Counting an infinite number of points: a testing ground for renormalization methods

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by  Li Guo

COUNTING AN INFINITE NUMBER OF POINTS: A TESTING GROUND FOR
RENORMALIZATION METHODS
LIGUO,SYLVIEPAYCHA,ANDBINZHANG
Abstract. Thisisaleisurelyintroductoryaccountaddressedtonon-expertsandbasedonprevious
5 work by the authors, on how methodsborrowedfrom physicscan be used to ”count” an infinite
1 numberofpoints. Webeginwiththeclassicalcaseofcountingintegerpointsonthenon-negative
0 real axis and the classical Euler-Maclaurinformula. As an intermediate stage, we count integer
2 pointson productconeswhere the roles played by the coalgebraand the algebraic Birkhofffac-

n torizationcan beappreciatedin a relativelysimplesetting. We then considerthe generalcase of
a (lattice)conesforwhichweintroduceaconilpotentcoalgebraofcones,withapplicationstorenor-
J
malizationofconicalzetavalues.Whenevaluatedatzeroargumentsconicalzetafunctionsindeed
2
”count”integerpointsoncones.

]
h Contents
p
- Introduction 2
h
t 1. Countingintegers 3
a
m 1.1. Approximatedsumsoverintegers 3
[ 1.2. Theone-dimensionalEuler-Maclaurinformula 4

1.3. Evaluatingmeromorphicfunctionsat poles 6
1
v 1.4. Thezetafunctionat non-positiveintegers 8
9 1.5. Conclusion 10
2
4 2. Countinglatticepointsonproductcones 10
0 2.1. Theexponentialsummationandintegrationmap onproduct cones 10
0
2.2. A complementmapon productcones 12
.
1
2.3. AlgebraicBirkhoff factorizationon productcones 13
0
5 3. From complementmapsto coproducts 15
1 3.1. Posets 15
:
v 3.2. Complementmaps onposets 16
i
X 3.3. A complementmapon convexcones 18
r 3.4. Coproductsderivedfromcomplementmaps 19
a
4. AlgebraicBirkhofffactorization onaconilpotentcoalgebra 22
4.1. Theconvolutionproduct 22
4.2. AlgebraicBirkhoff factorization 23
5. Applicationto renormalizedconical zetavalues 25
5.1. AlgebraicBirkhoff factorizationon cones 25
5.2. Meromorphicfunctionswithlinearpoles 26
5.3. Renormalized conicalzetavalues: thecaseofChen cones 27
5.4. Discussionsand outlook 30
References 31
Date:January5,2015.
2010MathematicsSubjectClassification. 11M32,11H06,16H15,52C07,52B20,65B15,81T15.
Key words and phrases. cones, coalgebra, renormalization, Birkhoff decomposition, Euler-Maclaurin formula,
meromorphicfunctions.
1
2 LIGUO,SYLVIEPAYCHA,ANDBINZHANG
Introduction
”Counting” an infinite number of points might seem pointless and a lost cause; it has never-
theless been the concern of many a mathematician as far back as Leonhardt Euler and Bernhardt
Riemann andrelates to renormalizationissuesin quantumfield theory.
We want to ”count” lattice points on rational polyhedral convex cones, starting from the one
dimensionalconeR with latticepointsgivenby the positiveintegers studiedin the first section.
