> logical work. [16]

I was struck by this quote of Rescher. Admittedly, it contains some insightful
observations. Firstly, Rescher points out that logic can be seen as an
unfinished discipline, which can be structured according to its applications.
A second interesting point comes from the observation that philosopher’s
interest in logic is triggered by those issues which have at least potential
utility for philosophical problems. Last but not least, logic is chiefly
viewed by Rescher as a tool for accomplishing extra-logical work. This last
remark is relevant with respect to teaching logic, if we include it among
those extra-logical activities.

In this short paper, I will be concerned with these remarks from the
perspective of modal logic in particular. It is clear that modal logic has
been articulated by considering its applications, as it is witnessed by the
alternative interpretations of the modal operators of necessity and
possibility. Thus, we are accustomed to talk nowadays about logical,
metaphysical, deontic, and epistemic modalities, among many other options.
Moreover, it is also easy to see that modal logic is central to the so-called
Philosophical Logic, wherein those logical issues of philosophical concern are
collected. Viewing modal logic as a tool for the accomplishment of extra-
logical work is less frequent, but it is not totally unusual, as it is
witnessed, for example, by certain applications in artificial intelligence.
Our primary concern will be, in this respect the potential use of modal logic
to improve people’s reasoning abilities, by furnishing them with formal tools
and normative standards. It has been said, however, that formal logic failed
in its attempt to accomplish this objective. Consequently, modal logic could
not reach this honour either. Contrariwise, I will argue that assuming and
elaborating certain perspective with respect to cognition and reasoning, modal
logic can still be fruitfully applied to improve people’s reasoning abilities.
Notwithstanding, it should not be misinterpreted. I will not offer here any
specific details about the concrete organization of a modal logic course
following this approach. I am only giving an overview, which can be exploited
and implemented in different ways.

##  2 Modal Logic: Philosophical Issues

I am going to take a look now at some themes which populated the relationships
between modal logic and philosophy during the twentieth century. I think that
these examples can serve to articulate a bunch of topics for teaching modal
logic. Admittedly, this is not an exhaustive enumeration, but it will suffice
to give an idea of the approach we have in mind. I will articulate these
issues in a pedagogical approach towards the end of the paper. A very useful
summary to start with is given by the comprehensive article by Linsdström and
Segerber in The Handbook of Modal Logic [9], where they point out some themes
concerning the relationship between modal logic and philosophy that are also
relevant from our point of view. They enumerates the followings:

  1. 1.

The back-and-forth between philosophy and modal logic: good examples are
Carnap and Prior among many others, who tried to use modal logic to throw
light on old philosophical questions.

  2. 2.

The interpretation problem. The problem of providing a certain modal logic
with an intuitive interpretation, which should not be conflated with the
problem of providing a formal system with a model-theoretic semantics.

  3. 3.

The fact that one may compare this situation with that in probability theory,
where definitions of concepts like ‘outcome space’ and ‘random variable’ are
orthogonal to questions about interpretations of the concept of probability.

  4. 4.

The value of formalization. Modal logic sets standards of precision, which are
a challenge to —-and sometimes a model for-— philosophy. Classical
philosophical questions can be sharpened and seen from a new perspective when
formulated in the framework of modal logic. On the other hand, representing
old questions in a formal garb has its dangers, such as simplification and
distortion.

  5. 5.

Why modal logic rather than classical (first or higher order) logic? The
idioms of modal logic seem better to correspond to human ways of thinking than
ordinary extensional logic. One can think, for example about counterfactual
reasoning and its central place in human thinking.

Let’s go a little bit more along this path.

###  2.1 The Readings of Modality

As it was said above, Carnap and Prior constitute two important philosophical
landmarks in the development of modal logic. Thence, it is worthwhile to
isolate the philosophical approach to modalities of these authors and to
introduce thereby other themes which are related with their work. Lindström
and Segerberg observe that Carnap’s project was not only to develop a
semantics (in the sense of Tarski) for intensional languages, but also to use
metalinguistic notions from formal semantics to throw light on the modal ones.
In "Modalities and quantification" from 1946, he writes:

> It seems to me … that it is not possible to construct a satisfactory system
> before the meaning of the modalities are sufficiently clarified. I further
> believe that this clarification can best be achieved by correlating each of
> the modal concepts with a corresponding semantical concept (for example,
> necessity with L-truth).

Concerning Prior, years ago, in a remarkably thoughtful article [10],
Christopher Menzel analyzed Prior’s approach to modal logic and his deep
philosophical concerns. Menzel points out that, for Arthur Prior, "the
construction of a logic was a supremely philosophical task". Being a clever
logician, Prior joined the virtue of appreciating "a finely crafted formal
system for its own sake", apart of its "meaning" or philosophical
significance, as well as the power of a genuine logic "to lays bare the nature
of those concepts that it purports to be a logic of". This last task can be
accomplished only by way of deep reflection and insightful philosophical
analysis. Menzel remarks that Prior’s search for the “true” modal logic is
where this sort of reflection and analysis is more evident222Particularly
Menzel considers system Q from Prior’s ”Tense Logic for Non-Permanent
Existents”[11]. Menzel’s paper in its turn is a nice guide "to trace the
course of Prior’s own struggles with the concepts and phenomena of modality,
and the reasoning that led him to his own rather peculiar modal logic Q". It
is also an advisable starting point to appreciate Prior’s intuitions and the
arguments that rest upon them.

Let us also consider Prior’s work under a new light. In a recent article about
Arthur Prior’s work on hybrid logic [2], Patrick Blackburn has considered
extensively those aspects of Prior’s philosophy which motivated his interest
on hybrid logic. This kind of logic departs from traditional logical
categories, allowing that formulas can be used as terms. Blackburn’s article
is also an excellent starting point to stimulate the reflection on
philosophical questions, which have motivated the development of modal
languages. Blackburn points out that Prior’s development of hybrid logic was
fuelled by “his most fundamental convictions concerning time and tense, namely
that the modal perspective, and in particular the internal view offered by
modal logic, was the way to capture genuinely temporal logic”. In the early
years of the last Century, the Hegelian philosopher John McTaggard posed a
tricky argument concerning the unreality of time. McTaggard’s argument is
based on the postulation of two alternative series, which respectively
constitute two ways of conceptualizing time: the A series and B series
conceptions. In his work Past, Present and Future, Prior discusses McTaggard
argument. The A series considers the flow of time from past, through present,
to future. It is an internal and situated conception of time, which reflects
the experience of temporality of human beings.

Under the B series, time is a collection of instants ordered by a relation of
temporal precedence, i.e. positions are ordered from earlier-than to later-
than relations. Particularly, Prior favours A series talk over B series talk,
because he thought that the external perspective underlying B series did not
reflect the way in which human beings experience time. This motivate, in its
turn the development of hybrid logic, since Prior believed that hybrid logic
was the A series logic underpinning everyday tensed logic. There are too many
connections between Prior’s temporal logic and philosophical problems raised
by Blackburn’s article. I cannot survey all of them here. Nonetheless, the
following quotations from [2] will be useful to illustrate an important point
concerning the changes experimented by the relationship between logic and
philosophy through the years:

> It is important to realize that Prior’s willingness, so clearly in evidence
> here, to base the core of a philosophical position at least on the differing
> properties of formal logical systems, contrasts sharply with the attitude of
> many, perhaps most, contemporary modal logicians. Nowadays logicians are
> rather more guarded about the direct relevance of the properties of
> particular formal systems to philosophy. (…) Much of this contemporary
> attitude stems from the model-theoretic perspective which underlies current
> practice.

Nonetheless, Patrick Blackburn also notices that

> (…) this does not mean that the model-theoretic conception rules out the
> possibility of drawing philosophical conclusions based on logical analysis.

On the contrary:

> (…) The most direct way that model-theoretically oriented logic helps
> philosophy is by demonstrating that there are more ontological possibilities
> than is apparent at first sight, or that certain possibilities can’t be made
> to fit together coherently. The model-theoretic perspective helps induce
> philosophical modesty.

This also holds for Prior’s work on tense and temporal logic.

Admittedly, unlike the case of mathematical modal logic, for teaching modal
logic in philosophy or broadly speaking in the humanities, it seems to be
crucial to count with a substantial reading of the box or the diamond.
Following ideas of an unpublished paper by John P. Burgess [3], the point to
be observed is that we have to admit throughout the whole history two
alternative readings of the box as "it is analytic that…", which seems the
official explanation of I. Lewis, and reading the box as "it could not have
failed to be the case that…" (a reading which doubtless influenced the
"intuitions" of some). It must also be admitted that these were two different
and incompatible readings. We’ll go back on this below.

###  2.2 Quine’s Interpretational Challenge

For the second item above mentioned, it is inevitable to think of Quine’s
criticism of quantified modal logic333The most known references at this
respect are [13], [12], [14], [15]. According to the article of Lindström and
Segerberg, this criticism comes in different strands. First, there is the
simple observation that classical quantification theory with identity cannot
be applied to a language in which substitutivity of identicals for singular
terms fails. The so-called Morning Star Paradox shows that we must abandon
either universal specification (and existential generalisation) or
indiscernibility of identicals. This premise goes to the seemingly
uncontroversial conclusion that classical quantification theory (with identity
and individual constants) cannot be mixed up with non-extensional operators
(those for which substitutivity of identicals for singular terms fail) without
being modified in some way. This weak claim already gives rise to the
challenge of extending quantification theory in a consistent way to languages
with non-extensional operators.444A very comprehensive treatment is given by
James Garson in [4] Perspicuously Lindström and Segerberg differentiate a much
stronger claim in addition to this former claim, which sometimes can be found
in Quine’s early works, relative to the fact that objectual quantification
into non-extensional (“opaque”) constructions simply does not make sense. The
cause is that occurrences of variables inside of opaque constructions do not
serve simply to refer to their objects, and cannot therefore be bound by
quantifiers outside of the opaque construction. Consequently, quantifying into
opaque contexts would be like trying to quantify into quotations. As Lindström
and Segerberg observe: "This claim is hardly credible in the face of the
multitude of quantified intensional logics that have been developed since
then, and it can be taken to be refuted by the work of Kaplan and Fine among
others. I think that Lindström and Segerberg’s distinction about Quine’s very
different thesis is a fact that must be appreciated and incorporated when
teaching modal logic in order to exploit this frequent confusions.

