Journal de Théorie des Nombres de Bordeaux

Jean-Pierre Kahane

Conditions pour que les entiers de Beurling aient une densité

Huei-Jeng Chen

Huei-Jeng CHEN
On shuffle of double Eisenstein series in positive characteristic
Tome 29, no 3 (2017), p. 815-825.
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Journal de Théorie des Nombres
de Bordeaux 29 (2017), 815–825
On shuffle of double Eisenstein series in positive
characteristic
par Huei-Jeng CHEN
Résumé. L’étude du présent article s’inspire du résultat de
Gangl, Kaneko et Zagier sur connexion entre les valeurs zêta
doubles et les formes modulaires. Nous introduisons la série
d’Eisenstein double Er,s en caractéristique positive avec les va-
leurs zêta doubles ζA(r, s) comme terme constant et calculons les
t-expansions de la série d’Eisenstein double. De plus, on en dé-
duit les relations de shuffle de deux séries d’Eisenstein qui corres-
pondent aux relations de shuffle des valeurs zêta doubles dans [4].
Abstract. The study of the present paper is inspired by Gangl,
Kaneko and Zagier’s result of the connection with double zeta val-
ues and modular forms. We introduce double Eisenstein series Er,s
in positive characteristic with double zeta values ζA(r, s) as their
constant term and compute the t-expansions of the double Eisen-
stein series. Moreover, we derive the shuffle relations of double
Eisenstein series which match the shuffle relations of double zeta
values in [4].
1. Introduction
The multiple zeta values (abbreviated as MZV’s) ζ(s1, . . . , sr), where
s1, . . . , sr are positive integers with s1 ≥ 2, are generalizations of zeta val-
ues. These numbers were first studied by Euler for the case of r = 2. He
derived certain interesting relations between double zeta values, such as
the sum formula
∑k−1
n=2 ζ(n, k − n) = ζ(k). The MZV’s have many connec-
tions with various arithmetic and geometric points of view, we refer the
reader to [1, 12, 13]. Gangl–Kaneko–Zagier [5] introduced and studied dou-
ble Eisenstein series Gr,s with double zeta value ζ(r, s) as its constant term
for r ≥ 3 and s ≥ 2. They showed that certain relations between double zeta
values can be “lifted” to relations between corresponding double Eisenstein
series.
Thakur [9] introduced the multizeta values ζA(s1, . . . , sr) (abbreviated
as MZV’s too) in positive characteristic and [11] showed the existence of
Manuscrit reçu le 23 août 2016, révisé le 21 octobre 2016, accepté le 2 décembre 2016.
2010 Mathematics Subject Classification. 11J91, 11M36.
Mots-clefs. Double zeta values, Eisenstein series, t-expansions, shuffle relations.
816 Huei-Jeng Chen
shuffle relations for multizeta values. These shuffle relations can be derived
from explicit formulas (see [4]). Unlike the classical double shuffle relation,
there is no other expression of the product of two single (Carlitz) zeta
values than the shuffle relation mentioned above at date. In the recent
work of Chang [3], he established an effective criterion for computing the
dimension of the space generated by double zeta values and a fundamental
period of the Carlitz module raised to the weight power in terms of special
points in the tensor power of the Carlitz module. One views his work as an
algebraic/arithmetic point of view.
In the present note the author also introduces double Eisenstein series
Er,s in positive characteristic with double zeta values ζA(r, s) as their con-
stant term. Moreover, the shuffle relations of double zeta values in [4] match
the shuffle relations of the corresponding double Eisenstein series. We men-
tion that the approach is from the idea of Gangl–Kaneko–Zagier, but the
proofs are entire different. We use Goss polynomials to compute the t-
expansions (an analogy to Fourier expansions) of the double Eisenstein se-
ries, and use partial fractional decomposition method to show the desired
relation.