+
Evaluating the Riemann zeta function at zero provides one way of ”counting” the positive inte-
gers. It indeed assigns a finite value ζ(0) = −1 to the ill-defined sum ” ∞ n0” by means of an
2 n=1
analyticcontinuationζ(z)of theregularized sum ∞ n−z. AlternativelyPthe ”number” 1 = 1− 1
n=1 2 2
of non-negative integers can be derived using an alternative approximation S(ε) = ∞ e−εn by
P n=0
anexponentialsum. Itsanalyticextension(denotedbythesamesymbolS)presentsasimplepole
P
atε = 0withresidue1sothatS(ε) = 1 +S (ε)whereS isholomorphicatzero. Coincidentally,
ε + +
∞
the ”polar part” 1 equals the integral I(ε) = e−εxdx leading to the Euler-Maclaurin formula
ε 0
S = I +µ which relates the sum and the integRral of the map x 7→ e−εx by means of the interpo-
lator µ = S . Using the terminology borrowed from physicists, we refer to the decomposition
+
S(ε) = 1 +S (ε) into a ”polar part” 1 and a holomorphic part S (ε) as the minimal subtraction
ε + ε +
scheme applied to S. For this particular function, it coincides with the Euler-Maclaurin formula
and wehaveS (0) = µ(0) = ζ(0)+1 = 1.
+ 2
The coincidence in the case of the discrete exponential sum, between the minimal subtraction
scheme and the Euler-Maclaurin formula, carries out to higher dimensions. The second section
is dedicated to ”counting” the lattice points Zk of a (closed) product cone Rk of dimension
≥0 ≥0
k ∈ N. One expects the ”number” of points of Zk to be the k-th power of the ”number” of
≥0
points of Z and this is indeed the case provided one ”counts carefully”. By this we mean
≥0
that one should not naively evaluate the ”holomorphic part” of the k-th power Sk(ε) at zero of
the exponential sum but instead take the k-th power Sk(0) of the holomorphic part S evaluated
+ +
at zero, which is a straightforward procedure in the rather trivial case of product cones. How-
ever there is a general algebraic construction which derives Sk from Sk, known as the algebraic
+
Birkhofffactorizationthatcanbeviewedasageneralizationtohigherdimensionsoftheminimal
subtraction scheme mentioned above. It relies on a coproduct on (product) cones built from a
complement map described in Section 3, which separates a face of the cone from the remaining
faces. When applied to the multivariableexponential sum S : (ε ,··· ,ε ) 7→ k S(ε) on the
k 1 k i=1 i
product cone Rk , the general algebraic Birkhoff factorization on coalgebras described in Sec-
≥0 e Q
tion 4 gives (ε ,··· ,ε ) 7→ k S (ε) as the ”renormalized holomorphic” part of the map S .
1 k i=1 + i k
ThisalgebraicBirkhoff factorizationcan also beinterpretedas an Euler-Maclaurin formulaforit
Q
e
factorizes thesumas a(convolution)productofintegralsand interpolatorson productcones.
We close this presentation by briefly mentioning the corresponding result on rational polyhe-
dral (lattice)cones, namely that theEuler-Maclaurinformula(first derivedin [BV], see also[B])
for the exponential sum is given by its algebraic Birkhoff factorization, leaving out the precise
statementforwhichwerefertoreaderto[GPZ3]. Renormalizedconicalzetavaluesassociatedto
a coneC correspond to the Taylor coefficients of the ”holomorphicpart” S (C) of the multivari-
+
able exponential sum S(C) on the cone. For Chen cones x ≤ ··· ≤ x they yield renormalized
k 1
multiplezetavalues;inSection5weillustrateourapproachwiththecomputationofrenormalized
multiplezeta valueswith 2 and 3 arguments. ThealgebraicBirkhoff factorizationon cones, seen
as a general device which renormalizes any conical zeta value at non-positiveintegers, therefore
yields a geometric approach to renormalize multiple zeta values at non-positive integers. This
COUNTINGANINFINITENUMBEROFPOINTSANDRENORMALIZATION 3
geometric approach contrasts with other approaches such as [MP] and [GZ] to the renormaliza-
tionofmultiplezetavaluesat non-positivearguments,wherethealgebraicBirkhofffactorization
is carried out on the summands (functions (x ,··· ,x ) 7→ x−s1 ···x−sk) rather than on the do-
1 k 1 1
main (the cones) of summation as in our present construction or [Sa] where a purely analytic
renormalizationmethodisimplemented,whichdoes notusealgebraicBirkhofffactorization.
