Comptes Rendus Mathématique

Sourav Das

Comptes Rendus
Mathématique
Sourav Das
Inequalities Involving q-Analogue of Multiple Psi Functions
Volume 358, issue 3 (2020), p. 327-332.
<https://doi.org/10.5802/crmath.44>
© Académie des sciences, Paris and the authors, 2020.
Some rights reserved.
This article is licensed under the
CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2020, 358, n 3, p. 327-332
https://doi.org/10.5802/crmath.44
Complex Analysis / Analyse complexe
Inequalities Involving q-Analogue of Multiple
Psi Functions
Inégalités impliquant des q-analogues des fonction psi
multiples
Sourav Das a
a Department of Mathematics, National Institute of Technology Jamshedpur,
Jharkhand-831014, India.
E-mails: souravdasmath@gmail.com, souravdas.math@nitjsr.ac.in.
Abstract. Logarithmic derivative of the multiple gamma function is known as the multiple psi function. In
this work q-analogue of multiple psi functions of order n have been considered. Subadditive, superadditive
and convexity properties of higher order derivatives of these functions are derived. Some related inequalities
for these functions and their ratios are also obtained.
Résumé. La dérivée logarithmique de la fonction gamma multiple est connue comme la fonction psi mul-
tiple. Dans ce travail, des q-analogues de fonctions psi multiples d’ordre n ont été considérés. Des propriétés
de sous-additivité, superadditivité et convexité des dérivées d’ordre supérieur de ces fonctions en découlent.
Certaines inégalités apparentées sont également obtenues pour ces fonctions et leur rapports.
2020 Mathematics Subject Classification. 33B15, 26D07, 26D15.
Manuscript received 18th August 2019, revised 13th February 2020, accepted 24th March 2020.
1. Introduction
The multiple gamma functions of Barnes [4, 5] introduced more than a century ago have been
taken up during the last decades because they enter in several different areas of modern mathe-
matics represented by e.g., S. N. M. Ruijsenaars [13] and M. F. Vignéras [16]. These functions are
useful in summation of series and infinite products and in the theory of elliptic functions and
theta functions [7]. The inverse of the double gamma function is the well known G-function [7].
Multiple gamma functions Γn are useful to study the determinant of Laplacians on the n-
dimensional unit sphere Sn [7]. The multiple gamma functions Γn (n ∈N) which are also defined
and denoted by Gn := Γ(−1)
n−1
n satisfy the following recurrence relation
Γn+1(z +1)Γn(z) = Γn+1(z), Γ1(z) = Γ(z), Γ(1) = 1, (n ∈N, z ∈C), (1)
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
328 Sourav Das
where Γ denotes the familiar Euler’s gamma function (see, e.g., [14, Sections 1.1 and 1.4]). Here
and throughout, let N and C be sets of positive integers and complex numbers, respectively. The
above relations (1) with the condition (−1)n d n+1
d xn+1 logΓn(x) ≥ 0, x > 0 are known as the conditions
of generalized Bohr–Mollerup theorem. Logarithmic derivative of the multiple gamma function
is known as the multiple psi function and is denoted byΨn(z) = Γ
′
n (z)
Γn (z)
. The poly multiple gamma
functionΨ(m)n is the m-th order derivative ofΨn .
Quantum calculus (or q-calculus), sometimes called calculus without limits plays important
role in mathematical physics, quantum physics, theoretical physics, approximation theory and
in many branches of applied science and engineering. Because of the nonlinear structure of the
quantum calculus, several problems remain unsolved. Nowadays, most of the researchers and
mathematicians [6, 8, 12, 14, 15] are interested in q-calculus due to its applications in various
fields of science and engineering.
