Journal de Théorie des Nombres de Bordeaux

Christophe Delaunay

Xavier-François Roblot

Regulators of rank one quadratic twists

Jean-Pierre Gazeau

Jean-Louis Verger-Gaugry

On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon

Ahmed Ait-Mokhtar

Abdelkader Necer

Alain Salinier

Endomorphismes d’algèbres de suites

Jacky Cresson

Dynamique des nombres et physique des oscillateurs

Jay Jorgenson

Cormac O’Sullivan

Unipotent vector bundles and higher-order non-holomorphic Eisenstein series

Jacques Martinet

Anne-Marie Bergé – In Memoriam

Tanguy Rivoal

On the distribution of Hawkins’ random “primes”

Marco Illengo

Marco ILLENGO
Cohomology of integer matrices and local-global divisibility on the torus
Tome 20, no 2 (2008), p. 327-334.
<http://jtnb.cedram.org/item?id=JTNB_2008__20_2_327_0>
© Université Bordeaux 1, 2008, tous droits réservés.
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bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord
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présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
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Journal de Théorie des Nombres
de Bordeaux 20 (2008), 327-334
Cohomology of integer matrices and local-global
divisibility on the torus
par Marco ILLENGO
Résumé. Soient p 6= 2 un nombre premier et G un p-groupe de
matrices dans SLn(Z), pour un nombre entier n. Dans cet article
nous montrons que, pour n < 3(p− 1), un certain sous-groupe du
groupe de cohomologie H1(G, Fnp ) est trivial. Nous montrons aussi
que cette affirmation peut être fausse pour n > 3(p− 1). Avec un
résultat de Dvornicich et Zannier (voir [2]), nous obtenons que
le principe local-global de divisibilité pour p vaut pour tout tore
algébrique de dimension n < 3(p− 1).
Abstract. Let p 6= 2 be a prime and let G be a p-group of ma-
trices in SLn(Z), for some integer n. In this paper we show that,
when n < 3(p− 1), a certain subgroup of the cohomology group
H1(G, Fnp ) is trivial. We also show that this statement can be
false when n > 3(p− 1). Together with a result of Dvornicich and
Zannier (see [2]), we obtain that any algebraic torus of dimension
n < 3(p− 1) enjoys a local-global principle on divisibility by p.
1. Introduction
Let G be a subgroup of SLn(Z), for some n. Then G acts on Zn and,
by projection, on Fnp , for some prime p. Consider the group cohomology
of the couple (G, Fnp ) and note that, for every subgroup C of G, there is
a well-defined restriction map H1(G, Fnp ) → H1(C, Fnp ). In this paper we
prove the following theorem.
Theorem 1. Let p 6= 2 be a prime and let n < 3(p− 1). For every p-
group G in SLn(Z) the projection H1(G, Fnp )
ϕ→
∏
H1(C, Fnp ), the product
being taken on all cyclic subgroups C of G, is injective.
We also prove that this statement is ‘best possible’ on n.
Proposition 2. Let p 6= 2 be a prime and let n > 3(p− 1). There exists
a p-group G in SLn(Z) such that the map H1(G, Fnp )
ϕ→
∏
H1(C, Fnp ), the
product being taken on all cyclic subgroups C of G, is not injective.
Manuscrit reçu le 21 janvier 2008.
328 Marco Illengo
Our Theorem 1 is motivated by a paper of Dvornicich and Zannier on
local-global divisibility for algebraic groups. In [2, Sections 4-5] they proved
that local-global divisibility by a prime p holds on every algebraic torus of
dimension n 6 max{3, 2(p− 1)}, but fails for at least one torus of dimension
n = p4 − p2 + 1. (We are using the additive notation for the torus: division
by p corresponds to taking p-th roots in the multiplicative group Gm.)
The authors also suggested that their proof of the condition n 6 2(p− 1)
in the case p 6= 2 could be adapted to prove local-global divisibility by p
under a weaker condition, so to reduce the gap of uncertainty for n. In
particular, in the first part of their proof they show that, for p 6= 2 and n
fixed, the injectivity of ϕ for any p-group G < SLn(Z) implies local-global