To conclude, ”counting” lattice points on cones which might a priori seem like a very spe-
cific issue, actually brings together i) renormalization methods a` la Connes and Kreimer [CK]
borrowedfromquantumfieldtheoryintheformofalgebraicBirkhofffactorization,ii)theEuler-
Maclaurinformulaonconesandhenceonpolytopesusedtostudythegeometryoftoricvarieties,
iii) number theory with the conical zeta values (introduced in [GPZ2]) that generalize multiple
zeta values [Ho, Za], and which arise in our context as the Taylor coefficients of the interpolator
intheEuler-Maclaurinformula. Wehopethatthispresentationwhichdoesnotclaimtobeneither
exhaustivenornewsinceitreliesonpreviousworkbytheauthors,willactasanincentiveforthe
lay readerto getfurtheracquainted withrenormalizationmethods.
1. Counting integers
We want to count the non-negativeinteger points i.e. to evaluate the ill-defined sum ”1 +1+
···+1+··· = ∞ n0”andmoregenerallythenobetterdefinedsum ∞ nk foranynon-negative
n=0 n=0
integerk.
P P
1.1. Approximated sums over integers. We first approximate these ill-defined sums; there are
at leastthreeways todo so1:
(a) Thecut-offregularization onlyconsidersafinitenumberoftermsofthesum. ForN ∈ N
weset S (N) := N nk;
k n=0
(b) The heat-kernel type regularization approximates the summand by an exponential ex-
P
pression. Forpositiveε weset
∞
(1) S(ε) := e−εn
Xn=0
and S (ε) := ∞ nke−εn = (−1)k∂ S(ε);
k n=0 k
(c) Thezeta-function typeregularizationapproximatesthesummandbyacomplexpower.
P
Foracomplexnumberz whosereal part islarger than1, theexpression
∞
S˜(z) := n−z =: ζ(z)
Xn=1
called the ζ-function converges and S˜ (z) := ∞ nk−z = ζ(z − k) converges for any
k n=1
complexnumberz whosereal part islarger thank+1.
P
ThesumsS andS˜ relateviatheMellintransform;foranypositivenumberλthemap f : ε 7→ e−λε
λ
defines aSchwartz functionwhoseMellintransform reads
1 ∞
M(f )(z) := εz−1f (ε)dε = λ−z.
λ Γ(z) Z λ
0
S˜(z) = ∞ M(f )(z) = M(S −1)(z). Thisextendsto anidentityofmeromorphicfunctions
n=1 n
P S˜ = M(S −1)
1Wereferthereaderto[P]foramoredetaileddescriptionofthesevariousregularizationmethods.
4 LIGUO,SYLVIEPAYCHA,ANDBINZHANG
with simple poles at integers smaller or equal 1. It turns out that the residue at z = 1 is one and
zero elsewhere.
ThesumS(ε) = 1 can beexpressedinterms oftheTodd function2.
1−e−ε
ε
(2) Td(ε) :=
eε −1
as
Td(−ε)
S(ε) = .
ε
The Todd function is the exponential generating function for the Bernoulli numbers3 that cor-
respondto theTaylorcoefficients4
∞ εn
(3) Td(ε) = B .
n
n!
Xn=0
Wehave
ε ε 1 ε
Td(ε) = = = = 1− +o(ε)
eε −1 ε+ ε2 +o(ε2) 1+ ε +o(ε) 2
2 2
so B = 1;B = −1. Since ε + ε = εeε2+e−2ε is an even function, B = 0 for any positive
0 1 2 eε−1 2 2eε2−e−2ε 2k+1
integerk.
Consequently,foranypositiveinteger K wehave
K
ε B
(4) Td(ε) = 1− + 2k ε2k +o(ε2K)
2 (2k)!
Xk=1
and
Td(−ε) 1 1 K B
(5) S(ε) = = + + 2k ε2k−1 +o(ε2K).