Throughout this paper we suppose 0 < q < 1. The q-deformed gamma function was first
introduced by F. Jackson in [10, 11] and is defined as [8]
Γq (x) = lim
n→∞
∫ [n]q
0
(
1− t
[n]q
)n
q
t x−1dq t =
∫ [∞]q
0
Eq (−t )t x−1dq t , (2)
where [x]q , [∞]q and [n]q ! are defined as
[x]q = 1−q
x
1−q , [∞]q =
1
1−q and [n]q ! = 1(1+q)(1+q +q
2) · · · (1+q +q2 +·· ·+qn−1)
and q-analogue of exponential function is defined as
Eq (x) =
∞∑
k=0
qk(k−1)/2
[k]q !
xk .
The q-binomial (Gaussian binomial) is defined as
(x + y)nq =
n∑
k=0
(
n
k
)
q
xk yn−k
with (
n
k
)
q
= [n]q !
[k]q ![n −k]q !
.
It is well known that
lim
q→1Γq (x) = Γ(x) =
∫ ∞
0
e−t t x−1dt .
The q-analogue of multiple gamma functions of order n are denoted by Γn(z; q) =
(Gn(z; q))(−1)
n−1
and satisfies a q-analogue of generalized Bohr–Mollerup theorem:
Theorem 1 ( [15]). There exists a unique hierarchy of functions which satisfy
(1) Gn(z +1; q) =Gn−1(z; q)Gn(z; q),
(2) Gn(1, q) = 1,
(3) d
n+1
dxn+1 logGn(x +1; q) ≥ 0 for x ≥ 0,
(4) G0(z; q) = [z]q .
A function f is called subadditive on a set I of real numbers if f (x + y) ≤ f (x) + f (y) for
all x, y ∈ I such that x + y ∈ I . If the inequality reverses, then f is called superadditive on I . If
f (x y) ≤ f (x) f (y) holds for all x, y ∈ I such that x y ∈ I , then f is known as submultiplicative. If the
inequality reverses then f is called supermultiplicative. These functions play vital role in number
theory, in the theory of differential equations and also in the theory of convex bodies.
H. Alzer and S. Ruscheweyh [3] proved that x 7→ (Γ(x))α is subadditive on (0,∞) if and only if
α∗ ≤ α ≤ 0, where α∗ ≈ −0.946850. . .. In [1], H. Alzer derived that Ψ(ex ) is strictly concave on R,
C. R. Mathématique, 2020, 358, n 3, 327-332
Sourav Das 329
where Ψ(x) = ddx logΓ(x) is known as the psi (digamma) function. Recently in 2007, H. Alzer [2]
proved the subadditive and superadditive properties of Euler’s gamma function and obtained the
following interesting inequality(
Γ(x + y + c)
Γ(x + y)
)1/α
<
(
Γ(x + c)
Γ(x)
)1/α
+
(
Γ(y + c)
Γ(y)
)1/α
.
The above inequality holds for all x, y > 0 if and only if α ≤ max(1,c), where 0 < c 6= 1. The
reverse inequality is valid for all positive x and y if and only if α ≤ min(1,c). In [9], B.-N. Guo,
F. Qi and Q.-M. Luo discussed the additivity of polygamma functions. In [12] T. Mansour and
A. Sh. Shabani derived several inequalities for the q-digamma function. Recently, N. Batir [6]
obtained monotonicity properties of q-digamma and q-trigamma functions.
The above results motivate us to derive subadditive, superadditive and convexity properties of
functions involvingΨn(x; q) = Γ
′
n (x;q)
Γn (x;q)
and its higher order derivatives (Ψ(m)n (x; q) = d
m
dxmΨn(x; q)).
Further we obtain some related inequalities for these functions and their ratios.
2. Inequalities for q-Poly Multiple Gamma Function
The q-analogue of multiple gamma functions of order n can also be defined as [15, (6)]
Γn(z; q) = (Gn(z; q))(−1)
n−1
, (3)
where
logGn(z +1; q) =−
(
z
n
)
log(1−q)−
∞∑
k=1
(
−k
n −1
)
log(1−q z+k )
+
n−1∑
j=0
Gn, j (z)
{ ∞∑
k=1
k j log(1−qk )
}
,
(4)
where (
z −u
n −1
)
=
n−1∑
j=0
Gn, j (z)u
j .