###  2.3 The Nature of Modality

It will be worthwhile to return here to some reflections about modal logic
recently introduced by John P. Burgess that we just referred to [3]. According
to Burgess’ approach, it is convenient to look backward to the origins of
modal logic. Originally logic used to "name an enterprise aiming to prescribe
the norms of deductive argumentation in philosophy and elsewhere". However,
"the pursuit of what were the original aims of logic is now for many
inseparable from philosophy generally, and philosophy of language
especially.[3] Thus, logic progressively has gone far away from its original
motivations. In fact, logic is no more restricted to the study of correct or
valid reasoning. Formal systems are studied in its own right. Burgess’
observation also enables to see that the original logic’s aim is now shared by
many disciplines other than logic. In consequence, in order to fulfil this
original aim one must cross some disciplinary borders, since it is no longer
in the domains of logic alone. Burgess also considers the case of modal logic
in particular and thinks that we must go back to the work of Aristotle to
appreciate the initial sense of modality that was getting lost as time went
by. Moreover, a neglected question at this point is what is meant by "modal"
here. And if there is a sense which is particular relevant to philosophy. What
is characteristic of modality is the expression of irreality of one kind or
another, as it is expressed by means of grammar. It is widely accepted the
distinction between deontic, epistemic and dynamic modalities. In the
beginnings of modal logic, H. von Wright also made an almost similar
distinction. In fact, all of them were of interest in philosophy, but
specially the third, which more or less corresponds to what is called
"metaphysical" modality. This is the conclusion of Burgess, which is worth to
consider in the setting of teaching modal logic in philosophy. The notions
associated with this type of modality are the necessary, in the sense, of what
is and would have been (could not failed to be) no matter what, and the
possible in the sense of what is or isn’t but potentially might (or could)
have been. The idea of a true modal logic endorsed by Burgess and other
philosophers must be interpreted as an effort to put modal logic again in the
right way. In the words of Burgess [3]:

> Very little of modal logic is devoted to the attempt to determine what are
> the valid forms of argument involving necessity and possibility in any
> sense, and vanishingly little is devoted to the original aim of modal logic,
> that of dealing with the logic of modality in our default sense.

The starting point in this regard is a distinction, which admits two
contrasting opinions about modality that have coexisted through the whole
history of modal logic. Together with this initial idea of necessity, there
would be an idea of necessity as analyticity, which is also unconsciously
influenced by thoughts about necessity in the "metaphysical" sense. According
to Burgess, the general pre-Kripkean failure to recognize both notions
explicitly and to distinguish between them naturally made it impossible to
appreciate that different logics may be appropriate to the different notions.
As it happens with the precedent observations in this paper, I believe that
this thesis about the confusion between two alternatives ideas of necessity
can be usefully incorporated in a strategy for teaching modal logic. Among
other confusions that one can take into account regarding the application of
modal logic, Burgess also considers "the modal distinction between the
contingent and the necessary". Let us consider the following examples also
from [3].

  1. 1.

There has been no female U. S. president.

  2. 2.

The number 29 has no nontrivial divisors.

Admittedly, though there has been no female U. S. president, there could have
been, whereas the number 29 not only has no nontrivial divisors but couldn’t
have had any. Thence, it is allowed to make counterfactual speculations
regarding the first proposition, but not with respect to the second. And the
most important question is how we know about the difference between these
propositions. This is a central problem for modal epistemology concerning the
bridge between ’is’ and ’could’. It is also an interesting problem to reflect
upon when you teach modal logic. A nice place to start is the classic
statement of the problem given by Immanuel Kant in the introduction to the
second or B edition of his Critique of Pure Reason that Burgess points out:

> Experience teaches us that a thing is so, but not that cannot be otherwise.

That is to say that experience can teach us that a necessary truth is true,
but it cannot teach us that it is necessary. Translated into the lexicon of
possible world semantics it means that we can access through our senses to the
actual world -the one we live in- whereas when we speak about what could or
couldn’t have been, we are making an assertion about how thinks are in other
worlds.

###  2.4 A Modal Puzzle

Let us close this section by considering a well-known tricky puzzle about
modality which can swell the list of teaching devices. The question concerning
the right modal logic for "metaphysical modality" has already prompted the
interest of philosophers and logicians. The most favoured systems have been S5
and S4, though there have been alternatives like McKinsey S4.1. Nathan Salmon
and others have developed an argument for the system T, or rather against S4.
As you know the characteristic law distinguishing S4 and T claims that if a
proposition P is necessary, it is necessarily necessary (or if P is possibly
possible), then it is possible. Salmon’s argument has became to be known as
the "modal paradox". Let us now describe the example which centres in the S4
axiom following the exposition of Burgess [3].

> The same principle [S4] would seem to apply to, say, a ship made of a
> thousand planks. There is an intuition X to the effect that the same ship
> could have been made of 999 of the same planks, arranged the same way, plus
> a replacement for plank 473; but at the same time there is an intuition Y to
> the effect that a ship made of a thousand different planks would have been a
> different ship. The puzzle is that these two intuitions, X and Y, seem to
> conflict.

In [3], the conflict referred above is described in Leibnizian-Kripkean
possible worlds language as follows.

> Here in the actual world A we have a ship, let us name it the good ship
> Theseus, made of 1000 planks. Our first intuition X is that the same ship
> could have been made of 999 of these planks plus a replacement for plank
> 473. In possible-worlds terms that means there is another world B where the
> same good ship Theseus exists with all but one plank the same as in our
> world A, and only plank 473 different. (…) [A]nd by the same sort of
> intuition, there must be another world C where the same good ship Theseus
> exists with all (…) but with plank 692 different. That means for us back in
> world A there is another world C where the good ship Theseus exists with all
> but two planks the same as in our world A, but with planks 473 and 692
> different (…). The same sort of considerations can then be used to argue
> that one could have three planks different, or four, or five, or all 1000.
> But that is contrary to our other intuition Y.

Salmon’s solution rejects S4 axiom to avoid treating identity as vague. It is
tantamount to recognize that though world B is a possibility for us in world
A, and world C is a possibility for those in world B, still C is not a
possibility for us in world A, but only a "possible possibility". That means
that transitiveness must be banned in this context. There are reasons,
however, as Burgess claims, to doubt of this analysis. Since the puzzle seems
to be parallel to another famous temporal one already discussed by Plutarch
and Thomas Hobbes, which involves the ship of Theseus. And in this last case,
transitivity cannot be excluded. It is admissible to conjecture that this
problem concerning the necessity of origins will have great significance and
may ultimately be relevant to discuss the status of the law that possibly
possible implies possible, when we have to teach about normal systems of modal
logic.

##  3 Dual-Process Theories

Dual-process theory in cognitive psychology is closely related to the study of
heuristics and biases in reasoning, as it was taken up for example in the
research of Kahneman and Tversky [8]. From this point of view, heuristics have
intrinsic limits. And it is claimed that heuristics inserted in the human mind
by the process of evolution can produce errors, although it is also given to
human beings to fix these failures. By means of different tasks, Stanovich
together with other researchers have shown the conflict between heuristics and
the so-called "analytic system". Precisely, the goal of these tasks is to
bring about "a heuristically triggered response against a normative response
generated by the analytic system" [18]. The idea of an "analytic system" is
crucial: it is one of the most important contributions of Kahneman and Tversky
that cognitive biases are not made by chance, since the human mind utilizes
"innate rules of thumb", which are autonomous and automatics, and which cannot
be totally banned by us when processing information. Moreover, Kahneman and
Frederick also have remarked that "[t]he persistence of such systematic errors
in the intuitions of experts implied that their intuitive judgements may be
governed by fundamentally different processes than the slower, more deliberate
computations they had been trained to execute" [7]. It is then possible to
design certain tasks or contexts where these natural heuristics bring out
responses which are not adequate, before being corrected by means of a second
system that is slower, more analytic, and more thoughtful. Thence, it is one
of the central intuitions of dual-process theories that the human mind is
composed by two different systems, the so-called system1 (S1) and system2
(S2). These models are heavily discussed in the recent literature of cognitive
science though. The collection of processes of S1 (the modules) are autonomous
and cognitively closed; we are not conscious of their working and we cannot
influence on them directly. These processes are experimented in our intuitions
and our spontaneous reactions. The S2 processes, on the contrary, are
voluntary and they are able to inhibit and to fix those automatic responses of
S1. Most of the time, these two systems get on well with each other, but there
are situations where the response elicited by S1 conflicts with that issued by
S2. Nevertheless, we can learn the rules and we can learn to apply them
rightly by using S2 processes. According to the advocates of dual-process
theories, for accomplishing a task it is necessary to identify first the bias
of S1 and then to fix it with the appropriate tools of S2. Moreover, we have a
tendency to utilize only S1, which in most cases gives us right answers, but
the processes of this system mislead us in other cases. Fortunately, we may
learn how to utilize S2 to detect these situations, where we are cheated, in
order to look for the normatively adequate answer. Stanovich [17] has stressed
the importance of acquiring the cognitive tools which make it possible to
attain the resolution of those problems which S1 has not answered adequately.
These are the kind of tools which are usually provided by logic courses.
However, as it has been emphasized, this is not enough to overcome this
obstacle: one has to have "the relevant logical rules and also [needs] to
recognize the applicability of these rules in particular situations" [6].
Thence it must be also detected that the automatic answer provided by S1 has
to be neutralized. Consequently biases’ correction is more complex and at
least involves two stages. This is the most important thing to be learned from
these studies concerning their pedagogical application. Particularly, the
complexity of this dual process serves also to explain the shouting failure of
standard logic courses with respect to improve the reasoning abilities of many
students. From this point of view, Guillaume Beaulac and Serge Robert [1] have
pointed out that we can rescue the normative role of logic, by considering it
in the light of a pedagogical reading of dual-process theories. The idea can
be applied also to modal logic. To that end efforts must be made to elaborate
cases which prompted the action of S1-systems that will have as normative
counterparts the current systems of modal logic. In this case, S1-systems will
serve to show up several drawbacks of modal reasoning and other puzzling
issues which stem from reasoning tasks performed in this context. I think that
the philosophical problems that have been shortly addressed in this paper, can
–at least partially– fulfil the requirements to confront the intuitions given
by S1-systems in this case and contribute to improve teaching modal logic
under these premises.