divisibility by p for every algebraic torus of dimension n.
Together with this result, Theorem 1 allows to replace the condition
n 6 2(p− 1) with the weaker condition n < 3(p− 1).
Theorem 3. Let p 6= 2 be a prime, k be a number field, and T be an
algebraic k-torus of dimension n < 3(p− 1). Fix any point P ∈ T (k); if for
all but a finite number of completions kν of k there exists a point Dν ∈ T (kv)
with pDν = P , then there exists a D ∈ T (k) such that pD = P .
Using the terminology of [2], we say that a cocycle Z on (G, Fnp ) sat-
isfies the local conditions if for every g ∈ G there exists a Wg ∈ Fnp such
that Zg = gWg −Wg. Note that the set of cocycles that satisfy the local
conditions is precisely the kernel of ϕ.
For p 6= 2 and n > 3(p− 1) the example in Proposition 2 allows, as Dvor-
nicich and Zannier pointed out in [2, Section 4] and [3, Section 3], to build
an algebraic torus of dimension n defined over some number field k and,
possibly extending the field k, a k-rational point on the torus for which the
local-global divisibility by p fails.
In Section 2 we shall prove Theorem 1, using some elementary results of
the geometry of numbers and of the theory of representations.
In Section 3 we shall prove Proposition 2 for the case n = 3(p− 1); the
general case can be obtained by means of a direct sum with the trivial
representation of dimension n− 3(p− 1).
Throughout this paper, whenever their orders are known, we shall denote
by I the identity matrix and by O the null matrix.
2. Proof of theorem
We begin the proof of Theorem 1 by an inspection of the p-group G. The
following result is slightly more general than needed.
Lemma 4. Let p be a prime and let G be a p-group of matrices in SLn(Q).
If n < p(p− 1) then G is isomorphic to (Z/pZ)b, for some b 6 n/(p− 1).
Cohomology of integer matrices and local-global divisibility on the torus 329
Proof. Note that any non-trivial element g of G is a matrix of multiplicative
order pm, for some positive integer m. Then at least one of the eigenvalues
of g is a pm-th primitive root of unity; since g is defined over Q, every pm-th
primitive root of unity must be an eigenvalue of g. This implies that the
number of eigenvalues of g, bounded by its order n < p(p− 1), is at least
φ(pm) = pm−1(p− 1). It follows that m = 1, i.e. that g has order p. Thus
G has exponent p.
Let now K be (Z/pZ)∗; we say that two elements, g and h, of G are
K-conjugate if there exists a k ∈ K such that gk and h are conjugate by an
element of G. By the theory of characters for finite representations (see [4,
Section 12.3]), the number of representations of G which are irreducible
over Q is equal to the number of K-conjugation classes of G. Now, let g
be a non-trivial element of G and assume that it is conjugate to gk, for
some k ∈ K. This means that there exists an element h in G such that
conjugation by h maps g to gk. This implies that conjugation by hp maps g
to gkp = gk; on the other hand hp is the neuter element, thus gk = g. This
shows that any two distinct powers of a same element are not conjugate,
and that every K-conjugation class of G - except the class of the identity
element - is the union of p− 1 distinct conjugation classes of G. In other
words, every Q-irreducible representation of G is equivalent to the direct
sum of the distinct conjugates of some C-irreducible representation of G.
Now, if the group G was non-commutative, its faithful representation G
would contain an irreducible representation of degree d > p, thus also a Q-
irreducible representation of degree (p− 1)d > (p− 1)p > n, which is not
possible. This implies that G is an abelian group.
By the classification of abelian groups, we obtain that G is isomorphic
to the direct product of b copies of Z/pZ, for some integer b. Note that any
faithful representation of G over C has order at least b, and that any faithful
representation of G over Q has order at least b(p− 1). Then b 6 n/(p− 1).