It can be noted that (
−k
n −1
)
= (−1)n−1
(
n +k −2
n −1
)
. (5)
Let f (z) = log(1−q z+k ). Then
f ′(z) =− ln q
∞∑
k=0
q z+k
1−q z+k =− ln q
∞∑
k=1
qkz
1−qk
=⇒ f (m+1)(z) =−(ln q)m+1
∞∑
k=1
km qkz
1−qk . (6)
From (3), we have
logGn(z; q) = (−1)n−1 logΓn(z; q)). (7)
Using (7), (3), (5) and (6) and differentiating (4) with respect to z, m +1 times (m ≥ n) we have
Ψ(m)n (z; q) = (ln q)m+1
∞∑
k=1
(
n +k −2
n −1
)
km qkz
1−qk , 0 < q < 1, m ≥ n.
Hence, clearly we have the following conclusions
(i) If m is odd (even), thenΨ(m)n (x; q) > (<)0 for 0 < q < 1 and x > 0.
(ii) If m is odd (even), thenΨ(m)n (x; q) is decreasing (increasing) for 0 < q < 1 and x > 0.
Thus we have the following results.
C. R. Mathématique, 2020, 358, n 3, 327-332
330 Sourav Das
Theorem 2. Ψ(m)n (x; q) is convex (concave) on (0,∞) if m is odd (even) for 0 < q < 1.
Corollary 3. Ψ(m)n (e
x ; q) is convex (concave) on R if m is odd (even) for 0 < q < 1.
Now we will discuss about subadditive and superadditive properties.
Theorem 4. For 0 < q < 1, a ≥ 0, x, y > 0 and m ≥ n, the following inequalities hold:
Ψ(m)n (a +x + y ; q) ≤Ψ(m)n (a +x; q)+Ψ(m)n (a + y ; q), n is odd;
Ψ(m)n (a +x + y ; q) ≥Ψ(m)n (a +x; q)+Ψ(m)n (a + y ; q), n is even.
Proof. Let f (x) =Ψ(m)n (a +x + y ; q)−Ψ(m)n (a +x; q)+Ψ(m)n (a + y ; q). Let m be odd. Then keeping
y fixed, we have
f ′(x) =Ψ(m+1)n (a +x + y ; q)−Ψ(m+1)n (a +x; q)
= (ln q)m+1
∞∑
k=1
(
n +k −2
n −1
)
km
1−qk
(
qk(a+x+y) −qk(a+x)
)
Since u ≥ v ⇒ qu ≤ q v for 0 < q < 1. Therefore, f ′(x) < 0. Hence, f is decreasing for x, y ∈ (0,∞)
and 0 < q < 1.
Clearly, limx→∞ f (x) > 0.
Consequently, f (x) > 0 for x, y > 0 and 0 < q < 1.
Similarly, if m is even we can show that f (x) < 0 for x, y > 0 and 0 < q < 1. which proves the
theorem. 
Now we have the following corollary explaining the additivity of Ψ(m)n (x; q) for x > 0 and
0 < q < 1.
Corollary 5. Ψ(m)n (x; q) is subadditive (superadditive) if m is odd (even) for x > 0 and 0 < q < 1.
Proof. a = 0 in Theorem 4 gives the proof of the corollary. 
Theorem 6. For 0 < q < 1, a ≥ 0, x, y > 0, the following inequalities hold[
Ψ(m)n (a +x + y ; q)
]2 ≤Ψ(m)n (a +x; q)Ψ(m)n (a + y ; q), if m is odd; (8)[
Ψ(m)n (a +x + y ; q)
]2 ≥Ψ(m)n (a +x; q)Ψ(m)n (a + y ; q), if m is even, (9)
for all m ≥ n.
Proof. Let m ≥ n. We will use the fact that u ≥ v ≥ 0 ⇒ qu ≤ q v for 0 < q < 1 to prove the theorem.