##  4 Conclusion

In this paper, I aimed at integrating from a pedagogical perspective, the
formal treatment of modalities with those philosophical problems which
appeared in the origin of modal logic and have also contributed to its
development. I have considered modal logic as a tool for treating and
elaborating many problems in philosophy and other connected areas, relegating
the mathematical issues. Concerning teaching, the dose of mathematical modal
logic to be introduced must be estimated from this point of view and it will
no more be the only motive of concern. On the contrary, it turns out to be
instrumental. Specifically I have not considered the mathematics of modal
logic, which I presupposed in a certain way. The central tenet may be
summarized by saying that things must be turned upside down. Instead of
focusing on the formal and mathematical aspects of modal logic, it is
convenient to return to the origins and search firstly for the philosophical
motivation behind the primeval ideas.

More controversially, the strategy for teaching modal logic that I am
proposing here would not be that new, but I confess that I have never seen a
full-fledged development of this approach. Certainly, one can find some
exemplifications or applications of modal logic to philosophical thesis
scattered in different logic texts. I have surveyed in this paper some of the
most relevant and more accessible ones in order to contribute to the
construction of an alternative perspective for teaching modal logic in
philosophy. It may sound perhaps too much oriented to philosophy ignoring
other uses of modal logic. That might be right, but I think that for teaching
modal logic in philosophy it must be taken into account the central
motivations of those people who have decided to study philosophy. And the
challenge for all of us involved both in modal logic and philosophy is to
articulate courses, which pay attention to these particular motivations.

In this regard, our position lies close to some recent trends in the
development of Mathematical Philosophy, which offers a new approach to
philosophical problems chiefly supported by the contribution of formal
methods. The most innovative feature of this proposal is the balanced
combination of formal methods with deep philosophical questions, which trigger
a productive reflection. This relationship suggests also a link between this
approach and the application of dual-process theories to the extra-logical
activity of teaching and learning the application of formal tools.

#### Acknowledgements

The author gratefully thank two anonymous referees of this publication for
their useful comments.

## References

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  * [2] Patrick Blackburn.  Arthur Prior and hybrid logic.  Synthese, 150(3), 2006.
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An equivalent criteria for irrationality of $\zeta(5)$

Shekhar Suman
M.Sc., I.I.T. Delhi, India

Email<EMAIL_ADDRESS>
Abstract: Defining a Beukers [1] like integral for $\zeta(5)$ as

$I_{n}:=\int_{(0,1)^{5}}\frac{(1-x_{3})^{n}(1-x_{4})^{n}P_{n}(x_{1})P_{n}(x_{2})}{1-(1-x_{1}x_{2}x_{3}x_{4})x_{5}}\
dx_{1}dx_{2}dx_{3}dx_{4}dx_{5}$

we prove that for each $n\in\mathbb{N}$

$I_{n}=\frac{p_{n}\zeta(5)+q_{n}\zeta(4)+r_{n}\zeta(3)+s_{n}}{d_{n}^{5}}$

where $p_{n},q_{n},r_{n},s_{n}$ are integers and
$d_{n}=\text{lcm}(1,2,...,n)$. We prove that the following are equivalent:
1\. $q_{n}\zeta(4)+r_{n}\zeta(3)-d_{n}^{5}I_{n}\notin\mathbb{Z}$ for each
natural number $n$.
2\. $q_{n}\zeta(4)+r_{n}\zeta(3)-d_{n}^{5}I_{n}\notin\mathbb{Z}$ for
infinitely many natural number $n$.
3\. $\zeta(5)$ is irrational.

Keywords and Phrases: Riemann zeta function, irrationality, Legendre
polynomials, Uniform convergence, Tonelli’s theorem.
2020 Mathematics Subject Classification: 11M06, 11J72.
1\. Introduction and Definitions The Riemann zeta function, $\zeta(s)$ is
defined as the analytic continuation of the Dirichlet series

$\zeta(s)=\sum_{n=1}^{{\infty}}\frac{1}{n^{s}}$

which converges in the half plane $\Re(s)>1$. The Riemann zeta function is a
meromorphic function on the whole complex plane, which is holomorphic
everywhere except for a simple pole at s = 1 with residue 1. Apery [2] proved
that, $\zeta(3)$ is irrational. Post Apery, Beuker’s [1] gave a proof of
irrationality of $\zeta(3)$ using multiple integrals. It was proved in [3,4]
that infinitely many numbers $\zeta(2n+1)$, $n\geq 1$ are irrational and
linearly independent over $\mathbb{Q}$. Zudilin [5] proved that one of the
numbers $\zeta(5)$, $\zeta(7)$, $\zeta(9)$, $\zeta(11)$ is irrational. Also
the same author proved in [6] that for any integer $n\geq 0$, there exists at
least one irrational number amongst the odd zeta values $\zeta(2n+3)$,
$\zeta(2n+5)$,…,$\zeta(16n+7)$.

2\. Main Theorems The goal of this article is to prove the following result:
For $n\in\mathbb{N}$, we define

$I_{n}:=\int_{(0,1)^{5}}\frac{(1-x_{3})^{n}(1-x_{4})^{n}P_{n}(x_{1})P_{n}(x_{2})}{1-(1-x_{1}x_{2}x_{3}x_{4})x_{5}}\
dx_{1}dx_{2}dx_{3}dx_{4}dx_{5}$

where $P_{n}(x)$ is the legendre polynomial as used in Beukers [1] and defined
as, $P_{n}(x):=\frac{d^{n}}{dx^{n}}\left(\frac{x^{n}(1-x)^{n}}{n!}\right)$.

Theorem : The following are equivalent:
1\. $q_{n}\zeta(4)+r_{n}\zeta(3)-d_{n}^{5}I_{n}\notin\mathbb{Z}$ for each
natural number $n$.
2\. $q_{n}\zeta(4)+r_{n}\zeta(3)-d_{n}^{5}I_{n}\notin\mathbb{Z}$ for
infinitely many natural number $n$.
3\. $\zeta(5)$ is irrational.
where $d_{n}=\text{lcm}(1,2,...,n)$.

Proof: Most of the integrals that appear in the proof are improper. The
manipulations with these integrals can be justified if we replace
$\int_{0}^{1}$ by $\int_{\varepsilon}^{1-\varepsilon}$ and let $\varepsilon$
tend to $0$. We denote

$\int_{0}^{1}\int_{0}^{1}...\int_{0}^{1}=\int_{(0,1)^{n}}$

So we have (see Appendix)

$\zeta(5)=-\frac{1}{4}\int_{(0,1)^{4}}\frac{\log(x_{1}x_{2}x_{3}x_{4})}{1-x_{1}x_{2}x_{3}x_{4}}\
dx_{1}dx_{2}dx_{3}dx_{4}$ (1)

Also

$-\frac{\log(x_{1}x_{2}x_{3}x_{4})}{1-x_{1}x_{2}x_{3}x_{4}}=\int_{0}^{1}\frac{1}{1-(1-x_{1}x_{2}x_{3}x_{4})x_{5}}\
dx_{5}$ (2)

So by (1) and (2)

$\zeta(5)=\frac{1}{4}\int_{(0,1)^{5}}\frac{1}{1-(1-x_{1}x_{2}x_{3}x_{4})x_{5}}\
dx_{1}dx_{2}dx_{3}dx_{4}dx_{5}$ (3)

We have

$I_{n}=\int_{(0,1)^{5}}\frac{(1-x_{3})^{n}(1-x_{4})^{n}P_{n}(x_{1})P_{n}(x_{2})}{1-(1-x_{1}x_{2}x_{3}x_{4})x_{5}}\
dx_{1}dx_{2}dx_{3}dx_{4}dx_{5}$ (4)

Integrating by parts $n$ times with respect to $x_{1}$, gives

$I_{n}=\int_{(0,1)^{5}}\frac{(1-x_{3})^{n}(1-x_{4})^{n}x_{1}^{n}(1-x_{1})^{n}x_{2}^{n}x_{3}^{n}x_{4}^{n}x_{5}^{n}P_{n}(x_{2})}{(1-(1-x_{1}x_{2}x_{3}x_{4})x_{5})^{n+1}}\
dx_{1}dx_{2}dx_{3}dx_{4}dx_{5}$ (5)

We use the transformation $x_{1}=x,x_{2}=y,x_{3}=u,x_{4}=v$ and

$x_{5}=\frac{1-w}{1-(1-xyuv)w}$ (6)

and we obtain after simplification

$I_{n}=\int_{(0,1)^{5}}\frac{(1-x)^{n}(1-u)^{n}(1-v)^{n}(1-w)^{n}P_{n}(y)}{1-(1-xyuv)w}\
dxdydudvdw$ (7)

Again integrating by parts $n$ times with respect to $y$, gives

$I_{n}=\int_{(0,1)^{5}}\frac{x^{n}(1-x)^{n}y^{n}(1-y)^{n}u^{n}(1-u)^{n}v^{n}(1-v)^{n}w^{n}(1-w)^{n}}{(1-(1-xyuv)w)^{n}}\
\frac{dxdydudvdw}{1-(1-xyuv)w}$ (8)

So we get by using AM-GM inequality $1-(1-xyuv)w\geq 2\sqrt{1-w}\sqrt{xyuvw}$,
$\sqrt{x}(1-x)\leq\frac{2}{3\sqrt{3}}$ and $w(1-w)\leq\frac{1}{4}$, the above
equation gives

$0<I_{n}\leq\left(\frac{4}{729}\right)^{n}\int_{(0,1)^{5}}\frac{dxdydudvdw}{1-(1-xyuv)w}$
(9)

By (3) and (9) we get

$0<I_{n}\leq 4\left(\frac{4}{729}\right)^{n}\zeta(5)$ (10)

For $n$ large, $d_{n}^{5}<e^{5n}$ (see [1]) and so
$\left(\frac{4e^{5}}{729}\right)^{n}$ tends to $0$ as $n$ tends to $\infty$
and we obtain by (10)