ε ε 2 (2k)!
Xk=1
1.2. The one-dimensional Euler-Maclaurin formula. As a consequence of formula (5), the
discretesumS(ε) := ∞ e−εk = 1 forpositiveε relates totheintegral
k=0 1−e−ε
P ∞ 1
(6) I(ε) := e−εxdx =
Z ε
0
by meansoftheinterpolator
K
1 1 B
µ(ε) := S(ε)−I(ε) = S(ε)− = + 2k ε2k−1 +o(ε2K) forall K ∈ N,
ε 2 (2k)!
Xk=1
whichis holomorphicat ε = 0. Thisinterpolationformulabetween thesumand theintegral
(7) S(ε) = I(ε)+µ(ε)
generalizes toother L1 functionsby meansoftheEuler-Maclaurinformula.
2Thereare two variantsof the Toddfunction;in topologyit is definedas the mapτ : ε 7→ ε , an alternative
1−e−ε
definitiononefindsintheliteratureisε7→τ(−ε),whichweoptforinthesenotes.
3TheywerediscoveredbyJakobBernoulliandindependentlybyaJapanesemathematicianSekiKo¨wa,bothof
whosediscoverieswereposthumouslypublished(in1712forSekiKo¨wa,inhisworkKatsuyoSampo,in1713for
Bernoulli,inhisArsConjectandi).
4TheyalsoariseintheTaylorseriesexpansionsofthetangentandhyperbolictangentfunctions.
COUNTINGANINFINITENUMBEROFPOINTSANDRENORMALIZATION 5
As a motivation for the Euler-Maclaurin formula with remainder, let us first derive a formal
Euler-Maclaurent formula using the Todd function Td(D) obtained by inserting the derivation
map D : f 7→ f′ onC∞(R)in formula(2).
Let ∇f(x) = f(x)− f(x−1) denote the discrete derivation. Using a formal Taylor expansion,
wehave
∞ (−1)k+1Dkf(x)
∇f(x) = f(x)− f(x−1) = = 1−e−D (f)(x)
k!
Xk=1 (cid:16) (cid:17)
and henceat anynon-negativeintegern
−1
∇−1f (n) = 1−e−D (f)(n)
(cid:16) (cid:17) (cid:16) (cid:17)
K
1 B
= D−1f (n)+ + 2k D2k−1f (n)+o(ε2K).
2 (2k)!
(cid:16) (cid:17) Xk=1 (cid:16) (cid:17)
Here ∇−1f (n) = n f(k)+C standsforthediscreteprimitiveof f definedmoduloaconstant;
k=0
it sati(cid:16)sfies (cid:17)∇◦∇−1Pf(n) = f(n) for any n ∈ Z . Similarly, D−1f (x) = x f(y)dy+C stands
≥0
0
for the con(cid:16)tinuous i(cid:17)ntegration map defined modulo a constan(cid:16)t; it sa(cid:17)tisfies RD◦D−1 f(x) = f(x)
forany x ∈ R. Thisgivesafirst formal expansion (cid:16) (cid:17)
b
f(n) = ∇−1f (b)− ∇−1f (a)
Xn=a (cid:16) (cid:17) (cid:16) (cid:17)
f(a)+ f(b)
= D−1f (b)− D−1f (a)+
2
(cid:16) (cid:17) (cid:16) (cid:17)
J B
+ 2j D2j−1f(b)−D(2j−1)f(a)
(2j)!
Xj=1 (cid:16) (cid:17)
b f(a)+ f(b)
= f(x)dx+
Z 2
a
J B
+ 2j f(2j−1)(b)− f(2j−1)(a) ,
(2j)!
Xj=1 (cid:16) (cid:17)
foranytwo non-negativeintegersaand b.