Case I. Let m be odd. Then (ln q)m+1 > 0. Therefore,
Ψ(m)n (a +x; q) = (ln q)m+1
∞∑
k=1
(
n +k −2
n −1
)
km qk(a+x)
1−qk
≥ (ln q)m+1
∞∑
k=1
(
n +k −2
n −1
)
km qk(a+x+y)
1−qk
≥Ψ(m)n (a +x + y ; q) > 0. (10)
Similarly,
Ψ(m)n (a + y ; q) ≥Ψ(m)n (a +x + y ; q) > 0. (11)
Now combining (10) and (11), we have
Ψ(m)n (a +x; q)Ψ(m)n (a + y ; q) ≥
[
Ψ(m)n (a +x + y ; q)
]2
.
C. R. Mathématique, 2020, 358, n 3, 327-332
Sourav Das 331
Case II. Let m be even. Then (ln q)m+1 < 0 for 0 < q < 1. Then
Ψ(m)n (a +x; q) = (ln q)m+1
∞∑
k=1
(
n +k −2
n −1
)
km qk(a+x)
1−qk
=⇒ 0 <−Ψ(m)n (a +x; q) ≤−Ψ(m)n (a +x + y ; q)
and 0 <−Ψ(m)n (a + y ; q) ≤−Ψ(m)n (a +x + y ; q).
Combining above inequalities, we have
Ψ(m)n (a +x; q)Ψ(m)n (a + y ; q) ≤
[
Ψ(m)n (a +x + y ; q)
]2
,
which proves the theorem. 
Theorem 7. Let 0 < q < 1, a ≥ 0 and 0 < x, y < 1. Then[
Ψ(m)n (a +x y ; q)
]2 >Ψ(m)n (a +x; q)Ψ(m)n (a + y ; q).
Proof. Let f (x) = [Ψ(m)n (a +x; q)]2. Then
f ′(x) = 2Ψ(m)n (a +x; q)Ψ(m+1)n (a +x; q) < 0.
which implies that f (x) is a decreasing on (0,1). Therefore,[
Ψ(m)n (a +x y ; q)
]2 > [Ψ(m)n (a +x; q)]2 ;[
Ψ(m)n (a +x y ; q)
]2 > [Ψ(m)n (a + y ; q)]2 .
Combining the above inequalities, we have[
Ψ(m)n (a +x y ; q)
]4 > [Ψ(m)n (a +x; q)Ψ(m)n (a + y ; q)]2 .
Since,Ψ(m)n (a +x; q)Ψ(m)n (a + y ; q) > 0. Therefore,[
Ψ(m)n (a +x y ; q)
]2 >Ψ(m)n (a +x; q)Ψ(m)n (a + y ; q).
Hence the theorem is proved. 
Now we will find the bounds for the ratio of q-analogue of multiple psi functions.
Theorem 8. Let a,b,c,d ,e, f be real numbers and f (x) be a function defined as
f (x) = Ψ
(m)
n (a +bx; q)c
Ψ(m)n (d +ex; q) f
, x ≥ 0, m ≥ n ≥ 1.
(a) If a,b,d ,e > 0, c ≤ 0 and f ≥ 0, then f (x) is increasing in [0,∞) and for all x ∈ [0,1] the
following inequality holds
Ψ(m)n (a; q)
c
Ψ(m)n (d ; q) f
≤ Ψ
(m)
n (a +bx; q)c
Ψ(m)n (d +ex; q) f
≤ Ψ
(m)
n (a +b; q)c
Ψ(m)n (d +e; q) f
.
(b) If a,b,d ,e > 0, c ≥ 0 and f ≤ 0, then f (x) is decreasing in [0,∞) and for all x ∈ [0,1] the
following inequality holds
Ψ(m)n (a +b; q)c
Ψ(m)n (d +e; q) f
≤ Ψ
(m)
n (a +bx; q)c
Ψ(m)n (d +ex; q) f
≤ Ψ
(m)
n (a; q)
c
Ψ(m)n (d ; q) f
.
