ON CELLULAR RATIONAL APPROXIMATIONS TO 5 FRANCIS BROWN AND WADIM ZUDILIN Abstract. We analyse a certain family of cellular integrals which are period

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ON CELLULAR RATIONAL APPROXIMATIONS TO ζ(5)

FRANCIS BROWN AND WADIM ZUDILIN

Abstract. We analyse a certain family of cellular integrals, which are period

integrals on the moduli space M0,8of curves of genus zero with eight marked

points, which give rise to simultaneous rational approximations to ζ(3) and ζ(5).

By exploiting the action of a large symmetry group on these integrals, we construct

inﬁnitely many eﬀective rational approximations p/q to ζ(5) satisfying

0<

ζ(5) −p

q

q0.86 .

1. Introduction

Following Ap´ery’s legendary proof [1] of the irrationality of ζ(3), the traditional

strategy for approaching the irrationality of ζ(5) is to try to construct linear forms

in 1 and ζ(5) with rational coeﬃcients and good arithmetic properties. At the time

of writing, it has not succeeded despite many years of eﬀort. In this paper we

propose a diﬀerent method for tackling this problem. It involves constructing small

linear forms in a larger set of multiple zeta values, which, after setting the unwanted

numbers to zero, leads to the approximations described in the abstract.

The starting point for this method is the recent work [6] of one of the authors which

revisits irrationality proofs from a new geometric perspective. It reproduces Ap´ery’s

irrationality proofs for ζ(2) and ζ(3), via Beukers’ famous re-interpretation [4] as

Euler-type double and triple integrals, and produces natural families of multiple

integrals of higher order which are linear forms in a controllable set of multiple

zeta values. Whilst a literal generalisation of Ap´ery’s approach for higher weight

zeta values remains a tough challenge, the machinery in [6] gives us hope to seek

alternative approaches.

In this manuscript we investigate just one particular family of such 5-fold integrals,

which was disguised as the family 8π∨

8and highlighted in [6, Example 7.5],

I(a) = I(a1, a2, a3, a4, a5, a6, a7, a8)

=Z· · · Z

0<t1<t2<···<t5<1

ta1

1(t2−t1)a2(t3−t2)a3(t4−t3)a4(t5−t4)a5(1 −t5)a6

(t3−t1)b24 tb14

3(1 −t4)b57 (t4−t2)b35 (t5−t2)b36

×dt1dt2dt3dt4dt5

(t3−t1)t3(1 −t4)(t4−t2)(t5−t2),(1)

Date: 11 October 2022.

2020 Mathematics Subject Classiﬁcation. Primary 11J72; Secondary 11M06, 20B35, 32G15,

33C90.

arXiv:2210.03391v2 [math.NT] 13 Oct 2022

2 FRANCIS BROWN AND WADIM ZUDILIN

where b24 =a2+a3+a6−a7−a8, b14 =a4+a7+a8−a2−a6,

b57 =a4+a5+a8−a2−a3, b35 =a2+a3−a8, b36 =a8

(2)

and all the parameters a1, . . . , a8are assumed to be integers. According to [6,

Sects. 5.1 and 5.2] the integral I(a) converges if and only if the following seventeen

linear forms in the aiare non-negative:

a1, a2, a3, a4, a5, a6, a7, a1+a5−a3, a3+a6−a8,

a4+a5+a7+a8−a2−a3−a6, a7+a8−a6, a4+a8−a2, a2+a3+a6−a4−a8,

a1+a8−a3, a1+a2−a4, a4+a5−a2, a4+a7+ 2a8−a2−a3−a6.

(3)

General results about periods of moduli spaces [5] imply a priori that the family

(1) of integrals is a Q-linear combination of multiple zeta values of weight ≤5,

namely: 1, ζ(2), ζ(3), ζ(4), ζ(5) and ζ(3)ζ(2). The cellular nature of this integral

(more precisely, Poincar´e duality) suggests that the term of subleading weight ζ(4)

vanishes, since it is dual to the non-existent ‘ζ(1)’. Cohomological considerations

furthermore imply that the coeﬃcients of the two terms ζ(5) and ζ(2)ζ(3) always

occur in the same proportion, i.e., there is a single period in leading weight, namely

ζ(5) + 2ζ(3)ζ(2).

The main interest of this family is that, additionally, the coeﬃcient of ζ(3) always

vanishes [6, Sect. 10.2.4]. Therefore, as hinted at in [6], the decomposition of I=

I(a) into a Q-linear combination of zeta values takes the very special form:

I=Q·(2ζ(5) + 4ζ(3)ζ(2)) −4ˆ

P·ζ(2) −2P(4)

for some Q, P, ˆ

P∈Q(in fact, Q∈Zsince it may be expressed as a 5-fold

residue of the integrand). The fact that these linear forms in ζ(5) + 2ζ(3)ζ(2),

ζ(2) and 1 are very small follows from bounds for the integrand along the domain

of integration. From this one may deduce that at least one of the two numbers

{ζ(2), ζ(5) + 2ζ(3)ζ(2)}is irrational, but since this is already known for ζ(2), one

cannot deduce any new irrationality result from I. In this regard, the number ζ(2)

is usually viewed as ‘parasitic’.