Wearenowready tostatetheEuler-Maclaurin formulawithremainder[Ha]
Proposition1.1. Foranyfunction f inC∞(R) andanytwointegersa < b,
b f(a)+ f(b) b
f(n) = + f(x)dx
2 Z
Xn=a a
J B
+ 2j f(2j−1)(b)− f(2j−1)(a)
(2j)!
Xj=1 (cid:16) (cid:17)
1 b
(8) − B (x) f(2J)(x)dx
2J
(2J)! Z
a
where J is any positive integer and B (x) := B (x−⌊x⌋) built from the Bernoulli polynomials
n n
(seee.g. [A]) B (x) := n n B xk and⌊x⌋ theintegralpartof x.
n k=0 k n−k
P (cid:16) (cid:17)
6 LIGUO,SYLVIEPAYCHA,ANDBINZHANG
In particular,for f(x) = xk and a = 0,b = N wehave f(2j−1)(x) = k! xk−2j+1 andhence
(k−2j+1)!
N
(9) S (N) : = nk
k
Xn=0
[k+1]
δ +Nk N 2 B k!
= k + xkdx+ 2j Nk−2j+1 −δ
2 Z 2j! (k−2j+1)! k−2j+1 !
0 Xj=1 (cid:16) (cid:17)
[k+1]
Nk+1 Nk 2 k B δ
(10) = + + 2j Nk−2j+1 −δ + k,
k+1 2 2j−1! 2j k−2j+1 2
Xj=1 (cid:16) (cid:17)
and werecoverthisway thewell-knownformulae
N N N(N +1) N N(N +1)(2N +1)
n0 = N +1; n = ; n2 =
2 6
Xn=0 Xn=0 Xn=0
usingthefact that B = 1 forthelast one. Moregenerally,itfollowsfromEq. (9)that
2 6
Corollary1.2. Thecut-offdiscretesumS (N)isapolynomialoforderk+1in N whichvanishes
k
atzerofor anypositiveintegerk.
1.3. Evaluating meromorphic functions at poles. Let Merk(C) be the set of germs of mero-
0
morphicfunctionsat zero 5withpolesat zero oforderno largerthan k, andlet
Mer (C) = ∪∞ Merk(C).
0 k=0 0
Let Hol (C)(also denotedby Mer0(C)) bethesetofgerms ofholomorphicfunctionsat zero.
0 0
If f inMerk(C)reads f(z) = ∞ azi,thenforany j ∈ {1,··· ,k}wesetResj(f) := a ,called
0 i=−k i 0 −j
the j-thresidue of f at zero.
P
Theprojectionmap
π : Mer (C) → Hol (C)
+ 0 0
k Resj(f)
f 7→ z 7→ f(z)− 0 for f ∈ Merk(C)
 zj  0
corresponds to what physicistscalla minimal sXuj=b1tractionscheme. Whereas π (f)corresponds
+
totheholomorphicpart of f,π (f) := (1−π )(f)corresponds tothe“polarpart”of f.
− +
Example1.3. Withthenotationofthepreviousparagraphs, wehave
(11) S := π ◦S(ε) = µ(ε); S := π ◦S(ε) = I(ε).
+ + − −
ThustheEuler-Maclaurinformula(7) amountsto theminimalsubtractionschemeappliedtoS:
(12) S = S +S = µ+I.
+ −
An easy computationfurthershowsthat
(13) π ◦S (ε) = (−1)kµ(k)(ε); π ◦S (ε) = (−1)kI(k)(ε).
+ k − k
5i.e.equivalenceclassesofmeromorphicfunctionsdefinedonaneighborhoodofzerofortheequivalencerelation
f ∼gif f andgcoincideonsomeopenneighborhoodofzero.
COUNTINGANINFINITENUMBEROFPOINTSANDRENORMALIZATION 7
The holomorphic part π (f g) of the product of two meromorphic functions f and g differs
+
from the product π (f)π (g) of the holomorphic parts of f and g by contributions of the poles
+ +
throughπ (f)and π (g)and wehave
− −
(14) π (f g) = π (f)π (g)+π (f π (g))+π (gπ (f)).