Proof. Let g (x) = ln f (x). Then
g ′(x) = bcΨ
(m)
n (d +ex; q)Ψ(m+1)n (a +bx; q)−e f Ψ(m)n (a +bx; q)Ψ(m+1)n (d +ex; q)
Ψ(m)n (a +bx; q)Ψ(m)n (d +ex; q)
. (12)
C. R. Mathématique, 2020, 358, n 3, 327-332
332 Sourav Das
Now,
Ψ(m)n (d +ex; q)Ψ(m+1)n (a +bx; q) < 0,Ψ(m+1)n (d +ex; q)Ψ(m)n (a +bx; q) < 0
andΨ(m)n (a +bx; q)Ψ(m)n (d +ex; q) > 0.
Hence, using the conditions of part (a), we have g ′(x) ≥ 0, which implies that g (x) is increasing
in [0,∞). Consequently, f (x) is increasing in [0,∞) and for 0 ≤ x ≤ 1, f (0) ≤ f (x) ≤ f (1), which
proves part (a) of the theorem.
Note that the condition (b) reverses the inequalities given in part (a). Hence, proceeding
similarly like part (a), part (b) of the theorem can be established. 
Acknowledgements
The author wishes to thank the reviewers for suggestions that helped to improve the paper.
References
[1] H. Alzer, “Sharp inequalities for the diagramma and polygamma functions”, Forum Math. 16 (2004), no. 2, p. 181-221.
[2] ——— , “Sub- and superadditive properties of Euler’s gamma function”, Proc. Am. Math. Soc. 135 (2007), no. 11,
p. 3641-3648.
[3] H. Alzer, S. Ruscheweyh, “A subadditive property of the gamma function”, J. Math. Anal. Appl. 285 (2003), no. 2,
p. 564-577.
[4] E. W. Barnes, “The theory of the G-function”, Q. J. Math 31 (1899), p. 264-314.
[5] ——— , “On the theory of the multiple Gamma function”, Cambr. Trans. 19 (1904), p. 374-439.
[6] N. Batir, “Monotonicity properties of q-digamma and q-trigamma functions”, J. Approx. Theory 192 (2015), p. 336-
346.
[7] J. Choi, “Multiple gamma functions and their applications”, in Analytic number theory, approximation theory, and
special functions, Springer, 2014, p. 93-129.
[8] W. S. Chung, T. Kim, T. Mansour, “The q-deformed gamma function and q-deformed polygamma function”, Bull.
Korean Math. Soc. 51 (2014), no. 4, p. 1155-1161.
[9] B.-N. Guo, F. Qi, Q.-M. Luo, “The additivity of polygamma functions”, Filomat 29 (2015), no. 5, p. 1063-1066.
[10] F. H. Jackson, “On q-functions and a certain difference operator”, Trans. R. Soc. Edinb. 46 (1908), no. 2, p. 253-281.
[11] ——— , “On a q-definite integrals”, Quart. J. 41 (1910), p. 193-203.
[12] T. Mansour, A. S. Shabani, “Some inequalities for the q-digamma function”, JIPAM, J. Inequal. Pure Appl. Math. 10
(2009), no. 1, article ID 12 (8 pages).
[13] S. N. M. Ruijsenaars, “On Barnes’ multiple zeta and gamma functions”, Adv. Math. 156 (2000), no. 1, p. 107-132.
[14] H. M. Srivastava, J. Choi, Zeta and q-Zeta functions and associated series and integrals, Elsevier, 2012.
[15] K. Ueno, M. Nishizawa, “The multiple gamma function and its q-analogue”, in Quantum groups and quantum spaces
(Warsaw, 1995), Banach Center Publications, vol. 40, Polish Academy of Sciences, 1995, p. 429-441.
[16] M.-F. Vignéras, “L’équation fonctionnelle de la fonction zeta de Selberg de groupe modulaire PSL(2;Z)”, in Journees
arithmétiques de Luminy (1978), Astérisque, vol. 61, Société Mathématique de France, 1979, p. 235-249.