$\lim_{n\to\infty}d_{n}^{5}I_{n}=0$ (*)

Now we have

$(1-x_{3})^{n}=\sum_{s_{3}=0}^{n}(-1)^{s_{3}}\binom{n}{s_{3}}x_{3}^{s_{3}}$
(11)
$(1-x_{4})^{n}=\sum_{s_{4}=0}^{n}(-1)^{s_{4}}\binom{n}{s_{4}}x_{4}^{s_{4}}$
(12)
$P_{n}(x_{1})=\sum_{s_{1}=0}^{n}(-1)^{s_{1}}\binom{n}{s_{1}}\binom{n+s_{1}}{n}x_{1}^{s_{1}}$
(13)
$P_{n}(x_{2})=\sum_{s_{2}=0}^{n}(-1)^{s_{2}}\binom{n}{s_{2}}\binom{n+s_{2}}{n}x_{2}^{s_{2}}$
(14)

Also using (2), (4) can be rewritten as

$I_{n}=-\int_{(0,1)^{4}}\frac{\log(x_{1}x_{2}x_{3}x_{4})(1-x_{3})^{n}(1-x_{4})^{n}P_{n}(x_{1})P_{n}(x_{2})}{1-x_{1}x_{2}x_{3}x_{4}}\
dx_{1}dx_{2}dx_{3}dx_{4}$ (15)

Using (11), (12), (13) and (14) in (15) we get

$I_{n}=-\sum_{[0,n]^{4}}a_{n,s_{1},s_{2},s_{3},s_{4}}\int_{(0,1)^{4}}\frac{\log(x_{1}x_{2}x_{3}x_{4})x_{1}^{s_{1}}x_{2}^{s_{2}}x_{3}^{s_{3}}x_{4}^{s_{4}}}{1-x_{1}x_{2}x_{3}x_{4}}\
dx_{1}dx_{2}dx_{3}dx_{4}$ (16)

where

$a_{n,s_{1},s_{2},s_{3},s_{4}}=(-1)^{s_{1}+s_{2}+s_{3}+s_{4}}\binom{n}{s_{1}}\binom{n}{s_{2}}\binom{n}{s_{3}}\binom{n}{s_{4}}\binom{n+s_{1}}{n}\binom{n+s_{2}}{n}$
(17)

Define in (16)

$J(s_{1},s_{2},s_{3},s_{4}):=-\int_{(0,1)^{4}}\frac{\log(x_{1}x_{2}x_{3}x_{4})x_{1}^{s_{1}}x_{2}^{s_{2}}x_{3}^{s_{3}}x_{4}^{s_{4}}}{1-x_{1}x_{2}x_{3}x_{4}}\
dx_{1}dx_{2}dx_{3}dx_{4}$ (18)

So by (16) and (18)

$I_{n}=\sum_{[0,n]^{4}}a_{n,s_{1},s_{2},s_{3},s_{4}}J(s_{1},s_{2},s_{3},s_{4})$
(19)
$J(s_{1},s_{2},s_{3},s_{4})=-\frac{d}{d\varepsilon}\int_{(0,1)^{4}}\frac{(x_{1}x_{2}x_{3}x_{4})^{\varepsilon}x_{1}^{s_{1}}x_{2}^{s_{2}}x_{3}^{s_{3}}x_{4}^{s_{4}}}{1-x_{1}x_{2}x_{3}x_{4}}\
dx_{1}dx_{2}dx_{3}dx_{4}\Biggr{|}_{\varepsilon=0}$ (20)

Write

$\frac{1}{1-x_{1}x_{2}x_{3}x_{4}}=\sum_{m=0}^{\infty}(x_{1}x_{2}x_{3}x_{4})^{m}$
(21)

So we get by (20) and (21)

$J(s_{1},s_{2},s_{3},s_{4})=-\frac{d}{d\varepsilon}\int_{(0,1)^{4}}\sum_{m=0}^{\infty}x_{1}^{m+s_{1}+\varepsilon}x_{2}^{m+s_{2}+\varepsilon}x_{3}^{m+s_{3}+\varepsilon}x_{4}^{m+s_{4}+\varepsilon}\
dx_{1}dx_{2}dx_{3}dx_{4}\Biggr{|}_{\varepsilon=0}$ (22)

Since
$f_{m}(x_{1},x_{2},x_{3},x_{4})=x_{1}^{m+s_{1}+\varepsilon}x_{2}^{m+s_{2}+\varepsilon}x_{3}^{m+s_{3}+\varepsilon}x_{4}^{m+s_{4}+\varepsilon}$
is non negative so using Tonelli’s theorem we can reverse the order of
integration and summation to get

$J(s_{1},s_{2},s_{3},s_{4})=-\frac{d}{d\varepsilon}\sum_{m=0}^{\infty}\int_{(0,1)^{4}}x_{1}^{m+s_{1}+\varepsilon}x_{2}^{m+s_{2}+\varepsilon}x_{3}^{m+s_{3}+\varepsilon}x_{4}^{m+s_{4}+\varepsilon}\
dx_{1}dx_{2}dx_{3}dx_{4}\Biggr{|}_{\varepsilon=0}$

We obtain

$\displaystyle J(s_{1},s_{2},s_{3},s_{4})$ $\displaystyle=$
$\displaystyle-\frac{d}{d\varepsilon}\sum_{m=0}^{\infty}\frac{1}{(m+\varepsilon+s_{1}+1)(m+\varepsilon+s_{2}+1)(m+\varepsilon+s_{3}+1)(m+\varepsilon+s_{4}+1)}\Biggr{|}_{\varepsilon=0}$

So we have

$J(s_{1},s_{2},s_{3},s_{4})=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\frac{1}{(p+\varepsilon+s_{1})(p+\varepsilon+s_{2})(p+\varepsilon+s_{3})(p+\varepsilon+s_{4})}\Biggr{|}_{\varepsilon=0}$
(23)

By (19), we split the sum over the following:
1\. All exponents are equal: $s_{1}=s_{2}=s_{3}=s_{4}$
2\. All exponents are pairwise distinct: $s_{i}\neq s_{j}$ for all $i\neq j$,
$1\leq i,j\leq 4$
3\. There are three distinct exponents:
3a. $s_{1}=s_{2}$, $s_{1}\neq s_{3}$, $s_{1}\neq s_{4}$, $s_{3}\neq s_{4}$
3b. $s_{1}\neq s_{2}$, $s_{1}=s_{3}$, $s_{1}\neq s_{4}$, $s_{2}\neq s_{4}$
3c. $s_{1}\neq s_{2}$, $s_{1}\neq s_{3}$, $s_{1}=s_{4}$, $s_{2}\neq s_{3}$
3d. $s_{1}\neq s_{2}$, $s_{1}\neq s_{4}$, $s_{2}=s_{3}$, $s_{2}\neq s_{4}$
3e. $s_{1}\neq s_{2}$, $s_{1}\neq s_{3}$, $s_{2}\neq s_{3}$, $s_{2}=s_{4}$
3f. $s_{1}\neq s_{2}$, $s_{1}\neq s_{3}$, $s_{2}\neq s_{3}$, $s_{3}=s_{4}$
4\. There are two distinct exponents appearing twice each:
4a. $s_{1}=s_{2}$, $s_{3}=s_{4}$, $s_{3}\neq s_{1}$
4b. $s_{1}=s_{3}$, $s_{2}=s_{4}$, $s_{4}\neq s_{3}$
4c. $s_{1}=s_{4}$, $s_{2}=s_{3}$, $s_{4}\neq s_{3}$
5\. There are two distinct exponents with one appearing three times:
5a. $s_{1}=s_{2}=s_{3}$, $s_{4}\neq s_{1}$
5b. $s_{2}=s_{3}=s_{4}$, $s_{2}\neq s_{1}$
5c. $s_{3}=s_{4}=s_{1}$, $s_{2}\neq s_{1}$
5d. $s_{4}=s_{1}=s_{2}$, $s_{3}\neq s_{1}$
So by the above cases (19) can be written as

$\displaystyle I_{n}$
$\displaystyle=\sum_{s_{1}=0}^{n}a_{n,s_{1},s_{1},s_{1},s_{1}}J(s_{1},s_{1},s_{1},s_{1})+\sum_{s_{i}\neq
s_{j},i\neq j}a_{n,s_{1},s_{2},s_{3},s_{4}}J(s_{1},s_{2},s_{3},s_{4})+$
$\displaystyle\sum_{s_{1}\neq s_{3},s_{1}\neq s_{4},s_{3}\neq
s_{4}}a_{n,s_{1},s_{1},s_{3},s_{4}}J(s_{1},s_{1},s_{3},s_{4})+\sum_{s_{1}\neq
s_{2},s_{1}\neq s_{4},s_{2}\neq
s_{4}}a_{n,s_{1},s_{2},s_{1},s_{4}}J(s_{1},s_{2},s_{1},s_{4})+$
$\displaystyle\sum_{s_{1}\neq s_{2},s_{1}\neq s_{3},s_{2}\neq
s_{3}}a_{n,s_{1},s_{2},s_{3},s_{1}}J(s_{1},s_{2},s_{3},s_{1})+\sum_{s_{1}\neq
s_{2},s_{1}\neq s_{4},s_{2}\neq
s_{4}}a_{n,s_{1},s_{2},s_{2},s_{4}}J(s_{1},s_{2},s_{2},s_{4})+$
$\displaystyle\sum_{s_{1}\neq s_{2},s_{1}\neq s_{3},s_{2}\neq
s_{3}}a_{n,s_{1},s_{2},s_{3},s_{2}}J(s_{1},s_{2},s_{3},s_{2})+\sum_{s_{1}\neq
s_{2},s_{1}\neq s_{3},s_{2}\neq
s_{3}}a_{n,s_{1},s_{2},s_{3},s_{3}}J(s_{1},s_{2},s_{3},s_{3})+$
$\displaystyle\sum_{s_{1}\neq
s_{3}}a_{n,s_{1},s_{1},s_{3},s_{3}}J(s_{1},s_{1},s_{3},s_{3})+\sum_{s_{1}\neq
s_{2}}a_{n,s_{1},s_{2},s_{1},s_{2}}J(s_{1},s_{2},s_{1},s_{2})+$
$\displaystyle\sum_{s_{1}\neq
s_{2}}a_{n,s_{1},s_{2},s_{2},s_{1}}J(s_{1},s_{2},s_{2},s_{1})+\sum_{s_{1}\neq
s_{4}}a_{n,s_{1},s_{1},s_{1},s_{4}}J(s_{1},s_{1},s_{1},s_{4})+$
$\displaystyle\sum_{s_{1}\neq
s_{2}}a_{n,s_{1},s_{2},s_{2},s_{2}}J(s_{1},s_{2},s_{2},s_{2})+\sum_{s_{1}\neq
s_{3}}a_{n,s_{1},s_{1},s_{3},s_{1}}J(s_{1},s_{1},s_{3},s_{1})+$
$\displaystyle\sum_{s_{1}\neq
s_{2}}a_{n,s_{1},s_{2},s_{1},s_{1}}J(s_{1},s_{2},s_{1},s_{1})$ (24)