Let DZk denote the Fq(θ)-vector space spanned by double zeta values of
weight k (In the classical case we denote by DZk the Q-vector space spanned
by double zeta values of weight k). It is a difficult problem in computing the
dimension of DZk. In the classical case, Gangl–Kaneko–Zagier [5] showed
that the structure of the DZk are well connected to the space of modular
formsMk of weight k for the full modular group Γ1 = PSL(2,Z) by means of
double Eisensetin series. With additional work, they bounded the dimension
of DZk in terms of the dimension of the space of cusp forms Sk of weight k
for Γ1. In a subsequent paper we shall explore the space DZk in connection
with a suitable space of weight k Drinfeld modular forms for GL2(Fq[θ]).
2. Preliminaries
2.1. Notation. We adopt the notations below in the following sections.
Fq := a finite field with q = pm elements.
K := Fq(θ), the rational function field in the variable θ.
∞ := the zero of 1/θ, the infinite place of K.
| |∞ := the nonarchimedean absolute value onK corresponding to∞.
K∞ := Fq((1/θ)), the completion of K with respect to | · |∞.
C∞ := the completion of a fixed algebraic closure ofK∞ with respect
to | · |∞.
A := Fq[θ], the ring of polynomials in the variable θ over Fq.
A+ := the set of monic polynomials in A.
Ad := the set of polynomials in A of degree d.
On shuffle of double Eisenstein series in positive characteristic 817
Ad+ := Ad ∩A+, the set of monic polynomials in A of degree d.
[n] := θqn − θ.
Dn :=
n−1∏
i=0
θq
n − θqi = [n][n− 1]q · · · [1]qn−1 .
Ln :=
n∏
i=1
θq
i − θ = [n][n− 1] · · · [1].
ln := (−1)nLn.
2.2. Shuffle relations of double zeta values. For any tuple
(s1, . . . , sr) ∈ Nr,
ζA(s1, . . . , sr) :=
∑
ai∈A+
degθ a1>···>degθ ar
1
as11 . . . a
sr
r
∈ K∞.
We mention that they are not only non-vanishing [10] but also transcen-
dental over K [2]. Here r is called the depth and w := s1 + · · ·+ sr is called
the weight of the presentation ζA(s1, . . . , sr). Recall that for the case r = 2,
we have the following shuffle relation of double zeta values.
Theorem 2.1 ([4, Theorem 3.1]). For any r, s ∈ N, we have
ζA(r)ζA(s) = ζA(r, s) + ζA(s, r) + ζA(r + s)
+
∑
i+j=r+s
q−1|j
[
(−1)r−1
(
j − 1
r − 1
)
+ (−1)s−1
(
j − 1
s− 1
)]
ζA(i, j).
3. Eisenstein series and Double Eisenstein series
3.1. Double Eisenstein series. Let Ω := C∞ − K∞ = {z ∈ C∞ | z /∈
K∞} be the Drinfeld upper half-plane, which has a natural structure as
a connected admissible open subspace of the rigid analytic space P1(C∞)
(see [7]).
Definition 3.1. Letm, a ∈ A and z ∈ Ω. We write “mz+a  0” ifm ∈ A+
or m = 0, a ∈ A+. Suppose that mz + a  0 and nz + b  0. We write
“mz + a  nz + b” if one of the following conditions holds:
(1) m ∈ A+, n = 0;
(2) m = n = 0, a, b ∈ A+ with deg a > deg b;
(3) m,n ∈ A+ with degm > degn.
Definition 3.2. For r, s, k ∈ N, let Ek be the Eisenstein series of weight k
defined on z ∈ Ω :
Ek(z) :=
∑
mz+a∈Az+A
mz+a0
1
(mz + a)k .
818 Huei-Jeng Chen
The double Eisenstein series Er,s is defined by
Er,s(z) :=
∑
mz+a,nz+b∈Az+A
mz+anz+b0
1
(mz + a)r(nz + b)s .
We note that since Ek(z) and Er,s(z) converge uniformly on an admissible
cover of Ω, they are both rigid analytic functions on Ω (cf. [8]).
