Infinite products over visible lattice points

INFINITE PRODUCTS OVER VISIBLE LA’I-I’ICE POINTS

GEOFFREY B. CAMPBELL MathematicsResearch Section

InstituteofAdvanced Studies School of Mathematical Sciences TheAustralianNationalUniversity

GPO Box

4, Canberra, Australia, 2601

(Received

August

10, 1992and in revised form

September

2,

1993)

ABSTRACT.

About fiftynewmultivariatecombinatorial identitiesaregiven,connected withpartition theory,primeproducts,and Dirichlet series. ConnectionstoLattice

Sums

inChemistryandPhysicsare alludedtoalso.

KEY

WORDS AND

PHRASES:

Combinatorial Identities, Lattice Points in Specified Regions, Riemann’s

Zeta

Function, Dirichlet Series and otherseriesexpansions.

1991 AMS

SUBJECT

CLASSIFICATION CODES:

05A19, llP21, llM06,30B50.

1.

INTRODUCTION

Theconceptofvisible-from-the-originlatticepointsin the

X- Y

planewasdescribedin

Apostol

[2],

where theirdensityin thespacewas calculated. Visiblefromtheorigin latticepoints have the

propertythattheircoordinates are relativelyprime. Recently,ahostofnewidentities werefound,based onsummation orproductstakenonthese points

(see

Campbell

[4],

[5],

[6]).

We presenthere someof the morenotable results fromthesethreepapers,thenextend themtohigherdimensionsgiving general theoremsaccordingly.

Firstly,wehave thethree"companion identities,"socalled becauseoftheirinterdependence,and

theirrelation to the visiblefromoriginlattice pointsinthefirstquadrant. Weusetheoperatorsdefined by

where9,isthesetofpositiveintegers less than andrelatively primetok. Thethree"companions" givenin

[4], [5],

and

[6]

are

(1-y),Hi(1

x:y*)

/

(l-Y)

1-)

(x,

1)

(1.1)

1-xy

1-1/1

)1/(1

-x)

(1-xy),Fl,(1-x

y

=(1-xy

(1.2)

0 x)

(1-y)(1-xy)I-I.(1-xy)’/(1-x

y

=(l-y)

t/tt-(1.3)

where

convergence

occursrespectivelyfor

xy

aridly

[<

1,

ix

aria

Ixy

I<

1,

Ix

and

ly

l<

1.

It

is worthnotingthat theleft side of

(1.3)

canbe rewritten

)l/b

(l-y) H

(1

x"y

(.,b)-a,b

It

is clear that the identities

(1.1),

(1.2)

and

(1.3)

belong naturally

togetherwhen we see that

any

one of themimpliesthe othertwo. The reader caneasily verify byiterativemultiplicationon

(1.1)

that

xy,

x2y

substitutedfor

y

leadsto

(1.3).

(1.3)

then divided on both sidesby

(1.1)

givesnext

(1.2).

If y/xis substitutedfor

y

in

(1.2)

weobtain

(1.3).

Later

we shall prove generalizations of

(1.1)

to

(1.3)

both in two dimensions and in higher dimensions. The "companion identity" idea isnot restrictedtojust the aboveformulae.

In [6],

for example,the newproductstaken overprimenumbers

p

weregivenasfollows

n),I-l(jm

+kn)

#’

I-I

I-P4"/"))

,

-p-"

(1.4)

m

+n

),IIi(km

+in)

0

li(l

_p_+,,

/,,))-u-,-’),

(1.5)

_,)-11-p-)

(1.6)

n

)m

+n

),Flijm

+ kn

)u’lon

+jn

)u

li

(I

p

validwith

p"

,

forrespectively

Re

m + n and

Re

n>l,Re mand

Re

m +n >l,Remandren >

I.

Theseidentitieswerederivedfrom

(I.I)

to

(1.3)

bysubstitution of x

-p-’,

y

-p-",

andformingthe

product

overprimes

p

oneach side.

(1.4)

to

(1.6)

ledtoresultsconcerningthe

multiplicative

function

co(re,k)-

{-I

1/(l-P2)

with k-lip,

-1 a -1

Aswiththeproducts

(1.1)

to

(1.3),

weshallgive generalizedversions of

(1.4)

to

(1.6),

or atleast methods fordoingso.

Theunderlyingtheoremsfor theproducts

(1.1)

to

(1.3)

were

THEOREM

1.1. If

(a,)

and

(b,)

are

sequences

of functions chosen so thefollowingseries

converge,

then

1-exp(bkq)

E

Sk,

(1.7)

,.a,

1-exp(b,q/k)

k-1

where

Sk

isdefinedby

ai, k- 1,

(1.8)

Sk.

jl

[.i

,a#,,,

e,,,

exp(bi,hq/k)

k>l

(1.9)

and

,

isthesetof

sitive

integele than andrelatively primetok.

For

a

proof

of is see

mpbell

[4].

(1.1)

derives

om

e

caseof

e

theorem where

a,

y*k

-,

b,

k

logx

q-.

condly,

a finite vemionof

eorem

1.1 was given in

mpbell

[5].

EOM

1.2.

(a,)

and

(b,)

are

arbitra

sequences

thorn so

thak

mgeer

with choiceof x,thefollowingnctionsareall defined then

1-exb)

[--]

a,

a + a.,exb.,jx/m)

(1.10)

1

exb/k)

where

In

denotesthe

eatest

integerinn.

questionsassociated withthe substitutedfunctionsas n ispermittedtoincrease. Either theorem could be used

(although

Theorem 1.2wasappliedin

Campbell

[5])

toobtainnewidentities for Jacobi theta functions. Thediversity of, say

(1.10),

is that ityieldsalso

proofs

of classical results such as

o_s(n

)(s

+

1)-

,

c(n

)k

-l,

Re

s>

k,,l

duetoRamanujan,where

os(n

isthe sumofsth

powers

of the divisors of n, and

c(n

is themultiplicative function

q(n)-

X

cos Also,in

[5],

further results were obtained, such as

and,the rather unusualformula for

lq

<1,

in which

lyl<l,

exp

4

q

2kctsin2kl

1-O,(z,q)=O(z)--

2q’

X

(-1)q’tk

+’)sin(

2k+

1)z.

k-o

Evidently thereisscopefor much research on thesetypesof indentities, and whilst Theorems 1.1 and 1.2

may

haveeasilystatedgeneralizationsinhigher dimensions, oreven"companions"in thesense of

(1.1),

(1.2),

and

(1.3),

in this

paper

weshallspecialize

although

still inhigherdimensions, and obtain numerousnew andsimpleidentities.