However, another hint from [6] suggests that the linear forms

I0=I0(a) = Qζ(5) −P,

obtained by ‘setting ζ(2) to zero’, are also reasonably small. A priori this operation

does not make sense, but can be justiﬁed either by cohomological arguments, or by

passing to motivic versions of the integral Iand motivic zeta values, for which it

does.

The fact that the I0are small implies a posteriori that the linear forms

I00 =I00(a) = Qζ(3) −ˆ

are small as well, since I= 2I0+ 4I00ζ(2). Thus the original cellular integral (4) is,

in disguise, a pair of simultaneous approximations I0, I00 to ζ(5) and ζ(3). It is our

principal goal here to quantify these hints from [6] as well as to analyse the arithmetic

properties of the coeﬃcients Q=Q(a), P=P(a) and ˆ

P=ˆ

P(a) of the simultaneous

ON CELLULAR RATIONAL APPROXIMATIONS TO ζ(5) 3

rational approximations to ζ(5) and ζ(3). As we will see below, there is a large

transformation group G(of order 7! = 5040) acting on normalised versions of the

integrals I(a) as well as on I0(a), I00(a) and on the coeﬃcients Q(a), P (a),ˆ

P(a); this

group allows us to sharply compute a denominator D=D(a)∈Zfor which DI0∈

Zζ(5) + Z. Such groups famously appear, and prove themselves to be arithmetically

useful, in constructions of rational approximations to ζ(2), ζ(3) and ζ(4) (and other

mathematical constants); see [13,16,17,19,24]. Our ‘group structure for ζ(5)’ shares

similarities with its predecessors but also features interesting novelties which we will

try to highlight in due course.

One outcome of our construction and analysis is the following result, for which we

need to recall a related concept of eﬀective rational approximations to a real number

α, as explained by Nesterenko in his paper [14]. A family of linear forms qnα−pn,

with pn, qn∈Q, is called eﬀective if it is given by the solution to a linear diﬀer-

ence equation with polynomial coeﬃcients (also known as an Ap´ery-type recursion).

Equivalently, its generating function satisﬁes a (Picard–Fuchs) diﬀerential equation

of geometric origin. We say that a family of rational approximations to αis eﬀective

if it can be written in the form pn/qn, where qnα−pnis eﬀective. Such eﬀective

rational approximations1pn/qnare distinguished from (ineﬀective!) solutions to

0<

α−p

q

in integers p, q whose existence follows from squeezing the rational number in ques-

tion via r/q ≤α≤(r+ 1)/q, with r=bqαc.

Theorem 1. There are inﬁnitely many eﬀective rational approximations p/q, with

p, q ∈Z, to ζ(5) such that

0<

ζ(5) −p

q

q0.86 .

Note that this result does not imply the (expected!) irrationality of ζ(5) which

would follow if the upper bound in the inequality were of the form 1/q1+εfor some

ε > 0. The ‘worthiness’ exponent 0.86 is nevertheless best possible when compared

to any other known constructions of eﬀective rational approximations to ζ(5) (see

the introduction in [14] for a related comparison in the case of Catalan’s constant).

The result and analysis in this paper also gives us conﬁdence that further exploration

of the cellular integrals from [6] will produce new arithmetic surprises.

A more general context for this subject is the study of Mellin transforms

Zγ

fa1

1· · · fan

nω

where f1, . . . , fn:X→Gmare morphisms from an algebraic variety Xdeﬁned over

Qto the multiplicative group Gm, the ai∈Care complex parameters, γ⊂X(C) is

a (locally ﬁnite) chain of integration, and ωis a diﬀerential form of degree equal to

1Be aware that passing from pn/qnto p/q with p, q ∈Zmay involve multiplying both pnand

qnby the same rational factor: rescaling is implicit in the deﬁnition.

4 FRANCIS BROWN AND WADIM ZUDILIN

the dimension of γ. In the present note, X=M0,n, the moduli space of Riemann

spheres with n= 8 marked points, the f1, . . . , f8are cross-ratios, the aiare integers,

and ωis the 5-form on the second line of (1). The combinatorial, arithmetic and

analytic structures that we have unearthed in this particular case may point to the

existence of general theorems for Mellin transforms in algebraic geometry. We hope

that our results may serve as inspiration for future research along these lines.