Remark 3.3. The modular Eisenstein series in [8] are defined by
Ek(z) =
∑
mz+a∈Az+A
1
(mz + a)k .
It can be checked that Ek(z) = 0 if q−1 - k and Ek(z) = −Ek(z) if q−1 | k.
In the latter case Ek(z) is a Drinfeld modular form of weight k and type m
(m is a class in Z/(q − 1)Z) for GL2(A).
3.2. t-expansions. Let (−θ)
1
q−1 be a fixed choice of (q− 1)-st root of −θ
and π̃ := (−θ)
q
q−1
∏∞
i=1(1− θθqi )
−1 be the fundamental period of the Carlitz
module. Let t(z) be the rigid analytic function on Ω given by
t(z) =
∑
b∈π̃A
1
(π̃z + b) .
We have t(z + a) = t(z) for any a ∈ A. The function t(z) serves as a
uniformizing parameter “at the infinity”, so that every function f(z) on
Ω satisfying the property f(z + a) = f(z) for any a ∈ A can be written
as the form f(z) = f̃(t(z)) with respect to t. This is analogous to q(z) =
exp(2π
√
−1z) in the classical case.
Let expπ̃A(z) := z
∏
λ∈π̃A,λ 6=0(1− zλ) =
∑
i≥0 αiz
qi be the lattice function
on C∞ with respect to the lattice π̃A. We refer to [6] for more details of
expπ̃A(z).
Proposition 3.4 ([7, Proposition 6.5, Proposition 6.6]). There exists a
sequence of polynomials Gk(X) ∈ K[X] for k ∈ N satisfying the following
properties:
(1) For z ∈ Ω,
π̃kGk(t(z)) =
∑
b∈A
1
(z + b)k .
(2) Gk(X) = X(Gk−1 + α1Gk−q + · · ·+ αiGk−qi), where i = [logq k].
(3) Gk(0) = 0 for all k.
Definition 3.5. We call Gk(x) the k-th Goss polynomial.
On the domain Ω, Ek(z) and Er,s(z) are invariant under the translations
z 7→ z + b for b ∈ A. We can use Proposition 3.4 to derive the expansions
of Eisenstein and double Eisenstein series with respect to t.
On shuffle of double Eisenstein series in positive characteristic 819
Proposition 3.6. The t-expansions of Ek(z) and Er,s(z) are given by
Ek(z) = ζA(k) +
∑
m∈A+
π̃kGk(t(mz))
and
Er,s(z) = ζA(r, s) +
∑
m∈A+
π̃rζA(s)Gr(t(mz))
+
∑
m,n∈A+
degm>degn≥0
π̃r+sGr(t(mz))Gs(t(nz)).
Proof. By Proposition 3.4(1) we have
∑
m∈A+,n∈A
1
(mz + n)k =
∑
m∈A+
π̃kGk(t(mz)).
It follows that
Ek(z) =
∑
m=0,n∈A+
1
nk
+
∑
m∈A+,n∈A
1
(mz + n)k
= ζA(k) +
∑
m∈A+
π̃kGk(t(mz)).
Er,s(z) =
∑
m=n=0,a,b∈A+
deg a>deg b
1
arbs
+
∑
m∈A+,a∈A;n=0,b∈A+
1
(mz + a)rbs
+
∑
m,n∈A+,a,b∈A
degm>degn
1
(mz + a)r(nz + b)s
= ζA(r, s) +
∑
m∈A+
π̃rζA(s)Gr(t(mz))
+
∑
m,n∈A+
degm>degn
π̃r+sGr(t(mz))Gs(t(nz)). 
4. Shuffle of double Eisenstein series
In this section we will derive the shuffle relation of a product of two
Eisenstein series.
820 Huei-Jeng Chen
Theorem 4.1. On the domain Ω, we have
Er(z)Es(z) = Er,s(z) + Es,r(z) + Er+s(z)
+
∑
i+j=r+s
q−1|j
[
(−1)r−1
(
j − 1
r − 1
)
+ (−1)s−1
(
j − 1
s− 1
)]
Ei,j(z).