2.

AN

INFINITE PRODUCT THEOREM IN N-SPACE

In

this section we give ageneraltheorem on multivariate infiniteproductsand a method forobtaining an infinitevarietyof similar theorems. The drawback from thisisthatmostof the other obtainable theoremswillrequireextraefforttoworkout. The ideasregardingcompanion identities as stated in

1

may

help

us to simply determinewhole classes ofinterdependent formulae in higher dimensional Euclidean

space. We

usethe notation

(a,a

,an)

tomean "the greatestcommon divisor ofall of

at,a2

an

together."

It

will beimportanttodistinguishbetween this and the ordered

n-tuple

givenby thevectornotation

(al,

a2,

...,an).

In

eithercase we will be concerned with latticepointsintherelevant

space,

henceanyvector orgcdwillbe overintegercoordinates.Theinfinite

product

(1.3)

isaparticular case of the

THEOREM

2.1. If 1,2 ,n, then for each

Ix,

<1,and b

C,

bit

FI

(1 x:’x2...Xn’)

"’

2

""

(2.1)

(al,a aa)-afZ

-exp,

lit,.)

provided

,-1

DEFINITION

2.2.

Any

Euclideanvector

(al, a2,...,a)

forwhich

(al,a2,...,a)-

wecall a

visiblepointvector, abbreviatedvpv.

LEMMA

2.3. Consider aninfiniteregionrayingoutof theorigininanyEuclideanvector

space.

Thesetofall latticepoint vectorsapart from the originin thatregion isprecisely thesetof positive integer multiples of the visiblepointvectors

(vpv’s)

in thatregion.

PROOF.

Eachvpvwillhaveintegercoordinateswhosegreatestcommondivisor isunity. Viewed fromtheorigin,all other latticepointsareobscured behindthe

vpv

endpoints. Ifxis a

vpv

in theregion then all vectors in thatregionfrom theoriginwith direction ofx preservedareenumerated bya

sequence

lx,2.x, 3x and thegreatestcommon divisorof thecomponentsof nx isclearlyn. This is because if the scalarn isnon-integeratleastoneof the coordinatesofnx would benon-integer. Therefore,if the

vpv’s

in theregionarecountablygivenbyxt,x2,x3 thenall lattice pointvectorsfromtheoriginin theregionare

lxt,

2x,

3x

lx:, 2x:,,

3x2,..., lx3, 2x,

3x etc.

Completionof theproofcomeswithrecognition that thesetof all

vpv’s

inany"rayedfrom the origin" region inanyEuclidean vectorspaceis acountableset. Proofof this last assertionisbyinduction on

thedimension, knowingthat the latticepointsare countable in

any

twodimensionalregion.

As

we count each latticepointvector in the desiredregionwe decide whetherit isa

vpv

simply by observingwhether its coordinates arerelativelyprime asawhole. End ofproof.

PROOF OF THEOREM

2.1.

We

startwith themultiplesum

which, dueto

Lemma

2.3,alsoequals (lettingb

,.

b,)

",.’

....

,.)-

1;

+ ₂ + ₃ + _{4 a a}b

...a,,"

y

’

"

’-1

log(1-

xlx

’ .x’)

(aa,a- .,a,,)- _{al a2}

...an"

if b-1.

Exponentiatingboth sides thenyieldsTheorem 2.1. End ofproof. The caseswithn 2and n 3 are stated easily,in theforms

FI

(1-xy)--’-’-exp

.

?

s+t-l"

(2.2)

(a,b

H

(l--xybzC,-"-’b-’c-’---exp{

X’

(a,b,c)-I

i-l?

1--

"

s+t +u -1;

(2.3)

where

Ix

l,

Y

l,

z <1,ands,t,u,arecomplexnumbers.

Thecaseof

(2.2)

with s 0 isequivalentto

(1.3). In

fact

(2.2)

is acompanionto aresultgivenin

Campbell

[4],

namely,forpositive integersm,

(R)Y’

(ml

iitx’)

(xY)i

g,,,(x)

d--(1

-e:)

-1

(2.5)

Iogx

then

(2.4)

holdsfor

xy

and

ly

l<

1.

Weshallinvestigatesuchcompanionidentities inthenext section.

As

indicatedin 1, such results

canimplya_{large range}ofnewandcuriousidentities, some ofwhichwerealludedtoinCampbell

[5].

However,

the classical methods ofasymptotics of infiniteproduct generatingfunctions asgiven in

Andrews

[1, Chapter 6]

will applytoeachof thenumerousinfiniteproductsstated andimpliedhere. Lemma2.3isalsoapplicabletomanyof thelattice sumsgiveninGlasserandZucker

[15]

andtothe

lattice sumspresentedwithapplicationstomathematicalphysicsas giveninFrankel, Glasser,Hughes

and Ninham

14].

ThelatterpaperappliessimpleMObiusand Mellininversionstolatticesums,obtaining selectedapplicationstoelectrostatics and statisticalmechanics.

In

thesequel here,we

may

also draw

parallel

conclusionstosomeof the ideas stated in Andrews

[1,

Chapter

12]

describing

partitions

into

vectors.

In

thepresent

paper

we haveexamplesof

"weighted"

vectorpartitions,andquiteadiversity of results.

The simplicity ofourproofofTheorem 2.1depended largelyon the factthatourchosen "raying fromorigin" regionwaslhe firsthyperquadrant,that is, theregionhaving allpositiveintegercoordinates inthen-spaceconcerned. This meant wehadnodifficultyinseeingthat themultiplesummation was simply aproductofsummationsof the samekindon the different variables.

For

other choices of"rayed fromorigin" regionsthis isclearlynotsuch astraightforwardmatter.

It

is not,however,animpossible

matter.