For the rest of this section, we shall assume the hypotesis of Theorem 1,
that is, we have a prime number p 6= 2, an integer n < 3(p− 1), and a
p-group G < SLn(Z).
We remark that, when G is a cyclic group, the theorem is trivially true.
Applying Lemma 4, we obtain that G is cyclic (and the theorem is proved),
except for the case G ∼= Z/pZ× Z/pZ, where 2(p− 1) 6 n < 3(p− 1). Let
us put ourselves in this case.
Note that the proof of Lemma 4 shows that the representation G is the
direct sum of two distinct Q-irreducible representations of order p− 1 and
(n− 2(p− 1)) copies of the trivial representation.
We remark that, after a base-change to the p-th cyclotomic field Q(ζp),
the representation G could be written in diagonal form, as a direct sum
330 Marco Illengo
of its irreducible subrepresentations. Also, after a base-change to Q, the
representation G could be written as a direct sum of its Q-irreducible sub-
representations. Since we are dealing with the action of G on Fnp , though,
we shall restrict to base-changes to Z, which are preserved under reduction
modulo p.
Consider the lattice N := Zn; it contains a sublattice M that is fixed
by G: it is the intersection of N with the subspace (Qn)G of vectors which
are invariant by G. We fix a Z-basis for M and we apply a result on lattices
(see [1, Cor. 3 to Thm. 1, Ch. 1]) to extend it to a basis of N: this splits
the lattice as N = M⊕ L. Now, let ρ be one of the two non-trivial, Q-
irreducible subrepresentations of G, and let H be its kernel. Repeating the
above argument on the restriction of H to L, we determine a basis for Zn
that allows us to write N in the form N(1) ⊕ N(2) ⊕ N(3). Using this new
basis, we can assume that every element g of G is of the form
g =
 I Ag BgO Mg Cg
O O Ng
 ,
where M and N are the two Q-irreducible representations of G of order
p− 1. In particular, we can choose generators σ and τ for G of the forms
σ =
 I Aσ BσO M Cσ
O O I
 ; τ =
 I Aτ BτO I Cτ
O O N
 .
Note that the eigenvalues of M are the p− 1 distinct p-th roots of unity.
This implies that the minimal polynomial of M is (xp − 1)/(x− 1) and that
the determinant of M − I is p.
Over Fp, the matrix M solves the polynomial (x− 1)p−1. Its minimal
polynomial is thus of the form (x− 1)s, for some s < p. This implies that
(M − I)s has all entries in pZ, so that p divides every column of (M − I)s.
Then pp−1 divides its determinant, det(M − I)s = ps; it follows that, over Fp,
the minimal polynomial of M is (x− 1)p−1 and M is a Jordan block. In
particular we deduce the following proposition.
Proposition 5. Let M be as above. For every two non-negative integers i
and j with i + j = p− 1, the image of (M − I)i is the kernel of (M − I)j,
i.e. for every vector1 A ∈ Zp−1
(M − I)jA ≡ O (mod p) ⇐⇒ ∃B ∈ Zp−1 | A ≡ (M − I)iB (mod p).
The same holds for N .
1This immediately extends to matrices (p− 1)×m, for any positive integer m.
Cohomology of integer matrices and local-global divisibility on the torus 331
We remark that a direct computation of στ = τσ provides
στ =
 I Aσ ?O M Cσ + Cτ
O O N