+ + + + − + −
Themapsπ andπ arebothRota-Baxter operators ofweight−1 on Mer (C), i.e.
+ − 0
π (f)π (g) = π (π (f)g)+π (f π (g))− π (fg).
± ± ± ± ± ± ±
Werefer thereader to[G]forasurveyonRota-Baxter operators.
Combining the evaluation map at zero ev : f 7→ f(0) on holomorphic germs at zero with the
0
mapπ providesa first regularizedevaluatorat zero onMerk(C). Themap
+ 0
evreg : Merk(C) → C
0 0
(15) f 7→ ev ◦π (f),
0 +
isalinearform thatextendstheordinaryevaluationmapev defined onthespaceHol (C).
0 0
Definition 1.4. We call a regularized evaluator any linear extension of the evaluation map ev
0
tothespaceMer (C).
0
Thefollowingresult providesaclassificationofregularized evaluators.
Proposition1.5. Regularizedevaluatorsat zeroonMerk(C)areof theform:
0
k
(16) λ = evreg + µ Resj
0 0 j 0
Xj=1
forsomeconstantsµ ,··· ,µ . In particular,regularizedevaluatorsatzeroonMer1(C)areofthe
1 k 0
form6
(17) λ = evreg +µRes .
0 0 0
forsomeconstantµ.
Proof. A linear form λ which extends ev coincides with ev on the range of π and therefore
0 0 0 +
fulfillsthefollowingidentity:
λ ◦π = ev ◦π = evreg.
0 + 0 + 0
Thus,forany f ∈ Merk(C), usingthelinearityofλ weget
0 0
k
λ (f) = λ (π (f))+λ (π (f)) = evreg + µ Resj(f)
0 0 + 0 − 0 j 0
Xj=1
wherewehavesetµ := λ (z−j). (cid:3)
j 0
Example1.6. Wehave
1
evreg(S) = µ(0) = = 1+ B .
0 2 1
Similarly, the higher Taylor coefficients of the holomorphic function µ at zero relate to the
valueofthezetafunctionat negativeintegers
B
evreg(S ) = (−1)kµ(k)(0) = − k+1
0 k k+1
6Theparameterµthatariseshereisrelatedtotherenormalizationgroupparameterinquantumfieldtheory.
8 LIGUO,SYLVIEPAYCHA,ANDBINZHANG
and yield the renormalized polynomial sums “ ∞ kn” on integer points of the one dimensional
k=0
closedcone[0,+∞).
P
1.4. The zeta function at non-positive integers. Let us start with some notation. Given α ∈ C
weconsidersmoothfunctions f onR withthefollowingasymptoticbehaviorat infinity
+
∞
(18) f(R) ∼ a Rα−j +b logR
R→∞ j
Xj=0
by whichwemean
N−1
f(R)− a Rα−j −blogR = o Rℜ(α)−N+ε
j
Xj=0 (cid:16) (cid:17)
for any positive ε and any positive integer N. We call such a function asymptotically log-
polyhomogeneous at infinity of logarithmic type 1. If b = 0 we call it asymptotically poly-
homogeneous at infinity; let us consider the class Sα(R ) of asymptotically polyhomogeneous
∞ +
functions at infinityoflogarithmictype1.
Example 1.7. The logarithmic function f : R 7→ logR is asymptotically log-polyhomogeneous
atinfinity,oflogarithmictype1. Physicistssaythattheintegral R 1 dx = logRhasalogarithmic
1 x
divergenceas R → ∞. R
TheHadamardfinite part of f atinfinity
a , ifα ∈ Z ,
fp f(R) := α ≥0
R→∞ ( 0, otherwise.
defines alinearmap
evreg : Sα(R ) −→ C
∞ ∞ +
f 7−→ fp f(R)
R→∞
which extendsthe ordinary limitat infinity wheneverit exists. We call such a linearextensionof
theordinarylimitaregularized evaluatoratinfinity.