To

demonstrate this, weapply

Lemma

2.3totheregionin

3-space

defined

by

<X

<

Y

<

Z,

noting thatitraysoutof theorigininthemannerrequired. The elements of thesetofvectorsfrom the origintothe latticepointsin thisregioncan be easily writtendown,and arecountable, aswewould expect. Thus we have

<1,2,3>,

(I,

2,

4), (I,

3,

4),

<2,

3,

4),

<I,

2,

5>,

<I,

3,

5>,

<I,

4,5>,

<2,

3,

5>, <2, 4,5>, <3, 4,5>,

<1,2,

6>, <1,3, 6>, <1,

4,

6>,

<1,5, 6>, <2,3, 6>, <2,

4,

6>, <2,5, 6>, <3,

4,

6>, <3,5, 6>,

<4,5, 6>,

and so on.

A

suitablemappingfrom thesevectorsresultsinthe summation

,

x’ybz

c-I

(2.6)

O<a<b<c a,b,c

which

formally

reduces to

z2/’

xy2(ll-Yt-x

1-x’Y’l

,-1

2+k 1 x

y

-"xy

which

converges

for z

l,

Y

z

l,

[xyz

I<

to

(for

non-zero

denominators)

--yz

+.xyz-+

log(1

yz)

+

(1 -x)(1

-y)(1 -xy)

(1

-x)(1

-y)

( -x)(

-xy)

log(1

x

y

z

(1

-y)(1 -xy)

log(1

z).

(2.7)

(2.8)

sum

E

xay

bZcC-1

(a,b,c)-O<a<b

a,b,c

convergent for thesame conditions asthe sum whichled to

(2.7).

ApplicationofLemma 2.3 then transforms

(2.8)

into

(2.7),

muchin thewayweproved Theorem 2.1. This leadsus to

THEOREM

2.4. If

x,y,xy

1, and z

I,

YZ

[,

[xyz

[<

1,then

FI

(1-x’ybzC)

-/c

(1

yz)"

...

(1

xyz)

’’-’’r

....

’(1

z)"

"G(x,y,z),

(2.9)

(a,b,c)-O<a<b<c

a,b,c

"

where

yz

G(x,y,z)-exp

-xyzl2"xyz

+-xyz

(1

-x)(1 -y)(1

-xy)

Althoughthistheorem looksquite impressive tothe reader newtosuch ideas, itisclearly only comparableto

say,

acorollaryof Theorem2.1whenapparent

depth

anddimensionalityare considered. Theproblemsfaced intryingtomakeTheorem 2.4acorollaryofTheorem 2.1involve: firstly,howto dividethehyperquadrantregionintosubregions,oneofwhich istheregionrelevanttoTheorem2.4; secondly,tothen find theappropriatetransformsothatnotjusttherightregion is covered, but that the

index"-l/c"in theleftsideproductisobtainable.

In many

casesitseems morepracticaltochoose a regionandapply

Lemma

2.3,butthere will be situations worthexaminingwhere thehyperspacedivides naturallyintoregions yieldingcompanionidentities.

In

ordertoclassifyallsuch identitiesit maybe necessarytouse

group

theoryand isometrics.

So

itseemsthat

(2.9)

is not inanyobvioussenseacompanion identityfor

(2.3),

norforacaseof

it. Similarly,this

appears

sofor thefollowing result,obtainedfromsummingonthe latticepointsin theinvertedright

square

pyramid

IX

and

Y

]<

Z

in the

X

Y Z

space.

THEOREM

2.4. Ifx, y,l-x,1-y,areallnon-zero, and each

oflz

l,

xz

I,

Iz/x

I,

lY

z

l,

Iz/Y

l,

IxY

z

I,

Iz/(xY)l,

Ixz/Y

I,

lY

z/x <1,then

(1 -z)(1

-xyz)

’

)"’-;)"-"

(1

-xz):(1

yz)

FI

(1

-xybzC)

/

(2.10)

(a,b,c

la IbI<cZ,"

a,bz,

Thisresultsuggeststhe existence of ananalogueof Theorem 2.1 where summation is taken over the

vpv’s

and latticepointsin theappropriatelydefinedhyperpyramidin Euclidean

n-space.

Clearly

thehyperquadrant

appears

not toeasilytransform into thehyperpyramid,and viceversa, thus atotally differentsetofcompanionidentitiesareimplied. There are alsoproblemsrelatedtothe consideration of thebranchingeffects of thecomplexrootsinvolved.

For

example,inexactlywhichsense do both sidesof

(2.10)

satisfythe functionalequation

f(x,y,z)

f(1/x,

I/y,

I/z)?

3.

NATURAL

CASES

AND

A FURTHER

N-SPACE THEOREM

There are natural andsimplecases ofTheorem 2.1toconsider.

Let

usfirstenlargethetheorem’s positivecoordinatehyperquadranttoincludelatticepointson each axisexceptfor thehighestornth dimension. Thuswecaneasilyobtain thefollowing

COROLLARY

3.1.

l/b

I-I

(1

-x"y

(1

y)

-x)

(3.1)

(a.t,) a,O,b,l

FI

(l-x’y’z)

/

=(l-z)

I/-’)-))

H

(I

v"xyza)

TM

(I

z)(-’)(-’)(-’)),

{a.b.c.d) a.b c=.O.d,l

H

(I

-vx’y’)

/

=(I

))(-’)(-’)()

-)(-’),

( b.c.d.e)= a,b,c.d O,e

eachvalid whenabsolute values of u,

,

x,y,andz are all less than unity. The

n-space

veionisthen

COROLRY

3.2. Iffor 1,2

n’lx

I<

,h

’

.x._z

=(l-z)

-’-’

H

-x x

(a,a..,a)-al.a2. ,an_ I0

al

At

this

im

wecanapply

mma

2.3to

rollary

3.2.

In

other words we gel ridofthe criterion onthe left sideproductfrom

(3.5)

that

(a,a2

a,)

I. We

shallratethe resultofthisas atheorem

sincethesubsequentcorollaries willbeof adistinctlydifferent kindfrom our earlier identities.

THEOM

3.3.

For

thesame conditions as

(3.5)

we have

H

(1-x

_{x2 ...x.[-z}

H(1-

(3.6)

a, a._=0 k=

PROOF.

Iftheleftsideof

(3.5)

isdefinedby

f(x,x

x

,z)

thenby

mma

2.3thistransforms

intothe leftsideof

(3.6)

by forming

k)l

Applingthe meoperationtotherightsideof

(3.5)

ields

therightsideof

(3.6).