and the relations
Aτ = O, (M − I)Cτ = −Cσ(N − I), Bσ = Aσ(M − I)−1Cσ.(1)
Let now Z̃ be a (G, Fnp )-cocycle that satisfies the local conditions. Then
for every g in G there exists a W̃g in Fnp such that Z̃g ≡ gW̃g − W̃g (mod p);
we choose representants Wg of W̃g in Zn and we define Zg := gWg −Wg for
every g in G. Note that Z̃g ≡ Zg (mod p) for every g in G.
Modulo a coboundary we can assume Zτ ≡ O (mod p). This implies,
by the cocycle relation, Zστ ≡ Zσ + σZτ ≡ Zσ (mod p). By definition, Zσ
and Zστ are: Z
(1)
σ
Z
(2)
σ
Z
(3)
σ
 =
 AσW
(2)
σ + BσW
(3)
σ
(M − I)W (2)σ + CσW (3)σ
O
 ;
Z
(1)
στ
Z
(2)
στ
Z
(3)
στ
 =
 ?(M − I)W (2)στ + (Cσ + Cτ )W (3)στ
(N − I)W (3)στ
 .
We remark that (N − I)W (3)στ ≡ O (mod p); by Proposition 5, this implies
that W (3)στ ≡ (N − I)p−2R̃ (mod p), for some R̃ with entries in Fp. It follows
that, modulo p, (M − I)p−2Z(2)στ is of the form
(M − I)p−1W (2)στ + (M − I)p−2(Cσ + Cτ )(N − I)p−2R̃.
Applying the second relation in (1) and (M − I)p−1 ≡ (N − I)p−1 ≡ O, we
obtain (M − I)p−2Z(2)στ ≡ O (mod p). Applying Proposition 5 to Z(2)σ (or
to Z(2)στ ) we obtain Z
(2)
σ ≡ (M − I)S̃ (mod p), for some S̃ with entries in Fp.
Let S be any representant of S̃ over Z; since the entries of Z(2)σ − (M − I)S
are all divisible by p and since (M − I) has determinant p, we may assume
Z
(2)
σ = (M − I)S. Thus we have
Z(1)σ = Aσ(M − I)−1Z(2)σ = AσS.
Taking V =
(
O
S
O
)
, we have Zσ = σV − V and Zτ ≡ τV − V (mod p).
This implies that Z̃ is a (G, Fnp )-coboundary, concluding the proof of The-
orem 1.
332 Marco Illengo
3. A counterexample
In this section we shall prove Proposition 2. Let p 6= 2 be a prime and
let n > 3(p− 1) be an integer. As we have said in Section 1, we can assume
n = 3(p− 1). We are going to define a p-group G of matrices in SLn(Z)
and a (G, Fnp )-cocycle Z that satisfies the local conditions without being a
coboundary.
Let M ∈ SLp−1(Z) be a matrix with minimal polynomial (xp − 1)/(x− 1)
(for instance, the Frobenius matrix of this polynomial). Note that M sat-
isfies Proposition 5, as in the previous section. Let now u and v be vectors
in Zp−1 such that
u 6≡ O (mod p), v 6≡ O (mod p);
(M − I)u ≡ O (mod p), vt(M − I) ≡ O (mod p).
We define the matrix X := 1pu× v
t, with entries in Q; note that its en-
tries are not all in Z. We also define the matrices A := (M − I)X and
B := X(I −M), with entries in Z.
Let G be the group generated by the matrices σ and τ defined as
σ =
M O AM A
I
 , τ =
I O BM A + B
M
 ;
it is easily verified that G is a subgroup of SLn(Z) and that the map
(i, j) 7→ σiτ j =
M i O M iX −XM jM i+j M i+jX −XM j
M j