Observe when all exponents are equal: $s_{1}=s_{2}=s_{3}=s_{4}$

$J(s_{1},s_{1},s_{1},s_{1})=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\frac{1}{(p+\epsilon+s_{1})^{4}}\Biggr{|}_{\varepsilon=0}$
(25)

Now since the series on the right of above equation is uniformly convergent,
so we can differentiate term by term and put $\varepsilon=0$ to obtain

$J(s_{1},s_{1},s_{1},s_{1})=4\sum_{p=1}^{\infty}\frac{1}{(p+s_{1})^{5}}$ (26)

This can be re-written as

$J(s_{1},s_{1},s_{1},s_{1})=4\ \zeta(5)-4\sum_{p=1}^{s_{1}}\frac{1}{p^{5}}$
(27)

So we have by (17) and (27)

$\sum_{s_{1}=0}^{n}a_{n,s_{1},s_{1},s_{1},s_{1}}J(s_{1},s_{1},s_{1},s_{1})=4\sum_{s_{1}=0}^{n}\binom{n}{s_{1}}^{4}\binom{n+s_{1}}{n}^{2}\left(\zeta(5)-\sum_{p=1}^{s_{1}}\frac{1}{p^{5}}\right)$
(28)

So there exists $A_{n},A_{n}^{\prime}\in\mathbb{Z}$ such that

$\sum_{s_{1}=0}^{n}a_{n,s_{1},s_{1},s_{1},s_{1}}J(s_{1},s_{1},s_{1},s_{1})=\frac{A_{n}\zeta(5)+A_{n}^{\prime}}{d_{n}^{5}}$
(29)

where by (28)

$A_{n}=4d_{n}^{5}\sum_{s_{1}=0}^{n}\binom{n}{s_{1}}^{4}\binom{n+s_{1}}{n}^{2}$
(30)

This completes the sum considering case 1.
Consider all exponents are pairwise distinct: $s_{i}\neq s_{j}$ for all $i\neq
j$, then by partial fractions, (23) gives

$\displaystyle J(s_{1},s_{2},s_{3},s_{4})$
$\displaystyle=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\Biggr{[}\frac{1}{(s_{1}-s_{2})(s_{2}-s_{3})(s_{2}-s_{4})(p+s_{2}+\varepsilon)}-$
$\displaystyle\frac{1}{(s_{1}-s_{3})(s_{2}-s_{3})(s_{3}-s_{4})(p+s_{3}+\varepsilon)}+\frac{1}{(s_{1}-s_{4})(s_{2}-s_{4})(s_{3}-s_{4})(p+s_{4}+\varepsilon)}$
$\displaystyle-\frac{1}{(s_{1}-s_{2})(s_{1}-s_{3})(s_{1}-s_{4})(p+s_{1}+\varepsilon)}\Biggr{]}\Biggr{|}_{\varepsilon=0}$
(31)

We obtain using the uniform convergence of the series on the RHS of above
equation (which is same as the series on RHS of (23)) for $s_{i}\neq s_{j}$,
we differentiate term by term to obtain

$\displaystyle J(s_{1},s_{2},s_{3},s_{4})$
$\displaystyle=\frac{1}{(s_{1}-s_{2})(s_{2}-s_{3})(s_{2}-s_{4})}\sum_{p=1}^{\infty}\frac{1}{(p+s_{2})^{2}}-$
$\displaystyle\frac{1}{(s_{1}-s_{3})(s_{2}-s_{3})(s_{3}-s_{4})}\sum_{p=1}^{\infty}\frac{1}{(p+s_{3})^{2}}+$
$\displaystyle\frac{1}{(s_{1}-s_{4})(s_{2}-s_{4})(s_{3}-s_{4})}\sum_{p=1}^{\infty}\frac{1}{(p+s_{4})^{2}}-$
$\displaystyle\frac{1}{(s_{1}-s_{2})(s_{1}-s_{3})(s_{1}-s_{4})}\sum_{p=1}^{\infty}\frac{1}{(p+s_{1})^{2}}$
(32)

So we get

$\displaystyle J(s_{1},s_{2},s_{3},s_{4})$
$\displaystyle=\frac{\zeta(2)-\sum_{p=1}^{s_{2}}\frac{1}{p^{2}}}{(s_{1}-s_{2})(s_{2}-s_{3})(s_{2}-s_{4})}-\frac{\zeta(2)-\sum_{p=1}^{s_{3}}\frac{1}{p^{2}}}{(s_{1}-s_{3})(s_{2}-s_{3})(s_{3}-s_{4})}+$
$\displaystyle\frac{\zeta(2)-\sum_{p=1}^{s_{4}}\frac{1}{p^{2}}}{(s_{1}-s_{4})(s_{2}-s_{4})(s_{3}-s_{4})}-\frac{\zeta(2)-\sum_{p=1}^{s_{1}}\frac{1}{p^{2}}}{(s_{1}-s_{2})(s_{1}-s_{3})(s_{1}-s_{4})}$
(33)

and we obtain

$\displaystyle J(s_{1},s_{2},s_{3},s_{4})$
$\displaystyle=\Biggr{(}\frac{1}{(s_{1}-s_{2})(s_{2}-s_{3})(s_{2}-s_{4})}-\frac{1}{(s_{1}-s_{3})(s_{2}-s_{3})(s_{3}-s_{4})}+$
$\displaystyle\frac{1}{(s_{1}-s_{4})(s_{2}-s_{4})(s_{3}-s_{4})}-\frac{1}{(s_{1}-s_{2})(s_{1}-s_{3})(s_{1}-s_{4})}\Biggr{)}\zeta(2)-$
$\displaystyle\frac{\sum_{p=1}^{s_{2}}\frac{1}{p^{2}}}{(s_{1}-s_{2})(s_{2}-s_{3})(s_{2}-s_{4})}+\frac{\sum_{p=1}^{s_{3}}\frac{1}{p^{2}}}{(s_{1}-s_{3})(s_{2}-s_{3})(s_{3}-s_{4})}-$
$\displaystyle\frac{\sum_{p=1}^{s_{4}}\frac{1}{p^{2}}}{(s_{1}-s_{4})(s_{2}-s_{4})(s_{3}-s_{4})}+\frac{\sum_{p=1}^{s_{1}}\frac{1}{p^{2}}}{(s_{1}-s_{2})(s_{1}-s_{3})(s_{1}-s_{4})}$
(34)

Observe that in (34), the coefficient of $\zeta(2)$ vanishes, that is

$\displaystyle\frac{1}{(s_{1}-s_{2})(s_{2}-s_{3})(s_{2}-s_{4})}-\frac{1}{(s_{1}-s_{3})(s_{2}-s_{3})(s_{3}-s_{4})}+\frac{1}{(s_{1}-s_{4})(s_{2}-s_{4})(s_{3}-s_{4})}-$
$\displaystyle\frac{1}{(s_{1}-s_{2})(s_{1}-s_{3})(s_{1}-s_{4})}=0$ (35)

So we get by (34) and (35)

$\displaystyle J(s_{1},s_{2},s_{3},s_{4})$
$\displaystyle=-\frac{\sum_{p=1}^{s_{2}}\frac{1}{p^{2}}}{(s_{1}-s_{2})(s_{2}-s_{3})(s_{2}-s_{4})}+\frac{\sum_{p=1}^{s_{3}}\frac{1}{p^{2}}}{(s_{1}-s_{3})(s_{2}-s_{3})(s_{3}-s_{4})}-$
$\displaystyle\frac{\sum_{p=1}^{s_{4}}\frac{1}{p^{2}}}{(s_{1}-s_{4})(s_{2}-s_{4})(s_{3}-s_{4})}+\frac{\sum_{p=1}^{s_{1}}\frac{1}{p^{2}}}{(s_{1}-s_{2})(s_{1}-s_{3})(s_{1}-s_{4})}$
(36)

So by (36), there exists $\tilde{A_{n}}\in\mathbb{Z}$ such that

$\sum_{s_{i}\neq s_{j},i\neq
j}a_{n,s_{1},s_{2},s_{3},s_{4}}J(s_{1},s_{2},s_{3},s_{4})=\frac{\tilde{A_{n}}}{d_{n}^{5}}$
(37)

This completes the sum considering case 2.
By (23) we have

$J(s_{1},s_{2},s_{3},s_{4})=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\frac{1}{(p+\varepsilon+s_{1})^{2}(p+\varepsilon+s_{3})(p+\varepsilon+s_{4})}\Biggr{|}_{\varepsilon=0}$
(38)

Using partial fractions we get

$\displaystyle J(s_{1},s_{2},s_{3},s_{4})$
$\displaystyle=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\Biggr{[}\frac{2s_{1}-s_{3}-s_{4}}{(s_{1}-s_{3})^{2}(s_{1}-s_{4})^{2}(p+\varepsilon+s_{1})}-$
$\displaystyle\frac{1}{(s_{1}-s_{3})^{2}(s_{3}-s_{4})(p+\varepsilon+s_{3})}-\frac{1}{(s_{1}-s_{4})^{2}(s_{4}-s_{3})(p+\varepsilon+s_{4})}+$
$\displaystyle\frac{1}{(s_{1}-s_{3})(s_{1}-s_{4})(p+\varepsilon+s_{1})^{2}}\Biggr{]}\Biggr{|}_{\varepsilon=0}$
(39)