End of

proof.

We

canllow in

eorem

3.3loobtain someimerestingresulm. Firstly,if

x,

-p

with p denotingthe ithprimewehave

COROLRY

3.4. If

Re

s>

and[z

[<

,,

(1

-xa)

(3.7)

Particularcasesofthisleadtothe Nrther

sin())

v

(1

(3.8)

H

(co*))

v*

H

(I

z)

4x

(3.9)

(sinh(’)

2

sinh(’)

co’))

la

Z6K)6K

n

(1

+

(3.o)

4z

The latter of these results isaffected with the aid of thefollowing product apparently due to

Ramanujan.

If xisanynon-zero

complex

number

x

sinh(2)-

2

sinh()

co)

1+-

4x3

(3.3)

(3.5)

Againemployingthe caseofTheorem 3.3with n and thistimeletting

x,

x’

wecanobtain

new relationsinvolving

p(n),

the number ofunrestricted partitionsof n.

It

iswellknown that

(see

Andrews

])

?i

Ix

I<

-0 k-I

Likewise, if

po(n

isthenumber ofpartitionsof nintooddparts

po(n)x

17I(1-x2t-)

-l’-go(x),

Ix

l<

1.

(3.12)

-0 -1

COROLLARY

3.6.

U

lx

and

lz

<

rl

(1

-xz’)

’)/

FI (1

-xX)

x

(3.13)

/,0 k.l

k:,l

1"I

(1

-xz)

’)-

F1

(1

-x)

’?

(3.13)

/a0

k,l

Clearly

thereisscopefor further researchintoidentitiessuchasthese, especiallygiven thealready existingwealthofmaterialon partitionidentities.

An

alternative versionof

(3.13)

for the sameconditions is

I’I

(1-xz*)

e)/’

l’I

I-I

(1-xa)

’(’/a)/a?

(3.15)

l,k,l k,l d]k

Another naturalconcept arising fromthe above workis

p.(j),

the number ofpartitions ofjintopartsat least oneof which isrelatively primetok.

In

other words

(a,a

a,,k)

1, orequivalentlyan

a

from and theform taken for such apartitionofjisthen

j

-Ala

+A2a

+

+A.a.,

(for

any m

1)

where the

A

are positiveintegerswithoutrestriction. Setting

pc.(0):--

1 andemployingour definition, thenallowingn in

Corollary

3.2 setting

x

-x,theresultfollows that

COROLLARY

3.7. If

Ix

land

Iz

l<

thenfor

x)as

in

(3.11),

),,o,,

n(1

x’z’

--(1-

(3.16)

io

kl

4.

COMPANION IDENTITIES AND VISIBLE POINT VECTOR PARTITIONS

In

the bookon Partitionsby

G.

E.

Andrews 1,

Chapter

12]

itisstated that

"we

donothave a device likeEuler’s pentagonalnumber theoremfor therapidcalculationof

any

multipartitefunctions." The preceding sectionsofthis

paper

anditspredecessors

[4

to

6]

indicatedevelopmentsin this direction, duetotheofteneasy expansionof the functionsconcerned,

e.g.

_{The right}sides of

(3.2)

throughto

(3.6)

areeasily expanded as

power

series in z. Andrewsmentionshow Carlitz and others

[7

to9, and 11,

12]

consideredproblemsof restrictedmultipartite partitions.

In

particular, k-partite partitions

(nl,

n

n,)

(m,i,mi

1-1

subjecttothedecreasingcondition

min(m,,

rnz,...,

rn,)

max(mL

,mz.

,

...,m,.

)

It

isshown in

[1]

that if

l-I(n,m)

isthe numberofpartitionsof

(n,m)

intosteadilydecreasing partsthen

E

H<n,m)x"y"

FI

(1

-x"y"-’)-(l-x"-’y")-(l

-xZ"yZ")

-n.m,O n,l

From

this ittbllows easilythat if

I-l<n,m>

isthenumber ofbipartitepartitions of

<n,m>

inwhichall parts are of the Ibrms

<2a,

2a

>or

<a

1, a or

<a,

a

1>

then forall n and m,

l-I<n

,m l-I

<n

,m

>.

Andrews

states thatthis resultisofadifferent kindthanthose arisingfrom thesimpleextensionsofpartition

idealsof order considered inhisbook.

In

thelightofourwork inthis

paper

itnowbecomes naturaltoconsiderpartitionsinto visiblepoint veclors

(vpv’s)

withinrayed-from-origin regionsofvectorspaces.

Our

notionof"companion identities," imprecisethoughitbeatthis state,duetoambiguitiesaspresented,willstillbeuseful whenweobtain

theoremsinvolving"the number ofvpv partitions"of various kinds ofvectors.

As

promised in 1,wenowpresentageneralizationof thecompanionidentities

(1.1), (1.2),

and

(1.3).

CONJECTURE

4.1.

Let

sand becomplexandsuch thats+ 1. Then

respectively

for

xy

aridly

I<

1,

Ix

and

]xy

]<

1,

Ix

and

ly

]<

1,

+

(4.2)

(1

-xy)erL(1

-xy)

-

--exp

-,.

(1

-xy),Fl,,(1--x"y)--’b-’(1

--xy

=exp

,.,

’’’7

(4.3)

whereit isunderstood that

any

divergent multilogarithmicpowerserieson therightsidescan bereplaced bythe function definedin

(2.5).

Partialproof.

(4.3)

isseentobeequivalentto

(2.2).

Similarly,

(2.4)

isthe case of

(4.1)

wherefor positiveintegersm, tn+1.

However, (4.1)

claimsconsiderablymorethan

(2.4);

namely,thatsand can be

complex

numbers.

(4.3)

isobviouslytruefor suchsand t,soitwould

appear

naturalfor

(4.1)

tofulfill theseconditions. Since

(4.2)

is asimple quotientof both sides of

(4.3)

over

(4.1)

it remains onlyfor ustoestablishthat

(4.1)

is trueforcomplexsandt.