Setting R = 1 in Eq. (18) with r > 0, and choosing β = −α, b = a , c = −b leads to smooth
r j j
functions f on(0,+∞)withthefollowinglog-polyhomogeneousasymptoticbehaviorat zero:
∞
(19) f(r) ∼ b rβ+j +c logr
r→0 j
Xj=0
by whichwemean
N−1
f(r)− b rβ+j −clogr = o rℜ(β)+N+ε
j
Xj=0 (cid:16) (cid:17)
for any positive ε and any positive integer N. We call such a function asymptotically log-
polyhomogeneous at zero of logarithmictype1. If c = 0 we call it asymptoticallypolyhomoge-
neousatzero; letusconsidertheclassSβ(R )ofasymptoticallypolyhomogeneousfunctionsat
0 +
zero oflogarithmictype1.
TheHadamardfinite part of f atzero
b , ifβ ∈ Z ,
fp f(r) := −β ≥0
r→0 ( 0, otherwise.
COUNTINGANINFINITENUMBEROFPOINTSANDRENORMALIZATION 9
defines alinearmap
evreg : Sβ(R ) −→ C
0 0 +
f 7−→ fp f(r)
r→0
whichextendstheordinarylimitatzerowheneveritexists. Wecallsuchalinearextensionofthe
ordinarylimitaregularized evaluatoratzero.
Werecall herewell knownresultson theMellintransform 7, seee.g. [Je].
Proposition 1.8. Let f be a Schwartz function in Sβ(R ) for some β ∈ C. Its Mellin transform
0 +
defines a holomorphic function on the half plane ℜ(z) + β > 0 which extends to a meromor-
phic function on the whole complex plane which is holomorphic at zero. We have M(f(k))(z) =
(−1)kM(f)(z−k)foranyk ∈ Z andthevalueat zeroisgiven by
≥0
(20) evreg ◦M(f) = M(f)(0) = evreg(f).
0 0
Proof. We splittheMellintransform
1 A ∞
M(f)(z) = εz−1f(ε)dε+ εz−1f(ε)dε
Γ(z) Z Z !
0 A
forsomepositivereal number A. Thefunction 1 isholomorphicat zero andwehave 1 ∼ z.
Γ Γ(z) 0
Since f is a Schwartz function, the second term in the bracket yields a holomorphic function
I : z 7→ 1 ∞εz−1f(ε)dεwhich vanishesat zero. For f(ε) = J b εβ+j and ℜ(z)+β+ j > 0,
2 Γ(z) A j=0 j
thefirst termRin thebracket givesriseto P
1 J A 1 J Az+β+j
IJ(z) := b εz+β+j−1dε = b
1 Γ(z) jZ Γ(z) jz+β+ j
Xj=0 0 Xj=0
whichextendsto ameromorphicfunctiondenoted bythesamesymbol. Hence
M(f)(z) := I (z)+IJ(z)+o εz+β+J for J ∈ N
1 2
(cid:16) (cid:17)
defines a meromorphic function on the whole plane. Integrating by parts k times and imple-
menting the property Γ(z) = (z − 1)Γ(z − 1) = (z − 1)···(z − k)Γ(z − k) gives M(f(k))(z) =
(−1)kM(f)(z−k).
Since Γ(z) ∼ 1, the value of IJ(z) at z = 0 is b if β ∈ Z ∩ [0,J] and zero elsewhere so
z 2 −β ≤0
the same holds for M(f)(z). Since fp f(ε) = b if β ∈ Z and zero elsewhere, this yields
ε=0 −β ≤0
Eq.(20). (cid:3)
The Mellin transform of the Schwartz function f : ε 7→ e−εn on R reads n−z = M(f )(z) for
n + n
anyn ∈ N.