provides an isomorphism G ∼= Z/pZ× Z/pZ.
Lemma 6. There exist vectors r, s and t in Zp−1 such that:
Bt ≡ (M − I)r 6≡ O (mod p),
(M − I)Bt ≡ O (mod p),
(A + B)t ≡ (M − I)s (mod p).
Proof. Assume B(M − I)p−2 ≡ O (mod p). Then by Proposition 5 there
exists an integer matrix X0 with B ≡ X0(M − I) (mod p); since (M − I)
has determinant p, this implies that X = −B(M − I)−1 is an integer ma-
trix, which is absurd. Thus B(M − I)p−2 6≡ O (mod p).
We take a vector t0 in Zp−1 with B(M − I)p−2t0 6≡ O (mod p) and we
define t = (M − I)p−2t0; then Bt 6≡ O (mod p).
By definition of A and B we have (M − I)B = −A(M − I). Together
with (M − I)p−1 ≡ O (mod p), this implies
(M − I)B(M − I)p−2 ≡ (M − I)p−2A(M − I) ≡ O (mod p).
Cohomology of integer matrices and local-global divisibility on the torus 333
Then (M − I)Bt ≡ O (mod p) and (M − I)p−2(A + B)t ≡ O (mod p); we
conclude by Proposition 5. 
Proposition 7. The vectors Z(1)σ := O and Z
(1)
τ := Bt define a (G, Fnp )-
cocycle Z ≡
(
Z(1)
O
O
)
(mod p) that is not a (G, Fnp )-coboundary.
Proof. To show that Z is a cocycle we only need to verify, on Z(1), the
cocycle conditions derived from the relations σp = I, τp = I and στ = τσ:
Z
(1)
σp − Z
(1)
I ≡ (M
p−1 + . . . + M + I)Z(1)σ ≡ O (mod p);
Z
(1)
τp − Z
(1)
I ≡ pZ
(1)
τ ≡ O (mod p);
Z(1)στ − Z(1)τσ ≡ (M − I)Z(1)τ ≡ O (mod p).
If Z was a coboundary, then there would exist a vector W in Zn such
that Zg ≡ (g − I)W (mod p) for every g in G; computing Zσ and Zτ , we
would obtain
Z(2)σ ≡ (M − I)W (2) + AW (3) (mod p),
Z(1)τ ≡ BW (3) (mod p),
Z(2)τ ≡ (M − I)W (2) + AW (3) + BW (3) (mod p),
which is absurd, since Z(2)τ ≡ Z(2)σ ≡ O (mod p) and Z(1)τ 6≡ O (mod p). 
It now remains to be shown that Z satisfies the local conditions, i.e. that
for every g in G there exists a Wg in Fnp such that Zg ≡ (g − I)Wg (mod p).
Over τ we have
(τ − I)
 O−s
t
 ≡
O O BO M − I A + B
O O M − I
  O−s
t
 ≡
Z(1)τO
O
 (mod p)
For every i ∈ F∗p we have Z
(1)
τ iσ
≡ iZ(1)τ + Z(1)σ ≡ iBt (mod p); then
(στ i − I)
irO
O
 ≡
M − I ? ?O ? ?
O O ?
 irO
O
 ≡
Z(1)στ iO
O
 (mod p)
Since τ and the στ i with i ∈ Fp are the generators of all non-trivial
cyclic subgroups of G, this shows that Z satisfies the local conditions. This
completes the proof of Proposition 2.
References
[1] J. W. S. Cassels, An introduction to the Geometry of Numbers. Springer, 1997.
[2] R. Dvornicich, U. Zannier, Local-global divisibility of rational points in some commutative
algebraic groups. Bull. Soc. Math. France 129 (2001), no. 3, 317–338.
334 Marco Illengo
[3] R. Dvornicich, U. Zannier, On a local-global principle for the divisibility of a rational
point by a positive integer. Bull. London Math. Soc. 39 (2007), 27–34.
[4] J.-P. Serre, Représentations linéaires des groupes finis. Hermann, 1967.
Marco Illengo
Scuola Normale Superiore
Piazza dei Cavalieri 7
56126 Pisa, Italia
E-mail : marco.illengo@sns.it