Proof. By Proposition 3.6 we have
Er(z)Es(z) = ζA(r)ζA(s) + ζA(r)
∑
n∈A+
π̃sGs(t(nz))
+ ζA(s)
∑
m∈A+
π̃rGr(t(mz)) +
∑
m,n∈A+
π̃r+sGr(t(mz))Gs(t(nz)).
Let Ci,jr,s := (−1)r−1
(j−1
r−1
)
+ (−1)s−1
(j−1
s−1
)
. Then we can expand Er,s(z) +
Es,r(z) + Er+s(z) +
∑
i+j=r+s
q−1|j
Ci,jr,sEi,j(z) as follows.
Er,s(z) + Es,r(z) + Er+s(z) +
∑
i+j=r+s
q−1|j
Ci,jr,sEi,j(z) = ∆1 + ∆2 + ∆3 + ∆4,
where
∆1 = ζA(r, s) + ζA(s, r) + ζA(r + s) +
∑
i+j=r+s
q−1|j
Ci,jr,sζA(i, j),
∆2 = ζA(s)
∑
m∈A+
π̃rGr(t(mz)) + ζA(r)
∑
n∈A+
π̃sGs(t(nz)),
∆3 =
∑
m,n∈A+
degn>degm≥0
π̃r+sGr(t(nz))Gs(t(mz))
+
∑
m,n∈A+
degn>degm≥0
π̃r+sGr(t(mz))Gs(t(nz)),
∆4 =
∑
m∈A+
π̃r+sGr+s(t(mz))
+
∑
i+j=r+s
q−1|j
Ci,jr,sζA(j)
∑
m∈A+
π̃iGi(t(mz))
+
∑
i+j=r+s
q−1|j
Ci,jr,s
∑
m,n∈A+
degn>degm
π̃r+sGi(t(mz))Gj(t(nz)).
On shuffle of double Eisenstein series in positive characteristic 821
By Theorem 2.1 we have ζA(r)ζA(s) = ∆1. So it suffices to show that
∞∑
l=0
∑
m,n∈Al+
π̃r+sGr(t(mz))Gs(t(nz)) = ∆4.
We will prove it by partial fraction decomposition. For any two distinct
elements a, b in an integral domain R (we adopt R for A or A[z] here) we
have
1
arbs
=
∑
i+j=r+s
1
(a− b)j
(
(−1)s
(j−1
s−1
)
ai
+
(−1)j−r
(j−1
r−1
)
bi
)
.
Consider
∞∑
l=0
∑
m,n∈Al+
π̃r+sGr(t(mz))Gs(t(nz))
=
∞∑
l=0
∑
m∈Al+
π̃r+sGr(t(mz))Gs(t(mz))
+
∞∑
l=0
∑
m,n∈Al+
m 6=n
π̃r+sGr(t(mz))Gs(t(nz)).
We rewrite the first part of the sum as∑
m∈A+
π̃r+sGr(t(mz))Gs(t(mz))
=
∑
m∈A+
a,b∈A
1
(mz + a)r(mz + b)s
=
∑
m∈A+
a=b∈A
1
(mz + a)r(mz + b)s +
∑
m∈A+
a6=b∈A
1
(mz + a)r(mz + b)s
=
∑
m∈A+
π̃r+sGr+s(t(mz))
+
∑
m∈A+
a6=b∈A
∑
i+j=r+s
1
(a− b)j
(
(−1)s
(j−1
s−1
)
(mz + a)i +
(−1)j−r
(j−1
r−1
)
(mz + b)i
)
=
∑
m∈A+
π̃r+sGr+s(t(mz))
+
∑
i+j=r+s
∑
m∈A+
a,f∈A,f 6=0
1
f j
(
(−1)s
(j−1
s−1
)
(mz + a)i +
(−1)j−r
(j−1
r−1
)
(mz + f + a)i
)
.