∞
Corollary 1.9. The function z 7→ n−z defined on the half-plane ℜ(z) > 1 extends meromor-
n=1
phicallyon thewhole plane to the zPeta function ζ, which has only onesimple poleat −1 and its
valueatnon-positiveintegersis expressed interms ofBernoullinumbers
1 B
(21) ζ(0) = evreg(S)−1 = − ; ζ(−k) = (−1)kevreg(∂kS) = − k+1 forallk ∈ N.
0 2 0 k+1
Proof. This follows from applying Proposition1.8 to thefunction ε 7→ S(ε)−1 = ∞ e−εn and
n=1
itsderivatives(−1)kS ,k ∈ N. (cid:3)
k P
7NotethatdefinitionsoftheMellintransformdifferaccordingtothereferencebyamultiplicativefactorΓ(z).
10 LIGUO,SYLVIEPAYCHA,ANDBINZHANG
1.5. Conclusion. By means oftheheat-kernel regularizationmethodweevaluated
∞ ∞
1
” n0 ” = ” n0 ”−1 = evreg ◦S −1 = µ(0)−1 = B −1 = − .
    0 1 2
In thispaper, BXn==1 1.By meaXnn=s0ofthezeta-function regularizationmethodwe evaluated
1 2
∞
1
” n0 ” = ev ◦S = ζ(0) = − ,
  0 2
sothesetwomethodsagreesintheXn=c1asek = 0. Thetwomethodsactuallycoincideforanyk ∈ Z .
≥0
∞ ∞
B
” nk ” = ” nk ” = evreg ◦S = ζ(−k) = − k+1 .
    0 k k+1
Moreover, combiningXEn=q0s. (21) andXn(=51) yields the Laurent expansion of the exponential sum in
termsofζ-valuesat non-positivearguments
K
1
(22) S(ε) = −ζ(0)− ζ(−(2k−1))ε2k−1 +o(ε2K) ∀K ∈ N.
ε
Xk=1
In contrast,thecut-offmethodgives
fp S (N) = P (0) = δ ,
N→∞ k k k
whereδ = 1 ifk = 0 and zero otherwise.
k
2. Counting latticepoints onproductcones
Given a positiveinteger k, we now want to ”count” the number ” ~n~0 ” of lattice points
~n∈Zk
≥0
~n ∈ Zk in the product cone Rk , where for~n = (n ,··· ,n ) ∈ Zk a(cid:16)nPd~r = (r(cid:17),··· ,r ) ∈ Zk we
≥0 ≥0 1 k ≥0 1 k ≥0
haveset~n~r = nr1 ···nrk. Wefirst describethealgebra ofproductcones.
1 k
2.1. The exponentialsummationandintegrationmaponproduct cones. GivenabasisB =
n
(e ,··· ,e )ofRn, let P (Rn)bethesetofproduct cones
1 n Bn
he i := R e, I ⊆ [n] := {1,··· ,n},
I ≥0 i
Xi∈I
viewed as subsets of Rn. Extending this basis to a basis B = (e ,··· ,e ) of Rn+1, a product
n+1 1 n+1
coneinRn can beviewedas aproduct coneinRn+1. SettingP (R0) = {0},wedefinetheset
B
0
P (R∞) := ∪∞ P (Rn)
B n=0 Bn
ofproductcones inR∞ equipped withabasisB = {e |n ∈ N}. Equivalently,
n
P (R∞) = {he i|I ⊂ Nfinite} withhe i := {0}.
B I ∅
It isZ -filtered by thedimensioncard(I) (here card standsforcardinal)oftheconehe i and itis
≥0 I
equippedwithapartial product
he i•he i := he i
I J I∪J
for two disjointsubsets I,J of N. This product is compatible with the filtration since the dimen-
sionoftheproductoftwocones isthesumoftheirdimensions.
Unless otherwise specified, we take B to be the canonical basis of R∞, in which case we drop
thesubscriptB inthenotation.