Cohomology of integer matrices and local-global divisibility on the torus

Hirofumi Tsumura

Hirofumi TSUMURA
On multiple analogues of Ramanujan’s formulas for certain Dirichlet series
Tome 20, no 1 (2008), p. 219-226.
<http://jtnb.cedram.org/item?id=JTNB_2008__20_1_219_0>
© Université Bordeaux 1, 2008, tous droits réservés.
L’accès aux articles de la revue « Journal de Théorie des Nom-
bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord
avec les conditions générales d’utilisation (http://jtnb.cedram.
org/legal/). Toute reproduction en tout ou partie cet article sous
quelque forme que ce soit pour tout usage autre que l’utilisation à
fin strictement personnelle du copiste est constitutive d’une infrac-
tion pénale. Toute copie ou impression de ce fichier doit contenir la
présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Journal de Théorie des Nombres
de Bordeaux 20 (2008), 219-226
On multiple analogues of Ramanujan’s formulas
for certain Dirichlet series
par Hirofumi TSUMURA
Résumé. Dans cet article, nous prouvons des analogues multiples
des célèbres formules de Ramanujan pour certaines séries de Di-
richlet, qui ont été présentées dans ses cahiers bien connus. De
plus, nous obtenons des versions multiples de formules semblables
à celles de Ramanujan qui ont été données par Berndt et d’autres
auteurs.
Abstract. In this paper, we prove multiple analogues of famous
Ramanujan’s formulas for certain Dirichlet series which were intro-
duced in his well-known notebooks. Furthermore, we prove some
multiple versions of analogous formulas of Ramanujan which were
given by Berndt and so on.
1. Introduction
Let N be the set of natural numbers, N0 = N∪{0}, Z the ring of rational
integers, Q the field of rational numbers, R the field of real numbers, and
C the field of complex numbers.
Ramanujan obtained the following fascinating formula:
α−N
{
1
2
ζ(2N + 1) +
∞∑
k=1
1
(e2kα − 1) k2N+1
}
(1.1)
= (−β)−N
{
1
2
ζ(2N + 1) +
∞∑
k=1
1
(e2kβ − 1) k2N+1
}
− 22N
N+1∑
k=0
(−1)k B2k
(2k)!
B2N+2−2k
(2N + 2− 2k)!
αN+1−kβk,
where N is any non-zero integer, α and β are positive numbers such that
αβ = π2, ζ(s) is the Riemann zeta-function, and {Bn} are the Bernoulli
Manuscrit reçu le 30 mars 2007.
This research is partially supported in part by Grand-in-Aid for Science Research (No.
17540053), Japan Society for the Promotion of Science.
220 Hirofumi Tsumura
numbers defined by
t
et − 1
=
∞∑
n=0
Bn
tn
n!
.
The formula (1.1) was introduced in Ramanujan’s notebooks [5, P. 275],
and its proof was given by several authors (for details, see, for example,
[1, 3, 4, 9]). Let
(1.2) X (s) :=
∞∑
k=1
1
(e2kπ − 1) ks
(s ∈ C).
If we put α = β = π in (1.1) and consider the case s = N for any odd
positive N , then the formulas
X (3) = −1
2
ζ(3) +
7
360
π3;
X (7) = −1
2
ζ(7) +
19
113400
π7;
X (11) = −1
2
ζ(11) +
1453
851350500
π11
(1.3)
can be derived. These results have been already studied by Lerch ([10]) and
Malurkar ([11]); see also [5].
On the other hand, if we consider the case of any odd negative N , then
X (−1) = 1
24
− 1
8π
;
X (−4m− 1) = B4m+2
8m + 2
(m ≥ 1)
(1.4)
can be derived. Furthermore, Ramanujan also obtained
X (1) = − π
12
− 1
2
log
(
$√
2π
)
;
X (−3) = Γ(1/4)
8
5120π6
− 1
240
=
1
80
(
$
π
)4
− 1
240
,
(1.5)
where
$ = 2
∫ 1
0
dx√
1− x4
.
Moreover, we consider
X1(s) =
∞∑
k=1
coth(kπ)
ks
(s ∈ C; <s > 1),(1.6)
Multiple analogues of Ramanujan’s formulas 221
where coth(x) =
(
e2x + 1
)
/
(
e2x − 1
)
. Then we see that for any s ∈ C with
<s > 1,
X1(s) =
∞∑
k=1
e2kπ − 1 + 2
(e2kπ − 1) ks
= 2X (s) + ζ(s).(1.7)
Hence X1(s) can be meromorphically continued to C. By combining (1.1)
and (1.7), we have
X1(4k + 3) =
(2π)4k+3
2
2k+2∑
j=0
(−1)j+1 B2j
(2j)!
B4k+4−2j
(4k + 4− 2j)!
(1.8)
for k ∈ N ∪ {0} (see also Ramanujan’s notebooks [5, (25.3) P. 293]).
Needless to say, the fascination of these formulas is common with that
of Euler’s well-known formula for ζ(2k). For these 15 years, as a multiple
analogue of ζ(s), the multiple zeta value
∞∑
k1,...,kr=1
1