C. R. Mathématique, 2020, 358, n 3, 327-332


Inequalities Involving $q$-Analogue of Multiple Psi Functions

Manuel Krannich

Oscar Randal-Williams

Comptes Rendus
Mathématique
Manuel Krannich and Oscar Randal-Williams
Mapping class groups of simply connected high-dimensional manifolds
need not be arithmetic
Volume 358, issue 4 (2020), p. 469-473.
<https://doi.org/10.5802/crmath.61>
© Académie des sciences, Paris and the authors, 2020.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2020, 358, n 4, p. 469-473
https://doi.org/10.5802/crmath.61
Group Theory, Differential Topology / Théorie des groupes, Topologie différentielle
Mapping class groups of simply connected
high-dimensional manifolds need not be
arithmetic
Manuel Krannicha and Oscar Randal-Williams∗, a
a Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK.
E-mails: krannich@dpmms.cam.ac.uk, o.randal-williams@dpmms.cam.ac.uk.
Abstract. It is well known that Sullivan showed that the mapping class group of a simply connected high-
dimensional manifold is commensurable with an arithmetic group, but the meaning of “commensurable” in
this statement seems to be less well known. We explain why this result fails with the now standard definition
of commensurability by exhibiting a manifold whose mapping class group is not residually finite. We do not
suggest any problem with Sullivan’s result: rather we provide a gloss for it.
Résumé. Il est notoire que Sullivan a démontré que le groupe de difféotopie d’une variété de haute dimension
simplement connexe est commensurable avec un groupe arithmétique, mais la signification du terme
« commensurable » dans son théorème semble bien moins connue. Nous expliquons la raison pour laquelle
ce résultat n’est plus vrai en utilisant la définition désomais standard de commensurabilité en exhibant une
variété dont le groupe de difféotopie n’est pas résiduellement fini. Il n’est pas question d’un problème avec le
théorème de Sullivan, mais plutôt d’y ajouter une glose.
2020 Mathematics Subject Classification. 57R50, 11F06, 20E26.
Funding. The authors were supported by the ERC under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No. 756444), and by a Philip Leverhulme Prize from the Leverhulme
Trust.
Manuscript received 25th October 2019, revised 17th March 2020 and 20th April 2020, accepted 26th April 2020.
In the landmark paper [13] Sullivan proves a structural result on the groups π0 Diff(M) of
isotopy classes of diffeomorphisms of a simply connected compact manifold M of dimension
at least 5, which has as consequence that these groups are “commensurable” with arithmetic
groups (Theorem 13.3 of loc. cit.).
Sullivan defines “commensurability” of groups (at the bottom of p. 307 of loc. cit.) as the
equivalence relation generated by passing to subgroups of finite index and taking quotients
by finite normal subgroups. This differs from the current common usage of this term as the
equivalence relation generated only by passing to subgroups of finite index; from now on let us
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
470 Manuel Krannich and Oscar Randal-Williams
reserve commensurable for the latter term, and refer to Sullivan’s notion as commensurable up to
finite kernel.
These two notions differ. Recall that a group is residually finite if the intersection of all its finite
index subgroups is trivial, or equivalently, if each nontrivial element remains nontrivial in some
finite quotient. As arithmetic groups are residually finite [11, p. 108], and residual finiteness is
clearly preserved by passing to finite index sub- and supergroups, a group commensurable to an
arithmetic group is in particular residually finite. However, work of Deligne [2] shows that there
are central extensions
0 −→ Z/n −→ Γ−→ Sp2g (Z) −→ 0
for which Γ is not residually finite. Such a Γ is clearly commensurable up to finite kernel with the
arithmetic group Sp2g (Z), but cannot be commensurable with an arithmetic group.
The aim of this note is to explain how Deligne’s example may be imported into manifold theory
to provide a mapping class group of a simply connected high-dimensional manifold that is not
residually finite, and hence not commensurable with an arithmetic group.
Theorem. For n ≡ 5 (mod 8) and g ≥ 5, the group π0 Diff(]g Sn ×Sn) is not residually finite.