822 Huei-Jeng Chen
For a fixed pair (i, j), we have that
∑
m∈A+
a,f∈A,f 6=0
1
f j
(−1)s
(j−1
s−1
)
(mz + a)i =
∑
η∈F∗q
∑
m∈A+
a∈A,f∈A+
1
(ηf)j
(−1)s
(j−1
s−1
)
(mz + a)i
=

0, if q − 1 - j;
−
∑
m∈A+
a∈A,f∈A+
1
f j
(−1)s
(j−1
s−1
)
(mz + a)i , if q − 1|j.
Similarly,
∑
m∈A+
a,f∈A,f 6=0
1
f j
(−1)j−r
(j−1
r−1
)
(mz + f + a)i =
∑
m∈A+
b,f∈A,f 6=0
1
f j
(−1)j−r
(j−1
r−1
)
(mz + b)i
=

0, if q − 1 - j;
−
∑
m∈A+
b∈A,f∈A+
1
f j
(−1)j−r
(j−1
r−1
)
(mz + b)i , if q − 1|j.
This implies that
∑
m∈A+
π̃r+sGr+s(t(mz))
+
∑
i+j=r+s
∑
m∈A+
a,f∈A,f 6=0
1
f j
(
(−1)s
(j−1
s−1
)
(mz + a)i +
(−1)j−r
(j−1
r−1
)
(mz + f + a)i
)
=
∑
m∈A+
π̃r+sGr+s(t(mz))
+
∑
i+j=r+s
q−1|j
∑
m∈A+
b∈A,f∈A+
1
f j
(−1)s−1
(j−1
s−1
)
+ (−1)j−r−1
(j−1
r−1
)
)
(mz + a)i
=
∑
m∈A+
π̃r+sGr+s(t(mz)) +
∑
i+j=r+s
q−1|j
Ci,jr,sζA(j)
∑
m∈A+
π̃iGi(t(mz)).
On shuffle of double Eisenstein series in positive characteristic 823
By using a similar argument, the second sum can be expressed as
∞∑
l=1
∑
m,n∈Al+,m 6=n
π̃r+sGr(t(mz))Gs(t(nz))
=
∞∑
l=1
∑
m,n∈Al+
m6=n,a,b∈A
1
(mz + a)r(nz + b)s
=
∞∑
l=1
∑
m,n∈Al+
m6=n,a,b∈A
∑
i+j=r+s
1
((m−n)z+(a−b))j
(
(−1)s
(j−1
s−1
)
(mz + a)i +
(−1)j−r
(j−1
r−1
)
(nz + b)i
)
=
∞∑
l=1
∑
i+j=r+s
∑
m∈Al+,n′∈A<l
n′ 6=0,a,b′∈A
(−1)s
(j−1
s−1
)
(n′z + b′)j(mz + a)i
+
∞∑
l=1
∑
i+j=r+s
∑
n∈Al+,m′∈A<l
m′ 6=0,a′,b∈A
(−1)j−r
(j−1
r−1
)
(m′z + a′)j(nz + b)i
=
∞∑
l=1
∑
i+j=r+s
∑
f∈F∗q
1
f j
∑
m,n′∈A+,a,b′∈A
degn′<degm=l
(−1)s
(j−1
s−1
)
(n′z + b′)j(mz + a)i
+
∞∑
l=1
∑
i+j=r+s
∑
f∈F∗q
1
f j
∑
m′,n∈A+,a′,b∈A
degm′<degn=l
(−1)j−r
(j−1
r−1
)
(m′z + a′)j(nz + b)i
=
∞∑
l=1
∑
i+j=r+s
q−1|j
∑
m,n′∈A+,a,b′∈A
degn′<degm=l
Ci,jr,s
1
(n′z + b′)j(mz + a)i
=
∞∑
l=1
∑
i+j=r+s
q−1|j
Ci,jr,s
∑
m,n′∈A+
degn′<degm=l
π̃r+sGi(t(mz))Gj(t(n′z)).