Hence we obtain using the uniform convergence of RHS of (39) (which is same as
the series on the RHS of (38)), we differentiate term by term to obtain

$\displaystyle J(s_{1},s_{2},s_{3},s_{4})$
$\displaystyle=\frac{2s_{1}-s_{3}-s_{4}}{(s_{1}-s_{3})^{2}(s_{1}-s_{4})^{2}}\sum_{p=1}^{\infty}\frac{1}{(p+s_{1})^{2}}-\frac{1}{(s_{1}-s_{3})^{2}(s_{3}-s_{4})}\sum_{p=1}^{\infty}\frac{1}{(p+s_{3})^{2}}+$
$\displaystyle\frac{1}{(s_{1}-s_{4})^{2}(s_{3}-s_{4})}\sum_{p=1}^{\infty}\frac{1}{(p+s_{4})^{2}}+\frac{2}{(s_{1}-s_{3})(s_{1}-s_{4})}\sum_{p=1}^{\infty}\frac{1}{(p+s_{1})^{2}}$
(40)

Above equation can be rewritten as

$\displaystyle J(s_{1},s_{2},s_{3},s_{4})$
$\displaystyle=\frac{(2s_{1}-s_{3}-s_{4})(\zeta(2)-\sum_{p=1}^{s_{1}}\frac{1}{p^{2}})}{(s_{1}-s_{3})^{2}(s_{1}-s_{4})^{2}}-\frac{\zeta(2)-\sum_{p=1}^{s_{3}}\frac{1}{p^{2}}}{(s_{1}-s_{3})^{2}(s_{3}-s_{4})}+$
$\displaystyle\frac{\zeta(2)-\sum_{p=1}^{s_{4}}\frac{1}{p^{2}}}{(s_{1}-s_{4})^{2}(s_{3}-s_{4})}+\frac{2(\zeta(3)-\sum_{p=1}^{s_{1}}\frac{1}{p^{3}})}{(s_{1}-s_{3})(s_{1}-s_{4})}$
(41)

We observe that the coefficient of $\zeta(2)$ in RHS of (41) vanishes, that is

$\frac{2s_{1}-s_{3}-s_{4}}{(s_{1}-s_{3})^{2}(s_{1}-s_{4})^{2}}-\frac{1}{(s_{1}-s_{3})^{2}(s_{3}-s_{4})}+\frac{1}{(s_{1}-s_{4})^{2}(s_{3}-s_{4})}=0$
(42)

So we get by (41) and (42)

$\displaystyle J(s_{1},s_{2},s_{3},s_{4})$
$\displaystyle=\frac{-(2s_{1}-s_{3}-s_{4})\sum_{p=1}^{s_{1}}\frac{1}{p^{2}})}{(s_{1}-s_{3})^{2}(s_{1}-s_{4})^{2}}+\frac{\sum_{p=1}^{s_{3}}\frac{1}{p^{2}}}{(s_{1}-s_{3})^{2}(s_{3}-s_{4})}-$
$\displaystyle\frac{\sum_{p=1}^{s_{4}}\frac{1}{p^{2}}}{(s_{1}-s_{4})^{2}(s_{3}-s_{4})}+\frac{2(\zeta(3)-\sum_{p=1}^{s_{1}}\frac{1}{p^{3}})}{(s_{1}-s_{3})(s_{1}-s_{4})}$
(43)

So there exists $B_{n},C_{n}\in\mathbb{Z}$ such that

$\sum_{s_{1}\neq s_{3},s_{1}\neq s_{4},s_{3}\neq
s_{4}}a_{n,s_{1},s_{1},s_{3},s_{4}}J(s_{1},s_{1},s_{3},s_{4})=\frac{B_{n}\zeta(3)+C_{n}}{d_{n}^{5}}$
(44)

where by (17) and (43)

$B_{n}=2d_{n}^{5}\sum_{s_{1}\neq s_{3},s_{1}\neq s_{4},s_{3}\neq
s_{4}}\frac{(-1)^{s_{4}+s_{4}}\binom{n}{s_{1}}^{2}\binom{n}{s_{3}}\binom{n}{s_{4}}\binom{n+s_{1}}{n}^{2}}{(s_{1}-s_{3})(s_{1}-s_{4})}$
(45)

Similarly proceeding by (23)

$J(s_{1},s_{2},s_{1},s_{4})=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\frac{1}{(p+\varepsilon+s_{1})^{2}(p+\varepsilon+s_{2})(p+\varepsilon+s_{4})}\Biggr{|}_{\varepsilon=0}$
(46)

So in (23) we replace the roles of $s_{1}$ by $s_{1}$, $s_{3}$ by $s_{2}$ and
there exists $D_{n},E_{n}\in\mathbb{Z}$ such that

$\sum_{s_{1}\neq s_{2},s_{1}\neq s_{4},s_{2}\neq
s_{4}}a_{n,s_{1},s_{2},s_{1},s_{4}}J(s_{1},s_{2},s_{1},s_{4})=\frac{D_{n}\zeta(3)+E_{n}}{d_{n}^{5}}$
(47)

where similarly by (17) and (43)

$D_{n}=2d_{n}^{5}\sum_{s_{1}\neq s_{2},s_{1}\neq s_{4},s_{2}\neq
s_{4}}\frac{(-1)^{s_{2}+s_{4}}\binom{n}{s_{1}}^{2}\binom{n}{s_{2}}\binom{n}{s_{4}}\binom{n+s_{1}}{n}\binom{n+s_{2}}{n}}{(s_{1}-s_{2})(s_{1}-s_{4})}$
(48)

Similarly proceeding by (23)

$J(s_{1},s_{2},s_{3},s_{1})=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\frac{1}{(p+\varepsilon+s_{1})^{2}(p+\varepsilon+s_{3})(p+\varepsilon+s_{2})}\Biggr{|}_{\varepsilon=0}$
(49)

In (23) we replace the roles of $s_{1}$ by $s_{1}$, $s_{3}$ by $s_{3}$ and
$s_{4}$ by $s_{2}$ so that there exists $F_{n},G_{n}\in\mathbb{Z}$ such that

$\sum_{s_{1}\neq s_{2},s_{1}\neq s_{3},s_{2}\neq
s_{3}}a_{n,s_{1},s_{2},s_{3},s_{1}}J(s_{1},s_{2},s_{3},s_{1})=\frac{F_{n}\zeta(3)+G_{n}}{d_{n}^{5}}$
(50)

where similarly by (17) and (43)

$F_{n}=2d_{n}^{5}\sum_{s_{1}\neq s_{2},s_{1}\neq s_{3},s_{2}\neq
s_{3}}\frac{(-1)^{s_{2}+s_{3}}\binom{n}{s_{1}}^{2}\binom{n}{s_{2}}\binom{n}{s_{3}}\binom{n+s_{1}}{n}\binom{n+s_{2}}{n}}{(s_{1}-s_{3})(s_{1}-s_{2})}$
(51)

Similarly proceeding by (23)

$J(s_{1},s_{2},s_{2},s_{4})=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\frac{1}{(p+\varepsilon+s_{1})(p+\varepsilon+s_{2})^{2}(p+\varepsilon+s_{4})}\Biggr{|}_{\varepsilon=0}$
(52)

In (23) we replace the roles of $s_{1}$ by $s_{2}$, $s_{3}$ by $s_{1}$ and
$s_{4}$ by $s_{4}$ so that there exists $H_{n},I_{n}\in\mathbb{Z}$ such that

$\sum_{s_{1}\neq s_{2},s_{1}\neq s_{4},s_{2}\neq
s_{4}}a_{n,s_{1},s_{2},s_{2},s_{4}}J(s_{1},s_{2},s_{2},s_{4})=\frac{H_{n}\zeta(3)+I_{n}}{d_{n}^{5}}$
(53)

where similarly by (17) and (43)

$H_{n}=2d_{n}^{5}\sum_{s_{1}\neq s_{2},s_{1}\neq s_{4},s_{2}\neq
s_{4}}\frac{(-1)^{s_{1}+s_{4}}\binom{n}{s_{2}}^{2}\binom{n}{s_{1}}\binom{n}{s_{4}}\binom{n+s_{1}}{n}\binom{n+s_{2}}{n}}{(s_{2}-s_{1})(s_{2}-s_{4})}$
(54)

Similarly proceeding by (23)

$J(s_{1},s_{2},s_{3},s_{2})=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\frac{1}{(p+\varepsilon+s_{1})(p+\varepsilon+s_{2})^{2}(p+\varepsilon+s_{3})}\Biggr{|}_{\varepsilon=0}$
(55)

In (23) we replace the roles of $s_{1}$ by $s_{2}$, $s_{3}$ by $s_{3}$ and
$s_{4}$ by $s_{1}$ so that there exists $J_{n},K_{n}\in\mathbb{Z}$ such that

$\sum_{s_{1}\neq s_{2},s_{1}\neq s_{3},s_{2}\neq
s_{3}}a_{n,s_{1},s_{2},s_{3},s_{2}}J(s_{1},s_{2},s_{3},s_{2})=\frac{J_{n}\zeta(3)+K_{n}}{d_{n}^{5}}$
(56)

where similarly by (17) and (43)

$J_{n}=2d_{n}^{5}\sum_{s_{1}\neq s_{2},s_{1}\neq s_{3},s_{2}\neq
s_{3}}\frac{(-1)^{s_{1}+s_{3}}\binom{n}{s_{1}}\binom{n}{s_{2}}^{2}\binom{n}{s_{3}}\binom{n+s_{1}}{n}\binom{n+s_{2}}{n}}{(s_{2}-s_{3})(s_{2}-s_{1})}$
(57)

Similarly proceeding by (23)

$J(s_{1},s_{2},s_{3},s_{3})=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\frac{1}{(p+\varepsilon+s_{1})(p+\varepsilon+s_{2})(p+\varepsilon+s_{3})^{2}}\Biggr{|}_{\varepsilon=0}$
(58)