The best known family of groups commensurable up to finite kernel with an arithmetic group
but not residually finite is Deligne’s, and the best known (to the authors, as we have studied them
elsewhere [3,6]) family of mapping class groups of simply connected high-dimensional manifolds
is that of ]g Sn × Sn . Fortunately these examples are often close to each other; the proof of the
Theorem will be to make this precise.
Remark. Sullivan also proves that the groups π0 hAut(X ) of homotopy classes of homotopy
equivalences of a simply connected finite CW complex X are commensurable up to finite ker-
nel with an arithmetic group [13, Theorem 10.3(i)]. However, as explained by Serre [11, p. 108],
in this case further results of Sullivan may be used to show that such groups are commensurable
to arithmetic groups in our sense. Indeed, the groups π0 hAut(X ) are residually finite as a con-
sequence of [12, Theorem 3.2], and among residually finite groups being commensurable up to
finite kernel and being commensurable agree: using the second description of residual finiteness
mentioned above, one can deduce that for any finite normal subgroup K ≤ G of a residually fi-
nite group G , there is a finite group C and a morphism G → C such that the induced morphism
G → (G/K )×C is injective and hence exhibits G as a finite index subgroup of (G/K )×C , so G and
G/K are commensurable.
As a point of terminology, the finite residual of a group is the intersection of all its finite index
subgroups; a group is residually finite precisely if its finite residual is trivial.
Deligne’s extensions
We begin by explaining the special case of Deligne’s work [2] mentioned above in a bit more detail
(see also [10]). The fundamental group of the Lie group Sp2g (R) can be identified with Z, so the
pullback of its universal cover to Sp2g (Z) yields a central extension of the form
0 −→ Z −→ S̃p2g (Z) −→ Sp2g (Z) −→ 0. (1)
As long as Sp2g (Z) has the congruence subgroup property, i.e. for g ≥ 2, Deligne’s work implies
that the finite residual of S̃p2g (Z) agrees with the subgroup 2 · Z ⊂ S̃p2g (Z), so S̃p2g (Z) is in
particular not residually finite. Moreover, this shows that for n ≥ 3 the quotients S̃p2g (Z)/(n ·Z)
are not residually finite either, and hence give rise to non-residually finite central extensions of
Sp2g (Z) as asserted in the introduction.
There is an alternative description of the extension (1), more convenient for our purposes,
which we now explain.
C. R. Mathématique, 2020, 358, n 4, 469-473
Manuel Krannich and Oscar Randal-Williams 471
The signature class
Let us describe Meyer’s signature class sgn ∈ H2(BSp2g (Z)), following [8]. For g ≥ 3 the group
Sp2g (Z) is perfect, i.e. the first homology H1(BSp2g (Z)) vanishes, so by the universal coefficient
theorem it suffices to specify a morphism
sgn: H2(BSp2g (Z)) −→ Z. (2)
To do so, recall that, as a consequence of classical work of Thom [14], the second integral
homology group of a space X is canonically isomorphic to the group of cobordism classes of
continuous maps f : S → X for oriented closed surfaces S. As a result, we may represent a
homology class in H2(BSp2g (Z)) by a map f : S → X of this kind and associate to it a symmetric
bilinear form given by the composition
H1(S; f ∗H )⊗H1(S; f ∗H ) ^−→ H2(S; f ∗(H ⊗H )) λ∗−−→ H2(S;Z) 〈[S],−〉−→ Z.
Here H denotes the local system H → BSp2g (Z) induced by the standard module Z2g of Sp2g (Z),
which comes equipped with a canonical fibrewise symplectic form λ : H ⊗ H → Z. In [8,
§3] Meyer notes that this form is nondegenerate after passing to the torsion free quotient of
H1(S; f ∗H ) (as a consequence of Poincaré duality). Moreover, he shows that the signature of the
induced non-degenerate form on the torsion free quotient depends only on the bordism class of
f (a a consequence of Lefschetz duality), so taking signatures yields a morphism of the form (2).