kp11 (k1 + k2)p2 · · · (k1 + · · ·+ kr)pr
(p1, . . . , pr ∈ N; pr ≥ 2)
(see, for example, [8, 13]) has been investigated actively in various branches
of mathematics and physics, and many fascinating formulas among multiple
zeta values were obtained. Therefore it seems to be interesting to consider
certain multiple analogues of X1. Indeed, in [12], Matsumoto and the author
gave some functional relation formulas, for example,∑
k,l∈Z\{0}
k+l>0
(−1)k+l+1
sinh(kπ)k2p(k + l)s
= −
∞∑
k=1
(−1)k+1
sinh(kπ)ks+2p
(1.9)
for s ∈ C, p ∈ Z.
In this paper, as r-ple analogues of X1, we consider
Xr(2p1, . . . , 2pr−1, s)
:=
∑
k∈Φr
coth(kπ)∏r−1
d=1
(∑d
j=1 kj
)2pd · (∑rj=1 kj)s
(1.10)
for r ∈ N with r ≥ 2 and p1, p2, . . . , pr−1 ∈ N, where
Φr =
k = (kj) ∈ (Z \ {0})r
∣∣∣∣ d∑
j=1
kj 6= 0 (1 ≤ d ≤ r − 1),
r∑
j=1
kj > 0
 .
(1.11)
Using the method similar to our previous one, we prove some functional
relations for these series. For example, we have
(1.12) X2(2p, s) = −X1(s + 2p) (s ∈ C; p ∈ N),
222 Hirofumi Tsumura
which is an analogue of (1.9). Putting p = 1 and s = 5 in (1.12), we have
X2(2, 5) = −X1(7) = −
19
56700
π7.
Also, putting s = −3 and using (1.4), we have
X2(2,−3) = −X1(−1) = −2X (−1)− ζ(−1) =
1
4π
in the non-convergent area. These can be regarded as certain multiple ana-
logues of Ramanujan’s formulas for the values not only at positive integers
but also at negative integers.
2. Key lemma
In this section, we prove the following lemma.
Lemma 2.1. For r ∈ N with r ≥ 2, p1, p2, . . . , pr ∈ N and θ ∈ R,
∑
k∈Φr
coth(k1π) sin
((∑r
j=1 kj
)
θ
)
∏r
d=1
(∑d
j=1 kj
)2pd
+ (−1)r
∞∑
k=1
coth(kπ) sin (kθ)
k2
∑r
d=1
pd
= 0.
(2.1)
Proof. We prove the assertion by induction on r ≥ 2. Let
Λr =
k = (kj) ∈ (Z \ {0})r−1 × Z
∣∣∣∣ d∑
j=1
kj 6= 0 (1 ≤ d ≤ r)
 .(2.2)
In the case r = 2, we see that
∑
(k1,k2)∈Λ2
coth(k1π) sin((k1 + k2)θ)
k2p11 (k1 + k2)2p2
=
∑
k1∈Z\{0}
∑
k2∈Z
k1+k2 6=0
coth(k1π) sin((k1 + k2)θ)
k2p11 (k1 + k2)2p2
=
∑
k1∈Z\{0}
coth(k1π)
k2p11
∑
n∈Z\{0}
sin(nθ)
n2p2
= 0
Multiple analogues of Ramanujan’s formulas 223
by putting n = k1 + k2 and changing the index n into −n. The left-hand
side is equal to
∑
k1∈Z\{0}
∑
k2∈Z\{0}
k1+k2 6=0
coth(k1π) sin((k1 + k2)θ)
k2p11 (k1 + k2)2p2
+
∑
k1∈Z\{0}
coth(k1π) sin(k1θ)
k2p1+2p21
= 2
∑
(k1,k2)∈Φ2
coth(k1π) sin((k1 + k2)θ)
k2p11 (k1 + k2)2p2
+ 2
∞∑
k1=1
coth(k1π) sin(k1θ)
k2p1+2p21
,
by changing the indices (k1, k2) into (−k1,−k2). Thus we have the assertion
in the case r = 2.
Now we assume the case of r − 1 holds and consider the case of r.
∑
(kj)∈Λr
coth(k1π) sin
((∑r
j=1 kj
)
θ
)
∏r
d=1
(∑d
j=1 kj
)2pd
=
∑
(kj)∈Λr−1
kr−1 6=0
coth(k1π)∏r−1
d=1
(∑d
j=1 kj
)2pd ∑
n6=0
sin (nθ)
n2pr
= 0
by putting n = k1+· · ·+kr and changing the index n into −n. The left-hand
side is equal to
2
∑
(kj)∈Φr
coth(k1π) sin
((∑r
j=1 kj
)
θ
)
∏r
d=1
(∑d
j=1 kj
)2pd
+ 2
∑
(kj)∈Φr−1
coth(k1π) sin
((∑r−1
j=1 kj
)
θ
)
∏r−2
d=1
(∑d
j=1 kj
)2pd · (∑r−1j=1 kj)2pr−1+2pr
by sparating kr 6= 0 or = 0 and by changing the indices (k1, . . . , kr) into
(−k1, . . . ,−kr). By the assumption of induction, the second term is equal
to
−2(−1)r−1
∞∑
k=1
coth(kπ) sin (kθ)
k2
∑r
d=1
pd
.
Thus we have the assertion in the case of r. By induction, we complete the
proof of this lemma. 
224 Hirofumi Tsumura
3. Main theorem and its proof
In this section, we prove the main theorem in this paper as follows.
Theorem 3.1. For r ∈ N with r ≥ 2 and p1, p2, . . . , pr−1 ∈ N,
Xr(2p1, . . . , 2pr−1, s) = (−1)r−1X1
s + 2 r−1∑
j=1
pj
(3.1)
holds for <s > 1. Furthermore the left-hand side of (3.1) can be continued
meromorphically to C by this equation.
First we quote the following fact which is one of the main result in our
previous paper [12].
Let Z(s) be the Dirichlet series defined by
(3.2) Z(s) =
∞∑
m=1
am
ms
,
where {an} ⊂ C. Let <s = ρ (ρ ∈ R) be the abscissa of convergence of
Z(s). This means that if <s > ρ then Z(s) is convergent and if <s < ρ
then Z(s) is not convergent. We further assume that 0 ≤ ρ < 1.
Theorem 3.2 ([12] Theorem 3.1). Assume that
(3.3)
∞∑
m=1
am sin(mt) = 0