In (23) we replace the roles of $s_{1}$ by $s_{3}$, $s_{3}$ by $s_{1}$ and
$s_{4}$ by $s_{2}$ so that there exists $L_{n},M_{n}\in\mathbb{Z}$ such that

$\sum_{s_{1}\neq s_{2},s_{1}\neq s_{3},s_{2}\neq
s_{3}}a_{n,s_{1},s_{2},s_{3},s_{3}}J(s_{1},s_{2},s_{3},s_{3})=\frac{L_{n}\zeta(3)+M_{n}}{d_{n}^{5}}$
(59)

where similarly by (17) and (43)

$L_{n}=2d_{n}^{5}\sum_{s_{1}\neq s_{2},s_{1}\neq s_{3},s_{2}\neq
s_{3}}\frac{(-1)^{s_{1}+s_{2}}\binom{n}{s_{1}}\binom{n}{s_{2}}\binom{n}{s_{3}}^{2}\binom{n+s_{1}}{n}\binom{n+s_{2}}{n}}{(s_{3}-s_{1})(s_{3}-s_{2})}$
(60)

This completes the sum considering case 3.
Now by (23) we have

$J(s_{1},s_{1},s_{3},s_{3})=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\frac{1}{(p+\varepsilon+s_{1})^{2}(p+\varepsilon+s_{3})^{2}}\Biggr{|}_{\varepsilon=0}$
(61)

Partial fraction decomposition of above equation gives

$\displaystyle J(s_{1},s_{1},s_{3},s_{3})$
$\displaystyle=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\Biggr{[}-\frac{2}{(s_{1}-s_{3})^{3}(p+\varepsilon+s_{3})}+\frac{1}{(s_{1}-s_{3})^{2}(p+\varepsilon+s_{3})^{2}}+$
$\displaystyle\frac{2}{(s_{1}-s_{3})^{3}(p+\varepsilon+s_{1})}+\frac{1}{(s_{1}-s_{3})^{2}(p+\varepsilon+s_{1})^{2}}\Biggr{]}\Biggr{|}_{\varepsilon=0}$
(62)

Hence we obtain

$\displaystyle J(s_{1},s_{1},s_{3},s_{3})$
$\displaystyle=-\frac{2}{(s_{1}-s_{3})^{3}}\sum_{p=1}^{\infty}\frac{1}{(p+s_{3})^{2}}+\frac{2}{(s_{1}-s_{3})^{2}}\sum_{p=1}^{\infty}\frac{1}{(p+s_{3})^{3}}+$
$\displaystyle\frac{2}{(s_{1}-s_{3})^{3}}\sum_{p=1}^{\infty}\frac{1}{(p+s_{1})^{2}}+\frac{2}{(s_{1}-s_{3})^{2}}\sum_{p=1}^{\infty}\frac{1}{(p+s_{1})^{3}}]$
(63)

So we get by (63)

$\displaystyle J(s_{1},s_{1},s_{3},s_{3})$
$\displaystyle=\frac{2}{(s_{1}-s_{3})^{3}}\left(\sum_{p=1}^{s_{3}}\frac{1}{p^{2}}-\sum_{p=1}^{\infty}\frac{1}{p^{2}}\right)+$
$\displaystyle\frac{2}{(s_{1}-s_{3})^{2}}\left(2\zeta(3)-\sum_{p=1}^{s_{1}}\frac{1}{p^{3}}-\sum_{p=1}^{s_{3}}\frac{1}{p^{3}}\right)$
(64)

So there exists $N_{n},O_{n}\in\mathbb{Z}$ such that

$\sum_{s_{1}\neq
s_{3}}a_{n,s_{1},s_{1},s_{3},s_{3}}J(s_{1},s_{1},s_{3},s_{3})=\frac{N_{n}\zeta(3)+O_{n}}{d_{n}^{5}}$
(65)

where by (17) and (64)

$N_{n}=2d_{n}^{5}\sum_{s_{1}\neq
s_{3}}\frac{\binom{n}{s_{1}}^{2}\binom{n}{s_{3}}^{2}\binom{n+s_{1}}{n}^{2}}{(s_{1}-s_{3})^{3}}$
(66)

Similarly proceeding by (23) we have

$J(s_{1},s_{2},s_{1},s_{2})=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\frac{1}{(p+\varepsilon+s_{1})^{2}(p+\varepsilon+s_{2})^{2}}\Biggr{|}_{\varepsilon=0}$
(67)

So in (61) we replace the roles of $s_{1}$ by $s_{1}$, $s_{3}$ by $s_{2}$ so
that there exists $P_{n},Q_{n}\in\mathbb{Z}$ such that

$\sum_{s_{1}\neq
s_{2}}a_{n,s_{1},s_{2},s_{1},s_{2}}J(s_{1},s_{2},s_{1},s_{2})=\frac{P_{n}\zeta(3)+Q_{n}}{d_{n}^{5}}$
(68)

where by (17) and (64)

$P_{n}=2d_{n}^{5}\sum_{s_{1}\neq
s_{2}}\frac{\binom{n}{s_{1}}^{2}\binom{n}{s_{2}}^{2}\binom{n+s_{1}}{n}\binom{n+s_{2}}{n}}{(s_{1}-s_{2})^{3}}$
(69)

Similarly proceeding by (23) we have

$J(s_{1},s_{2},s_{2},s_{1})=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\frac{1}{(p+\varepsilon+s_{1})^{2}(p+\varepsilon+s_{2})^{2}}\Biggr{|}_{\varepsilon=0}$
(70)

So there exists $P_{n},Q_{n}\in\mathbb{Z}$ such that

$\sum_{s_{1}\neq
s_{2}}a_{n,s_{1},s_{2},s_{2},s_{1}}J(s_{1},s_{2},s_{2},s_{1})=\frac{P_{n}\zeta(3)+Q_{n}}{d_{n}^{5}}$
(71)

where $P_{n}$ is given by (69). This completes the sum considering case 4.
Finally by (23) we have

$J(s_{1},s_{1},s_{1},s_{4})=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\frac{1}{(p+\varepsilon+s_{1})^{3}(p+\varepsilon+s_{4})}\Biggr{|}_{\varepsilon=0}$
(72)

Partial fraction decomposition gives

$\displaystyle J(s_{1},s_{1},s_{1},s_{4})$
$\displaystyle=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\Biggr{[}\frac{1}{(s_{1}-s_{4})^{3}}\left(\frac{1}{p+\varepsilon+s_{4}}-\frac{1}{p+\varepsilon+s_{1}}\right)-$
$\displaystyle\left(\frac{1}{(s_{1}-s_{4})^{2}(p+\varepsilon+s_{1})^{2}}+\frac{1}{(s_{1}-s_{4})(p+\varepsilon+s_{1})^{3}}\right)\Biggr{]}\Biggr{|}_{\varepsilon=0}$
(73)

So we obtain using the uniform convergence of RHS of (73) (which is same as
the series on the RHS of (72))

$\displaystyle J(s_{1},s_{1},s_{1},s_{4})$
$\displaystyle=\frac{1}{(s_{1}-s_{4})^{3}}\left(\sum_{p=1}^{\infty}\frac{1}{(p+s_{4})^{2}}-\sum_{p=1}^{\infty}\frac{1}{(p+s_{1})^{2}}\right)+$
$\displaystyle\frac{2}{(s_{1}-s_{4})^{2}}\sum_{p=1}^{\infty}\frac{1}{(p+s_{1})^{3}}+\frac{3}{(s_{1}-s_{4})}\sum_{p=1}^{\infty}\frac{1}{(p+s_{1})^{4}}$
(74)

Hence we obtain

$\displaystyle J(s_{1},s_{1},s_{1},s_{4})$
$\displaystyle=\frac{1}{(s_{1}-s_{4})^{3}}\left(\sum_{p=1}^{s_{1}}\frac{1}{p^{2}}-\sum_{p=1}^{s_{4}}\frac{1}{p^{2}}\right)+\frac{2}{(s_{1}-s_{4})^{2}}\left(\zeta(3)-\sum_{p=1}^{s_{1}}\frac{1}{p^{3}}\right)+$
$\displaystyle\frac{3}{(s_{1}-s_{4})}\left(\zeta(4)-\sum_{p=1}^{s_{1}}\frac{1}{p^{4}}\right)$
(75)

So by (75) there exists $R_{n},S_{n},T_{n}\in\mathbb{Z}$ such that

$\sum_{s_{1}\neq
s_{4}}a_{n,s_{1},s_{1},s_{1},s_{4}}J(s_{1},s_{1},s_{1},s_{4})=\frac{R_{n}\zeta(4)+S_{n}\zeta(3)+T_{n}}{d_{n}^{5}}$
(76)

where by (17) and (75)

$R_{n}=3d_{n}^{5}\sum_{s_{1}\neq
s_{4}}\frac{(-1)^{s_{1}+s_{4}}\binom{n}{s_{1}}^{3}\binom{n}{s_{4}}\binom{n+s_{1}}{n}^{2}}{s_{1}-s_{4}}$
(77)

and

$S_{n}=2d_{n}^{5}\sum_{s_{1}\neq
s_{4}}\frac{(-1)^{s_{1}+s_{4}}\binom{n}{s_{1}}^{3}\binom{n}{s_{4}}\binom{n+s_{1}}{n}^{2}}{(s_{1}-s_{4})^{2}}$
(78)

Similarly by (23)

$J(s_{1},s_{2},s_{2},s_{2})=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\frac{1}{(p+\varepsilon+s_{2})^{3}(p+\varepsilon+s_{1})}\Biggr{|}_{\varepsilon=0}$
(79)

So in (72) and (75), we replace the roles of $s_{1}$ by $s_{2}$ and $s_{4}$ by
$s_{1}$ so that there exists $U_{n},V_{n},W_{n}\in\mathbb{Z}$ such that

$\sum_{s_{1}\neq
s_{2}}a_{n,s_{1},s_{2},s_{2},s_{2}}J(s_{1},s_{2},s_{2},s_{2})=\frac{U_{n}\zeta(4)+V_{n}\zeta(3)+W_{n}}{d_{n}^{5}}$
(80)

where by (17) and (75)