In Campbell

[4]

the

proof

of

(2.4)depended

onan ruth derivative withrespecttoz logxof

f(n,

y

+

,,

f(n,x

y’)k

(f(n,

y

f(n,xy

))/(1 -,x

(4.4)

wherentE

C

and

f(n,y)-

E

y’n-"

with y

l<

1. Thisledto

k

f(n

+

k,xy)g+

(x),

(4.5)

for non-negative integersm and

g,(x)

asin

(2.5).

Allowingn in thisyields

(2.4)

and hence

(4.1)

and

(4.2)

for the values tn +

Ix. To

fully prove conjecture4.1 wouldrequire

complex

iteration on thederivativeprocessapplied aboveto

(4.4).

End ofpartial proof.

It

seems worthnotingat thisstage,having justfailedtoprove conjecture 4.1 completely,that under the conditionsspecifiedthere,

Lemma

2.3whenapplied directly gives

,H,,(1-x"Y’)-"-’’-’-exp

1-+

"

2,)

+

+2’+

7

+.-.

.)4-’,-’

y

y

x y

y__

(l-xy)H,(l-x"y

--expxy+

and of course,

(4.3).

Thisappearsto addfurther weighttothe likelihoodof conjecture 4.1 beingtrue.

It

also shows that the summationsexponentiatedon therightsidesof

(4.6)

and

(4.7)

are eachvaluable in termsof a finite sum ofmultilogarithms

(or

polyiogarithms, see Lewin

[16],

and

[17])

whenever tn+ fornon-negative integerm, with s + 1. Thiscouldnodoubt be

developed

further inalater

paper.

At

this stage, let usreturn tothe idea ofvectorpartitions.

More

specifically,weconsider

vpv’s

in the firstquadrantof the

X

Y

plane. Basically,the

vpv’s

can be divided into threeclasses:

a)

those above the line

X

Y,

theupper

vpv’s,

b)

those belowand on theline

X

Y,

thelower

vpv’s,

c)

thoseof the entirequadrant,that is, all

vpv’s.

Weshalltherefore lookat

U(n, m), l.,(n, m), A(n, m);

the number of partitions of

(n,m)

intorespectively,

upper,

lower,andall

vpv’s.

UsingthetechniquesinAndrews

],

itisasimplematter toshow that

+

U(n,m)x’y"

,FI(I -xy’)

-

(4.8)

+

E

ln,m)x"Y’-(1-xy)-,Fl,(1-x*Yi)

-,

(4.9)

+,,.,.

A<n,m>x"y"

(1

xy)-Fl(1 x’y*)-(1

xy)

-

(4.10)

valid for the samerespectiveconditions onx and

y

asimposedon

(1.1),

(1.2)

and

(1.3).

Each sideof

(4.8)

satisfiesthe functionalequation

f(x,y)= f(1/x,xy),

(4.11)

provided

f(1/x,

y

is firsttakenbefore proceedingsto

f(1/x,xy),

theoperations beingnotcommutative. The coefficients of

y"

in

(4.8)

arethepolynomials

m-I

Y.

U(n,m)x

(4.12)

n-I

It

therefore followsnaivelybutsignificantly that,

THEOREM

4.2.

For

allpositive integers n,m,

U(m-

n,m>

U(n,m>.

Thereasoningbehindthistheorem canequallybeappliedtonumerousotherresultsinvolving

vpv

identities.

For

example,

(4.9)

and

(1.2)

bothsatisfythe functionalequation

1-x

f(x,

y)

-"x

f(xy,

1/y)

(4.13)

where inthiscase

f(x,

1/y)

precedesarrival at

f(xy, 1/y). For

thisreason weformalize thefollowing

DEFINITION 4.3.

Let

p

bea mapping appliedto

U(n,m), L(n, m), A(n,m),

respectivelytoyield

p<n,

m>,

pt<n,

m>,

p.<n,

m>.

If

+

E

<p..,m>x"y"

F(x,y),

(4.14)

+

E

<Pt,

m>xny

=F,.(x,y),

(4.14)

"I-

n,,

(0,,

m)x"y"

F(x,

y)

(4.14)

then

F(x,

y),F(x, y),F(x, y),

arerespectivelycalled

"upper vpv",

"lower

vpv",

"all

vpv"

functions; and likewise

(4.14), (4.15),

(4.16),

called

"upper vpv",

"lower

vpv",

"all

vpv"

identities.

F(x,xy)

FL(x,

y

(4.17)

F^(x,

y

,

Fu(x,xy)

(4.18)

F(x,

y

)F(x,

y

F(x,

y

(4.19)

In

otherwords, if givena suitable

"upper",

"lower",or"all"vpv identity,we canmanufactureasetof companionidentities.

(1.1),

(1.2)

and

(1.3)

are essentially anexampleof this. Rather than labor this point, wesimply presentthe"all

vpv"

identity resultingfromapplying

(4.18)

to

(2.4).

THEOREM

4.4. If

Ix

and

y

[<

1,

g(x)

asin

(2.5),

andhisany positive integer,then

y,

h

l(

FI

(1-x’y")

a"/"

-(1-)exp-g,(x),E

j,

’(1

x’)

+

g,(x),

j,.

l_xi

a,b* d(b)

Thistheorem isanalmostunrecognizable generalizationof

(1.3),

which wouldcoffespondto

e

caseh 0.

It

is distinctfrom the othertypesof "all

vpv"

identitiesgivensofar.

e

caswithh

andh 2are,with

f(n, y)

asin

(4.4),

H

(1

-xy)

/

exp{xf(E,y)(1

-x)-2

(1

-xy)-

-)

(4.21)

(a.b)-a,b

H

(1

-x’y)a%’-exp{-(x

+

x:)(1

-x)-3f(3,y)

x(1

-x)-:f(2,y)}

(1

-xy)-t-)

(4.22)

(a.b)-a,b

Yet

another method

(and

rhaps the most

obvious)

for manufacturing related

"companions"

involvessimplythe

o

equations

Fx,y)-F(,y),

and

F(x,y)-F(x,).