is boundedly convergent for t > 0 and that, for ρ < s < 1,
(3.4) lim
λ→∞
∞∑
m=1
am
∫ ∞
λ
ts−1 sin(mt)dt = 0.
Then Z(s) can be continued meromorphically to C, and actually Z(s) = 0
for all s ∈ C.
Now we aim to apply this theorem to the situation mentioned above. For
this aim, we further quote the following result.
Lemma 3.3 ([12] Lemma 7.6). For r ∈ N and p1, . . . , pr ∈ N,∑
k∈Ωr
1∏r
d=1
(∑d
j=1 kj
)2pd ≤ r∏
d=1
{2ζ(2pd)} < ∞,
where
Ωr :=
{
k = (kj) ∈ (Z \ {0})r
∣∣∣∣ d∑
j=1
kj 6= 0 (1 ≤ d ≤ r)
}
.
Multiple analogues of Ramanujan’s formulas 225
Proof of Theorem 3.1. By Lemma 3.3, we see that the left-hand side of
(2.1) satisfies the conditions (3.3) and (3.4) because of the absolute conver-
gency. Hence, by Theorem 3.2, we see that
(3.5) Xr(2p1, . . . , 2pr−1, 2pr + s) + (−1)rX1
s + 2 r∑
j=1
pj
 = 0
holds for any s ∈ C, where the left-hand side of (3.5) is continued mero-
morphically to C. Since X1(s) is continued meromorphically to C, so is
Xr(2p1, . . . , 2pr−1, 2pr + s). Hence, by placing 2pr + s with s, we have the
assertion of Theorem 3.1. 
Example 1. By Theorem 3.1, we have, for example,
X2(2, s) = −X1(s + 2);(3.6)
X3(2, 6, s) = X1(s + 8);(3.7)
X4(4, 4, 2, s) = −X1(s + 10).(3.8)
Putting s = 3 in (3.7) and s = 5 in (3.8), we respectively obtain
X3(2, 6, 3) = X1(11) =
1453
425675250
π11;
X4(4, 4, 2, 5) = −X1(15) = −
13687
390769879500
π15.
Also, by putting s = −3 in (3.6) and by using (1.5) and (1.7), we have
X2(2,−5) = −X1(−3) = −2X (−3)− ζ(−3)
= −Γ(1/4)
8
2560π6
= − 1
40
(
$
π
)4
.
We can regard these results as multiple analogues of Ramanujan’s formulas
for the values not only at positive integers but also at negative integers.
Acknowledgements. The author would like to express his sincere grati-
tude to Professor Bruce C. Berndt for his valuable comments and encour-
agements during his stay in Kyoto. The author also thanks Professors Kohji
Matsumoto and Yasushi Komori for their valuable suggestions about ana-
lytic properties of multiple Dirichlet series. Furthermore the author greatly
thanks the referee for his or her helpful comments and suggestions.
References
[1] B. C. Berndt, Generalized Dedekind eta-functions and generalized Dedekind sums. Trans.
Amer. Math. Soc. 178 (1973), 495–508.
[2] B. C. Berndt, Generalized Eisenstein series and modified Dedekind sums. J. Reine Angew.
Math. 272 (1974), 182–193.
226 Hirofumi Tsumura
[3] B. C. Berndt, Modular transformations and generalizations of several formulae of Ra-
manujan. Rocky Mountain J. Math. 7 (1977), 147–189.
[4] B. C. Berndt, Analytic Eisenstein series, theta-functions, and series relations in the spirit
of Ramanujan. J. Reine Angew Math. 303/304 (1978), 332–365.
[5] B. C. Berndt, Ramanujan’s Notebooks, part II. Springer-Verlag, New-York, 1989.
[6] B. C. Berndt, Ramanujan’s Notebooks, part V. Springer-Verlag, New-York, 1998.
[7] K. Dilcher, Zeros of Bernoulli, generalized Bernoulli and Euler polynomials. Memoirs of
Amer. Math. Soc. 386 (1988).
[8] M. E. Hoffman, Multiple harmonic series. Pacific J. Math. 152 (1992), 275–290.
[9] K. Katayama, On Ramanujan’s formula for values of Riemann zeta-function at positive
odd integers. Acta Arith. 22 (1973), 149–155.
[10] M. Lerch, Sur la fonction ζ(s) pour valeurs impaires de l’argument. J. Sci. Math. Astron.
pub. pelo Dr. F. Gomes Teixeira, Coimbra 14 (1901), 65–69.
[11] S. L. Malurkar, On the application of Herr Mellin’s integrals to some series. J. Indian
Math. Soc. 16 (1925/1926), 130–138.
[12] K. Matsumoto, H. Tsumura, A new method of producing functional relations among
multiple zeta-functions. Quart. J. Math. 59 (2008), 55–83.
[13] D. Zagier, Values of zeta functions and their applications. In “Proc. First Congress of
Math., Paris”, vol. II, Progress in Math. 120, Birkhäuser, 1994, 497–512.
Hirofumi Tsumura
Department of Mathematics and Information Sciences
Tokyo Metropolitan University
Minami-Ohsawa, Hachioji
192-0397 Tokyo, Japan
E-mail : tsumura@tmu.ac.jp


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