$U_{n}=3d_{n}^{5}\sum_{s_{1}\neq
s_{2}}\frac{(-1)^{s_{1}+s_{2}}\binom{n}{s_{2}}^{3}\binom{n}{s_{1}}\binom{n+s_{1}}{n}\binom{n+s_{2}}{n}}{s_{2}-s_{1}}$
(81)

and

$V_{n}=2d_{n}^{5}\sum_{s_{1}\neq
s_{2}}\frac{(-1)^{s_{1}+s_{2}}\binom{n}{s_{2}}^{3}\binom{n}{s_{1}}\binom{n+s_{1}}{n}\binom{n+s_{2}}{n}}{(s_{2}-s_{1})^{2}}$
(82)

Similarly by (23)

$J(s_{1},s_{1},s_{3},s_{1})=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\frac{1}{(p+\varepsilon+s_{1})^{3}(p+\varepsilon+s_{3})}\Biggr{|}_{\varepsilon=0}$
(83)

So in (72) and (75), we replace the roles of $s_{1}$ by $s_{1}$ and $s_{4}$ by
$s_{3}$ so that there exists $X_{n},Y_{n},Z_{n}\in\mathbb{Z}$ such that

$\sum_{s_{1}\neq
s_{3}}a_{n,s_{1},s_{1},s_{3},s_{1}}J(s_{1},s_{1},s_{3},s_{1})=\frac{X_{n}\zeta(4)+Y_{n}\zeta(3)+Z_{n}}{d_{n}^{5}}$
(84)

where by (17) and (75)

$X_{n}=3d_{n}^{5}\sum_{s_{1}\neq
s_{3}}\frac{(-1)^{s_{1}+s_{3}}\binom{n}{s_{1}}^{3}\binom{n}{s_{3}}\binom{n+s_{1}}{n}^{2}}{s_{1}-s_{3}}$
(85)

and

$Y_{n}=2d_{n}^{5}\sum_{s_{1}\neq
s_{3}}\frac{(-1)^{s_{1}+s_{3}}\binom{n}{s_{1}}^{3}\binom{n}{s_{3}}\binom{n+s_{1}}{n}^{2}}{(s_{1}-s_{3})^{2}}$
(86)

Similarly by (23)

$J(s_{1},s_{2},s_{1},s_{1})=-\frac{d}{d\varepsilon}\sum_{p=1}^{\infty}\frac{1}{(p+\varepsilon+s_{1})^{3}(p+\varepsilon+s_{2})}\Biggr{|}_{\varepsilon=0}$
(87)

So in (72) and (75), we replace the roles of $s_{1}$ by $s_{1}$ and $s_{4}$ by
$s_{2}$ so that there exists $\alpha_{n},\beta_{n},\gamma_{n}\in\mathbb{Z}$
such that

$\sum_{s_{1}\neq
s_{2}}a_{n,s_{1},s_{2},s_{1},s_{1}}J(s_{1},s_{2},s_{1},s_{1})=\frac{\alpha_{n}\zeta(4)+\beta_{n}\zeta(3)+\gamma_{n}}{d_{n}^{5}}$
(88)

where by (17) and (75)

$\alpha_{n}=3d_{n}^{5}\sum_{s_{1}\neq
s_{2}}\frac{(-1)^{s_{1}+s_{2}}\binom{n}{s_{1}}^{3}\binom{n}{s_{2}}\binom{n+s_{1}}{n}\binom{n+s_{2}}{n}}{s_{1}-s_{2}}$
(89)

and

$\beta_{n}=2d_{n}^{5}\sum_{s_{1}\neq
s_{2}}\frac{(-1)^{s_{1}+s_{2}}\binom{n}{s_{1}}^{3}\binom{n}{s_{2}}\binom{n+s_{1}}{n}\binom{n+s_{2}}{n}}{(s_{1}-s_{2})^{2}}$
(90)

This completes the sum considering case 5.
So we get by (37),(44),(47),(50),(53),(56),(59),(65),(68),(76),(88) for each
$n\in\mathbb{N}$, there exists $p_{n},q_{n},r_{n},s_{n}\in\mathbb{Z}$ such
that

$I_{n}=\frac{p_{n}\zeta(5)+q_{n}\zeta(4)+r_{n}\zeta(3)+s_{n}}{d_{n}^{5}}$ (91)

where by (30),

$p_{n}=4d_{n}^{5}\sum_{s_{1}=0}^{n}\binom{n}{s_{1}}^{4}\binom{n+s_{1}}{n}^{2}$
(92)

Also by (44), (45), (47), (48), (50), (51), (53), (54), (56), (57), (59),
(60), (65), (66), (68), (69), (71), (76), (77), (78), (80), (81), (82), (84),
(85), (86), (88), (89) and (90)

$r_{n}=B_{n}+D_{n}+F_{n}+H_{n}+J_{n}+L_{n}+N_{n}+P_{n}+S_{n}+V_{n}+Y_{n}+\beta_{n}$
(93)

and

$q_{n}=R_{n}+U_{n}+X_{n}+\alpha_{n}$ (94)

So by (91)

$q_{n}\zeta(4)+r_{n}\zeta(3)-d_{n}^{5}I_{n}=-p_{n}\zeta(5)-s_{n}$ (95)

So we get by (95), $q_{n}\zeta(4)+r_{n}\zeta(3)-d_{n}^{5}I_{n}\in\mathbb{Z}$
if and only if $\zeta(5)=\frac{a}{b}$ for some $a,b\in\mathbb{Z}$ where
$b|p_{n}$ and $p_{n}$ is given by (92). Also we have by (*)

$\lim_{n\to\infty}d_{n}^{5}I_{n}=0$ (96)

We have $(1)\implies(2)$ is trivial. To prove $(2)\implies(3)$, we are given
that $q_{n}\zeta(4)+r_{n}\zeta(3)-d_{n}^{5}I_{n}\not\in\mathbb{Z}$ for
infinitely many natural number $n$. So, if on the contrary, $\zeta(5)=a/b$ for
some $a,b\in\mathbb{N}$ and choose any $n\geq b$ then $b|d_{n}$ which means
$b|d^{5}_{n}$, so by (92) $b|p_{n}$, so
$q_{n}\zeta(4)+r_{n}\zeta(3)-d_{n}^{5}I_{n}\in\mathbb{Z}$, which is a
contradiction. So, if
$q_{n}\zeta(4)+r_{n}\zeta(3)-d_{n}^{5}I_{n}\not\in\mathbb{Z}$ for infinitely
many natural number $n$ then $\zeta(5)$ is irrational. To prove
$(3)\implies(1)$, we first prove that if
$q_{n}\zeta(4)+r_{n}\zeta(3)-d_{n}^{5}I_{n}\in\mathbb{Z}$ then $\zeta(5)$ is
rational. Since by (95),
$q_{n}\zeta(4)+r_{n}\zeta(3)-d_{n}^{5}I_{n}\in\mathbb{Z}$ gives
$-p_{n}\zeta(5)-s_{n}\in\mathbb{Z}$ and since $s_{n}\in\mathbb{Z}$ so we have
$p_{n}\zeta(5)\in\mathbb{Z}$. Since by (92), $p_{n}$ is a positive integer, so
we obtain $\zeta(5)\in\mathbb{Q}$. So we get, if $\zeta(5)$ is irrational,
then $q_{n}\zeta(4)+r_{n}\zeta(3)-d_{n}^{5}I_{n}\not\in\mathbb{Z}$ for each
natural number $n$. This settles the proof of the Theorem.

Appendix

Denote

$I=-\frac{1}{4}\int_{(0,1)^{4}}\frac{\log(x_{1}x_{2}x_{3}x_{4})}{1-x_{1}x_{2}x_{3}x_{4}}\
dx_{1}dx_{2}dx_{3}dx_{4}$ (97)

So we get

$I=\frac{1}{4}\int_{(0,1)^{4}}\sum_{m=0}^{\infty}-\log(x_{1}x_{2}x_{3}x_{4})\
(x_{1}x_{2}x_{3}x_{4})^{m}\ dx_{1}dx_{2}dx_{3}dx_{4}$ (98)

Since
$F_{m}(x_{1},x_{2},x_{3},x_{4})=-\log(x_{1}x_{2}x_{3}x_{4})(x_{1}x_{2}x_{3}x_{4})^{m}$
is non negative so using Tonelli’s theorem we can reverse the order of
integration and summation to get

$I=-\frac{1}{4}\sum_{m=0}^{\infty}\int_{(0,1)^{4}}\log(x_{1}x_{2}x_{3}x_{4})\
(x_{1}x_{2}x_{3}x_{4})^{m}\ dx_{1}dx_{2}dx_{3}dx_{4}$ (99)

which can be re-written as

$I=-\frac{1}{4}\sum_{m=0}^{\infty}\int_{(0,1)^{4}}(\log x_{1}+\log x_{2}+\log
x_{3}+\log x_{4})\ (x_{1}x_{2}x_{3}x_{4})^{m}\ dx_{1}dx_{2}dx_{3}dx_{4}$ (100)

By integration by parts we have

$\int_{0}^{1}x_{1}^{m}\log x_{1}\ dx_{1}=-\frac{1}{(m+1)^{2}}$ (101)

So we get by (100) and (101)

$I=\frac{1}{4}\sum_{m=0}^{\infty}\frac{4}{(m+1)^{5}}$ (102)

So we obtain

$I=\zeta(5)$ (103)

By (97) and (103)

$\zeta(5)=-\frac{1}{4}\int_{(0,1)^{4}}\frac{\log(x_{1}x_{2}x_{3}x_{4})}{1-x_{1}x_{2}x_{3}x_{4}}\
dx_{1}dx_{2}dx_{3}dx_{4}$ (104)

which completes the proof of equation (1)

Acknowledgement The authors are thankful to the Referee and the Editor for
their useful comments and processing of the article.

References

[1] Beukers, F., A note on the irrationality of $\zeta(2)$ and $\zeta(3)$,
Bull. London. Math. Soc., 11 (1979),268-272.

[2] Apéry, R., Irrationalité de $\zeta(2)$ et $\zeta(3)$, Astérisque. (1979)
61: 11–13.

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