(4.23)

Whenusingthismethodanadjustmentmaybe madeto

e

firstteofanidentitytotidy ings up. Applying

(4.23)

tothe "all

vpv"

(2.2)

givesthe

"upper"

and"lower" vpvidentities, evenough

ey

aretrivially interdependent.

cononnv

4.s.

u

and are=ompl=xand +

,

hit and

Y

< for

(4.Z);

and

l

[<

for

(4.25),

then

,H,(I-xly’)-’

"-i’

exp{(i,)

(i,)

t

(4.24,

(4.)

j’

/j"

is

corolla

hi.lightsthe

ssibilities

forwritingdown differentelatesof companion identities, each tof "companions" having often strikinglydifferent

appearance,

evenwhen

o

distinctcompanion

COROLLARY

4.6.

For

respectiveconditionsonxandyasin

(1.1),

(1.2),

and

(1.3)

wehave

(1

+

y),l-lj(1

+xy)

/

(1

--xY-f-(1-

+

Y)]

vt-’)

(1-y)’(l+xy)/

1/(1 2)

(1

+

xy

),FI.,(1

+

x’y)

l/j

..x..y

(1 -xy)

1/(1 2)

(4.26)

(4.27)

_/ l+y

(1 +y)(1

+xy),I-li(1

+x’y’)l/’(1

+x

y

--,

(l_y)

Applicationof the same methodto

(2.10)

yields:

COROLLARY

4.7.

For x, y,

andzas in

(2.10),

FI

(1

+x=yz)

/

(1

+z)’(1

+xyz)b(1

--X)’(1

--yz)

(.b,)-

(1-z)’(1-xyz)r(1

+xz)(l

+y)

a,b

where a-1,

b-x2y

,

e-x2y(1

+x)(1

+y)-x

2,

d-

y2x(1

+x)(1

+y)_yZ,

e-(1 +x)(1

+y)-1,

rxy(l

+x)(

+

y)-xy

,

gx:,h-

y

.

It

should be cleartothereader that

(4.29)

can,like

(2.10),

be derived

om

firstprinciples by taking analtematingsum on thelatticepointsand

vpv’s

in therayed-from-origin regionboundedbythe inveed right

square

defined in

X

Y

Z

space

by

Xl

and

Y

I"

Z.

In

mpbell

[4],

similarsuggestionsforobtainingcorollariesweregiven. otherneat

corollary

isobtainedby swappingsand in

(4.25),

thendividingboth sides of this into

(4.24),

sothen

COROLRY 4.8.

n

sand are

complex

with s+ 1,

Ix

and

ly

<1,

H{

l_xy

"t-r’

x’y

exp((,

(

(,x’

..k, )).

4.30

e

identitiesof this section show that

many

of thesimple propertiesof thecompanionidentities

appear

tobe wohformalizing.

For

example,

(4.30)

indicatesthat thequotient ofan

"upper

v"

with a"lower

vpv"

identity

yields

(or

is

expected

to

yield)

an"all

vpv"

identity.

We

continue this sectionby givingasetofcompanionidentities in three dimensional

space.

In

thisinstance,weconsiderthe firsthyperquadrantof the

X

Y

Z

plane including

X

and

Y

axeswhilst excludingthe

Z

axis. fact,if the first

hyperquadrant

istreated as aninfinitely extended cube,then the identities here

coespond

tothe latticepointsin:

a)

theextended tetrahedron boundedby

X,Y <Z,

b)

theextendedtetrahedronbounded

by

X,Z

Y,

c)

the extended tetrahedronbounded

by

Y,Z X,

d)

the entirehyperquadrantwhich isthe unionof

a), b),

and

c).

TEOM

4.9. ffa andb eachsum overthenon-negativeintegersand c thepositive integers, suchthatfora,b>0,

(a,

b,

c)

1 then

H

(1--xaybgC)

-/

((1-xz)(1-yz))

1/((1-x)(1-y))

,,b,,

(1-z)(1-z)

(4.31)

H

(1

x’yz)

-/

((1-z

/(-)t-)

(4.32)

....

(1

yz

(4.28)

II

x"

y’z)

-/"

z

)-/o

-,,) -,)

(4.34)

a,b,c

where the conditions for

convergence

arerespectively,

lyzl,

Ixyz

<

Il,

lyl,

with variables chosen for non-zero denominator.

PROOF.

In

the usualmannerwe consider thefourlatticesums:

a,b b b, a,b,

The last of thesewehave already givenin

(3.2)

asacaseof

(2.3)

basically.

e

second and ird of theseareeentiallythe me as eachother, by interchange ofxand

y.

ey

easilyreduceto

e

logarithm of therightsidesof

(4.32)

and

(4.33)

bye of the summation

(I

u

and[z

I<

1)

"( z’)

-1 as

pan

ofthe

proce.

e

first lattice sum is

((11

-)(1-.)

(4.35)

z

(1-x

*)(1 y)

log

-

(1-x)(1-y)

"(1-x)(1-y)

-z)(1-z)

whichisthelogarithmof therightsideof

(4.31).

Now

eachof the four lattice sums considered

coespond

tothe aforementioned

"infinitely

extended" tetrahedra which fittogethertomakeupthe first

hyper-quadrant.

such,each tetrahedron is a

"rayed

om

origin" region

as

ains

to

mma

2.3.

erefore,

applying

mma

2.3toourfourlatticesums, we have forexampleused the first sum that itffansfos into

(x"y"’)’/()-

(-1/)og(-x"y"z’),

(a,b,e)-

(a,b,e)-and is establishes

(4.31).

Similarreasoning appliedto

e

other

ree

latticesumsgives

(4.32 (4.33)

and

(4.34).

Endof

proof.

ere

is aninterestingresemblance between

eorems

4.9and2.4.

e

sum

(4.35)

shows

at

identities like

(4.31)

caneasilybegeneralizedto nimensional

space. Using

sums like

(4.35)

e

next few s are

,.,.,,

(

z)(

)(

)(

_)

(4.a)

FI

(1-

v"w

’x"y

az

")

-/"

a,b,c,d

(a,b,c,d,e)

a,b,c,da0;

(1

vz)(1

wz)(1 -xz)(1

yz)(1

vwxz)

(1

vwyz)

(1

vxyz

(.1.

.-_

wxyz!l

...

"

(1 -z)(1

vwz) (1

vxz) (1

vyz)

(1

wxz) (1

wyz)

(1

x’yz)

(1

vwxyz)

]

(4.37)

where

G (v,

w,x,

y)

((1

v) (1

w) (1

x) (1

y))-l"

(4.36)

and

(4.37)

needtheabsolute valuesofeach ofz, vz,wz,xz vwz vwxz

vwxyz

tobeappropriatelyless thanunity.