Since our main task is to emphasise the ideas and methods behind the analysis

of the integrals (1), we have tried to stay non-technical in our exposition as far as

possible, and details of proofs which are either obtainable by ﬁnite calculation, or

have been veriﬁed by computer computation, are omitted. Each section presents

diﬀerent mathematical features and structures underlying the integrals (1) and ends

with some suggestions for generalisation. We leave it to the reader to judge whether

our narrative style is suﬃciently clear, self-contained and reader-friendly.

2. Totally symmetric case

We ﬁrst focus our attention on the ‘totally symmetric’ case when all the parame-

ters a1, . . . , a8are equal,

a1=· · · =a8=n.

This also means that all the exponents in (1) including b24, b14, b57, b35, b36 are equal

to n:

In=I(n, . . . , n) = Z· · · Z

0<t1<t2<···<t5<1t1(t2−t1)(t3−t2)(t4−t3)(t5−t4)(1 −t5)

(t3−t1)t3(1 −t4)(t4−t2)(t5−t2)n

×dt1dt2dt3dt4dt5

(t3−t1)t3(1 −t4)(t4−t2)(t5−t2).

It is the simplest possible choice of the parameters; as we will witness later, the

transformation group Gacts trivially in this case.

The integrals Inare eﬀectively computed (up to n= 10) using Panzer’s Hyper-

Int [15]. We ﬁnd out that we indeed have

In=Qn·(2ζ(5) + 4ζ(3)ζ(2)) −4ˆ

Pn·ζ(2) −2Pn(5)

for this range; more speciﬁcally,

Q0= 1, Q1= 21, Q2= 2989,

P0= 0,ˆ

P1=101

4,ˆ

P2=344923

96 , P0= 0, P1=87

4, P2=1190161

384

ON CELLULAR RATIONAL APPROXIMATIONS TO ζ(5) 5

for n= 0,1,2. Then Koutschan’s HolonomicFunctions [11] produces a third order

Ap´ery-type recursion for the integrals In:

2(2n+ 1)(41218n3−48459n2+ 20010n−2871)(n+ 1)5Qn+1

−(97604224n9+ 178061760n8+ 72005308n7−48634688n6−39076836n5

+ 2622730n4+ 7581006n3+ 920112n2−543402n−120582)Qn

−2n(3874492n8−2617900n7−3144314n6+ 2947148n5+ 647130n4−1182926n3

+ 115771n2+ 170716n−44541)Qn−1

+n(41218n3+ 75195n2+ 46746n+ 9898)(n−1)5Qn−2= 0,where n= 2,3,...,

which is also satisﬁed by the rational coeﬃcients Qn,ˆ

Pn, Pn. This already proves

the decomposition (5), so that In= 2I0

n+ 4I00

nζ(2) with I0

n=Qnζ(5) −Pnand

I00

n=Qnζ(3) −ˆ

Pn. The characteristic polynomial of the recurrence equation is

4λ3−2368λ2−188λ+ 1 .

λ1= 0.00500378 . . . , λ2=−0.08438431 . . . and λ3= 592.07938053 . . .

denote its roots (ordered according to their absolute value), then a standard locali-

sation procedure leads to the asymptotics

lim

n→∞

log |In|

n= log |λ1|=−5.29756135 . . . ,

lim

n→∞

log |I0

n= lim

n→∞

log |I00

n= log |λ2|=−2.47237372 . . .

and

lim

n→∞

log |Qn|

n= lim

n→∞

log |ˆ

Pn|

n= lim

n→∞

log |Pn|

n= log |λ3|= 6.38364071 . . . .

Finally, based on an extensive computation of the rational coeﬃcients Qn,ˆ

Pn, Pnwe

observe experimentally that

Qn, d2

nd2nˆ

Pn, d5

nPn∈Zfor n= 0,1,2,..., (6)

where dndenotes the least common multiple of 1,2, . . . , n. As we show later,

Qn=

k1=0 n+k1

nn

k12n

k2=0 n+k2

nn

k22n+k1+k2

n(7)

implying in particular that the coeﬃcients Qnare integral. The validity of this

formula can be independently established by verifying, again on the basis of [11],

that the double sum on the right-hand side satisﬁes the above recursion.

The approximating forms are similar in spirit to the ones for 1, ζ(2), ζ(3) con-

structed in [25, Section 2] (although their characteristic polynomials have two roots

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ONCELLULARRATIONALAPPROXIMATIONSTO(5)FRANCISBROWNANDWADIMZUDILINAbstract.Weanalyseacertainfamilyofcellularintegrals,whichareperiodintegralsonthemodulispaceM0;8ofcurvesofgenuszerowitheightmarkedpoints,whichgiverisetosimultaneousrationalapproximationsto(3)and(5).Byexploitingtheactionofalargesymmetr...

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ON CELLULAR RATIONAL APPROXIMATIONS TO 5 FRANCIS BROWN AND WADIM ZUDILIN Abstract. We analyse a certain family of cellular integrals which are period

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