Combining the first sum with the second sum, we have
∞∑
l=0
∑
m,n∈A+
degm=degn=l
π̃r+sGr(t(mz))Gs(t(nz))
824 Huei-Jeng Chen
=
∞∑
l=0
∑
m∈A+
degm=l
π̃r+sGr(t(mz))Gs(t(mz))
+
∞∑
l=0
∑
m,n∈A+
degm=degn=l,m6=n
π̃r+sGr(t(mz))Gs(t(nz))
=
∑
m∈A+
π̃r+sGr+s(t(mz)) +
∑
i+j=r+s
q−1|j
Ci,jr,sζA(j)
∑
m∈A+
π̃iGi(t(mz))
+
∞∑
l=1
∑
i+j=r+s
q−1|j
Ci,jr,s
∑
m,n′∈A+
degn′<degm=l
π̃r+sGi(t(mz))Gj(t(n′z))
= ∆4. 
Remark 4.2. From Proposition 3.4(3), we see that the constant terms in
the t-expansions of Er(z)Es(z) and
Er,s(z) + Es,r(z) + Er+s(z)
+
∑
i+j=r+s
q−1|j
[
(−1)r−1
(
j − 1
r − 1
)
+ (−1)s−1
(
j − 1
s− 1
)]
Ei,j(z)
are ζA(r)ζA(s) and
ζA(r, s) + ζA(s, r) + ζA(r + s)
+
∑
i+j=r+s
q−1|j
[
(−1)r−1
(
j − 1
r − 1
)
+ (−1)s−1
(
j − 1
s− 1
)]
ζA(i, j)
respectively. It follows that the relation in Theorem 4.1 can be viewed as a
“lifting” of the shuffle relation in Theorem 2.1.
References
[1] F. Brown, “Mixed Tate motives over Z”, Ann. Math. 175 (2012), no. 2, p. 949-976.
[2] C.-Y. Chang, “Linear independence of monomials of multizeta values in positive character-
istic”, Compos. Math. 150 (2014), no. 11, p. 1789-1808.
[3] ———, “Linear relations among double zeta values in positive characteristic”, Camb. J.
Math. 4 (2016), no. 3, p. 289-331.
[4] H.-J. Chen, “On shuffle of double zeta values over Fq [t]”, J. Number Theory 148 (2015),
p. 153-163.
[5] H. Gangl, M. Kaneko & D. Zagier, “Double zeta values and modular forms”, in Auto-
morphic forms and zeta functions (Tokyo, 2004), World Scientific, 2006, p. 71-106.
[6] E.-U. Gekeler, “On the coefficients of Drinfeld modular forms”, Invent. Math. 93 (1988),
no. 3, p. 667-700.
[7] D. Goss, “The algebraist’s upper half-plane”, Bull. Am. Math. Soc. (1980), no. 2, p. 391-415.
On shuffle of double Eisenstein series in positive characteristic 825
[8] ———, “π-adic Eisenstein series for function fields”, Compos. Math. 41 (1980), p. 3-38.
[9] D. S. Thakur, Function field Arithmetic, World Scientific, 2004, xv+388 pages.
[10] ———, “Power sums with applications to multizeta and zeta zero distribution for Fq [t]”,
Finite Fields Appl. 15 (2009), no. 4, p. 534-552.
[11] ———, “Shuffle relations for function field multizeta values”, Int. Math. Res. Not. 2010
(2010), no. 11, p. 1973-1980.
[12] D. Zagier, “Values of zeta functions and their applications”, in First European Congress
of Mathematics Vol. II (Paris, 1992), Progress in Mathematics, vol. 120, Birkhäuser, 1994,
p. 497-512.
[13] J. Zhao, Multiple zeta functions, multiple polylogarithms and their special values, Series on
Number Theory and Its Applications, vol. 12, World Scientific, 2016, xxi+595 pages.
Huei-Jeng Chen
Taida Institute for Mathematical Sciences
No.1, Sec.4, Roosevelt Road
Taipei, Taiwan
E-mail: hjchen1204@ntu.edu.tw


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