THEOREM

4.10. If

Xo(xt,x2

xn)

and

X,(xi,x

xn)

arethesetsofrespectivelyoddandeven number ofproductsof dissimilarxi,

...,x,,

then for 1,2 ,nwehave

(assuming

X,

actingonthe

empty

set)

’’

’

.x,’z

)-/"

rI

l-x

x₂

ex.l-

(4.38)

(at.a

.a,)-ai.a, ,a,,0,a

validfor absolute values of "each elementproductof

Xo

and

X,

multipliedby

z"

tobe lessthanunity, and also z

l<

1. Alsononeofthe

x,

areunity.

Thistheorem has aninterestingconnection topartition theoryif we permit ntoincreaseindefinitely. After this,letting

x,

x’,

pt(0)

1,

p,(0)

0(k

>

1)

I,-,’)(,.

2)(,

_,,)..

n

-x’z’

-"’’

(

(

-#’(-xzf’’"

,.0.l

(1 -z)(1

=’0(1

-x2z’...

(4.39)

where

pO’)

isthenumber ofpartitionsinto

paas

<k,atleast one

pa

from

1,

Po)

isthenumber ofpaaitions ofjintoan odd number ofdistinct

paas,

p,)

isthe number ofpaitionsofjintoan even number of distinct

pas.

However,

Euler’s

ntagonal

number theorem

(see

drews

[1,

pp.

10])

statesthat

Applyingthistotherightsideof

(4.39)

gives

COROLRY

4.11.

fflx

and

Iz

l<

1,

(4.40)

It appears

that the results of this ction arenotthe final word on thetopicofcompanionidentities and

vpv

paifions.

e

connectionwiththe

own

heo

ofpaitions highligMed

by

results such as the latter

corollary

also indicatesarichpotentialforNherdevelopment.

5.

EULER PRODUCT IDENTITIES AND MULTIPLICATIVE FUNCTIONS

Themethod outlined in

1

for

obtaining

(1.4), (1.5),

and

(1.6)

is,as surmised fi’om

Campbell [6],

capable

of

applying

toalmost allofthe identities in our

subsequent

sections.

Basically

here we revisit

[6]

andgivenatural extensions of the results of that

paper. We

startwiththe

DEFINITION

5.1. If

FI[.

jT’

istheprime decompositionofpositive integer kthenfornon-zero

ml, m2,

(5.2)

In [6]

the functionfrom

(5.1)

was showntonaturallyoccur in the Dirichlet seriesresulting from

(1.4),

(1.5),

and

(1.6).

For

example,

(1.6)

isequivalenttothestatement

FI

(j’m

+kn)

-la’

.

h(k)Oal(m,k)

Rern

and Ren>l.

(5.4)

o$, k

In

this

X(k)=

I-I;.

t(-1)"

isLiouville’s arithmetic function

(for

an account of this function see Chart-drasekharan

10]).

Given the results of ourpresentpaper,wecanderiveDirichletseriesgeneratingfunctions for the multiplicativefunctions ofDefinition5.1. Firstly applyingtheproduct operator

H

toeachsideof theidentities inCorollaries3.1 and3.2,wehave, letting each variable,sayx bereplaced by

p-*

beforeemployingtheproduct operator,

THEOREM

5.2. If

Re

tn and

Re

n > then

andgenerally

H

(arni

+bn)-/b=

Y.

k)tot(mt,

k)

O,b,,1 k-I k

(a,b)-El

(amt

+bin

+cn)

-t/*

,

X(k)(mt’m2"k)

a,b.0;c k"

(a,b,c)-I-[

(am

_+bin +cm

+dn)

-I/a

k(k)J3(tnl’m2"m3’k)

a,b,e.0;d

(a,b,c,d)

(5.5)

(5.6)

(5.7)

H

(arn

+brn:,

+cn)

-/

rI(1

-p-)/((-’"’)(-’-’))

a,b O,c p

(a,b,)

From

here wesimplyexpandtherightsideof

(5.9)

intotheappropriateDirichlet seriesleadingto

(5.6).

Thismethod can bejustaswellappliedtothe other identities in

Corollary

5.2. End of

proof.

Identities suchas

(5.9)

can be written down immediatelyfrom

many

of the identities already presented here;forexample

(4.36)

implies

(5.9)

onboth sidesgives

H

H(1

-p

a,b,0; p

(a,b,)-which isequivalentto

II(1-p-*)

El

(alm

+a:zm

+..._+ahm

+an)

-/

,

X(’k)ch(m’m2’""m"k).

(5.8)

a O’,a k

(at,a.2.,

I-I

(a.b.c,d)-a,b,cl O,d

(ami

+ bm.,+ cm +dn

)-t/a.

I-I(

(I -p-)(l

-p-"

+"

+))(I

p-"’

+"

/"))

’

(1

p-0,,, +"))

(1

p-"

+"))

(1

(1

p4,,,/,,,,

/,,))

(I

-p’+"’

*"

/"’

/"))

(5.10)

valid for real parts ofthe following: n,mt+n,m2+n,m3+n,

m

_{+ m2 +}n,

m

+m3 +n, m2+ m3 + n, mt+_{m2 +}m +n, allgreaterthanunity. Alsoml,m2,m3,n are_*0.

In [6],

summationssuch as

(5.5)

weregivensomeinterpretation, showingthat our function

cot(re,k)

occursnaturallyineach ofthecompanionidentities

(1.4),

(1.5), (1.6). In

orderto

properly

find inter-pretationsof the"prime

products" presented

itisnecessarytouse the

THEOREM

5.3. If

Re(ira

+

kn)

> forpositive integerk then

O’m

+

kn)a/

=,.

f,(r)

r,,,+,

(5.11)

thenwith

r)

the Liouville function,

(-1/kl=

r)

(-’/kl

(5.12)

.,(r)

1.1I1(-1

)’

a,

,.

a,

’

where

{I

p istheprime decompositionofpositive integerr.

For

a

proof

of this see

[6],

however the

proof

iselementaryand based on nothing more than the

binomialtheorem and theEulerproductformof the Riemannzetafunction. Theorem5.3holdsan

important keytointerpretingmanyof the identities of our present work.

It

isclear from

(5.12)

that

,,(r)

isreally onlyafunction oftype

f(r),

andsoforinstancethe function under theproduct operation

onleft sideof

(5.10)

is

(ami

+

bm2

+

cm

+

dn)-t/a

]’(r)

(5.13)

am

+bm + cm + dn

In

applyingthemethods of

[6]

tothehigherdimensional caseswestraight

away

runintoformal difficultiesunless we chooseappropriateidentitiestoexamine.

It

seems that the identitiesofCorollaries 3.1 and3.2 againarise, further supportingour assertion in

{}3

thatthey representa classof"natural

cases"

ofTheorem2.1. Although thej

of,,(r)

in

(5.12)

isseento besuperfluousweshall findituseful

tokeepas a means ofidentifyingthat this function arisesfor the(jm +

kn)

function.

We

willdistinguish thenbetween functionsgenerated by

(l"m

+

kn)

and

(j2m

+

kn).

Indeed,for thehigherdimensional cases we will introducefurther parametersto

any

.,(r),

makingit

say

f,,,:,i,(r),

when arising for

(am

+

bm2

+ cm3 +jm+

kn)/.

Thisallowsustoidentifythesource of the Dirichletgenerated coef-ficientsshouldtheyhave same values as other coefficient functions in theproducts.

At

thispointwe

giveourtheorem,whichessentially equatesDirichlet coefficientsof

(5.8).

We

firstwill needtogivea definition.

DEFINITION

5.4.Consider the latticepointarrivedatfrom theoriginbythevector

(ai,

a2,

...,a).

t(N (N

o

(m

t,

m

m,N

I-I

f,,l.,

(N<a’

_a2’

ah))N<al’

a2

ah)-"l’l

h-l"-l)

(5.14)

a,, .a_ 0

(a,%,

,a)-with the summatio over all solutionsof

al,, ,ah_ 0

ah

where allbut afinit,umberof the

N<a,

a

a

areunityandonlythenon-unityfactors are considered for

(5.14) (

istheMobius

function).

PROOF.

isssentially parallelsthe

proof

of thanalogousrsult

iven

in

[6].

W

start

om

the finiteproducton the

int8r

seuenc

(i),

1, j h,

)/’

This iseasily

proved

byinduction on s. All paaial productsontherightsideof

(5.14)

areofthis kind

sincethelattice

ints

inthehyperquadrantconcernedarecountable

(recall

ourproofof

mma

2.3in

2).

Ifk isany positive integerthe coefficientofk

"

in

(5.16)

isdeteined

om

those tes

r[

"

whichare factors of k

-.

otherwords,the Dirichlet

generated

coefficientof k

"

in

wherethe summation is over each solution ofk

H.

r,

’.

In (5.16)

we

may

choose the a

sequence

toconform withthe lattice

ints

in the

product

operators of

(5.14)

and

(5.15),

owingtothecountabilityof the latticepointsinthis

"rayed

om

origin" region.

Hence

the product operatorof

(5.1

transforms into that of

(5.14)

and

(5.15),

andwecanallow sto

increase indefinitely.

e

Mobius Nnction onthe leftsideof

(5.14)

isdue toour considering

e

reciprocal identityof

(5.8)

inordertobetterapply

(5.11)

totheanalysis. is is

e

formal

proo e

stipulationin the theorem that"allbut a finitenumber of the

N(a,a,...,a

are

unity"

simplymeans that,whilstanyteunderthe

product

operator

may

benon-unity,be it the millionthteor

e

first

teinthepaaialproduct,eachproductunder the summation in

(5.14)

is considered finite. Of coume theleft side of

(5.14)comes

om

thereciprocal identity

produced

om

(5.8)

ingMobius inveion. Endof

proof.

It

is important to undetand that

eorem

5.5 is essentially a multidimensional factorization theorem where each visible lattice

int

intheregionisallowed anarbitrary

sitive

integervalue raised

toa cemin

power,

and all

ssible

productsof such numbe areconsidered,

g

in

eorem

5.5werake the

example

where

N

is aprime,thecomplexityreducestoasimpleresult.

(5.14)

isbasicallya mul-tidimensional Dirichlet summationrepresentationofthe arithmetic Nnction on its left side.

e

case whereh 2of

eorem

5.5wasgivenas

paa

ofatheorem in

[6].

6.

CONCLUDING

REMARKS

thatafullclassificationofthe identities hereinwouldamount toaclassification ofsymmetricrotations inarbitrarydimensionallattices;notasimplematter. Nevertheless,thevpvideaexemplifiedin

Lemma

2.3 isapplicabletoany rayed-from-origin Euclidean region. Wehaveshown that classesof

interde-pendent vpvidentities existprobablyinanydimension,andthereappearstobeavarietyof methods for obtainingthese.

A

consistent theoretical approachtoclassification of "companion identities" would seemdesirableatthisstage,giventhatwehavenowopenedthedoortoasensible multivariateworld.

In

thispaperwehave refrainedfromentering deeplyintoparticular logarithmicderivatives of the products. Alsowehavenotfully exploredtheinteresting

convergence boundary

casesofmanyresults. Wehavenotyetfollowedup higherdimensionalanaloguesin the moregeneralsenseofTheorems 1.1 and1.2, whichleadus tonumerous results onRamanujan typearithmetic functions on the onehand, and Jacobithetafunctionresults on theother.

Future

work onvpv identitiesmayinvolvelatticesumsofchemistryaspresentedinGlasserand Zucker

[15],

applicationstothetheoryofpartitions,orsome of thevpvworkmayfindapplicationin

additive numbertheory. Thislatter seems a naturalapplication since additive functions are invariant

undersomeofthetransformsin thispaper. Assuch,vpvmethodsmaybe linked tosomeof theproblems given in Elliott 13,pp.330-341

].

ACKNOWLEDGEMENT.

Theauthor is indebtedto: ProfessorR.J.

Baxter,

who kindly arranged

Visitor

Status

and accessatAustralian NationalUniversity;Professor

G. E.

Andrews,whosethoughtful and positiveresponsehas beenacatalyst;

Dr.

Peter Forrester

whowasalwaysin therightmathematical place,besidesbeingasourceof encouragement.

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