The signature class is closely related to the extension (1): by construction, this extension is
pulled back from the central extension of BSp2g (R) ' BU(g ) that, under the classification of
central extensions in terms of second cohomology, corresponds to a generator in H2(BSp2g (R)) ∼=
Hom(π1 Sp2g (R),Z) ∼= Z. This group is also generated by the first Chern class c1 ∈ H2(BSp2g (R)), so
up to possibly rechoosing the identification π1 Sp2g (R) ∼= Z, the extension (1) is classified by the
pullback c1 ∈ H2(BSp2g (Z)) along the inclusion Sp2g (Z) ⊂ Sp2g (R). It is a consequence of Meyer’s
signature formula [8, §4] (see also [9, p. 246]) that
sgn =−4 · c1 ∈ H2(BSp2g (Z)). (3)
The theta subgroup
The mapping class groups we shall consider are closely connected to a certain finite index sub-
group Spq2g (Z) ≤ Sp2g (Z), sometimes called the theta group, which consists of those symplectic
matrices that preserve the standard quadratic refinement, i.e. the function q : Z2g → Z/2 mapping∑
i (ai ·ei +bi · fi ) to
∑
i ai bi , where (ei , fi )1≤i≤g is the usual symplectic basis. By [3, Lemma 7.5], the
signature morphism (2) becomes divisible by 8 when restricted to H2(BSp
q
2g (Z)), but the resulting
morphism sgn/8: H2(BSp
q
2g (Z)) → Z does not uniquely determine a class in H2(BSp
q
2g (Z)), be-
cause in contrast to Sp2g (Z), the subgroup Sp
q
2g (Z) is not perfect. To specify such a class, we use
that the abelianisation of Spq2g (Z) is by [4, p. 147] cyclic of order 4 and generated by
Xg = diag
((
0 −1
1 0
)
,
(
1 0
0 1
)
, . . . ,
(
1 0
0 1
))
as long as g ≥ 3. This implies that the morphism Z/4 → Spq2g (Z) defined by Xg induces an
isomorphism on torsion subgroups of second cohomology, so induces a splitting
H2(BSp2g (Z))
∼=−→ Hom(H2(BSp2g (Z)),Z)⊕H2(BZ/4), (4)
which we can use to define a class
µ ∈ H2(BSpq2g (Z))
C. R. Mathématique, 2020, 358, n 4, 469-473
472 Manuel Krannich and Oscar Randal-Williams
by declaring it to agree with the divided signature sgn/8 on the first summand and to be trivial
on the second. This class is related to the pullback of c1 ∈ H2(BSp2g (Z)) by
8 ·µ=−4 · c1 ∈ H2(BSpq2g (Z)), (5)
since both sides agree in the first summand of (4) as a result of (3) and in the second because
H2(BZ/4) is annihilated by 4. By Deligne’s result the finite residual of the central extension (1)
classified by c1 is 2 ·Z, which implies that the analogous central extension of Sp2g (Z) classified by
−4 · c1 has finite residual 8 ·Z, using that
(i) for a central extension 0 → A → E →G → 0 classified by a classλ ∈ H2(BG; A), the multiple
k ·λ ∈ H2(BG; A) for k ∈ Z classifies the pushout of this extension along the multiplication
map k · (−) : A → A (see [1, p. 94]), and that
(ii) taking finite residuals is functorial with respect to all group homomorphisms, and maps
inclusions of finite index to isomorphisms.
Since Spq2g (Z) ≤ Sp2g (Z) has finite index and we just argued that the central extension of Sp2g (Z)
classified by −4 ·c1 ∈ H2(BSp2g (Z)) has finite residual 8 ·Z, we conclude that the central extension
of Spq2g (Z) classified by −4 · c1 ∈ H2(BSpq (Z)) also has finite residual 8 ·Z. Using (i) and (ii) once
more, we derive from (5) that the central extension
0 −→ Z −→ Eg −→ Spq2g (Z) −→ 0 (6)
classified by µ has finite residual Z.
Remark. It does not matter for our argument, but the relation (5) cannot be divided by 4. By
definition µ vanishes on BZ/4, but c1 restricts to a generator of H2(BZ/4) ∼= Z/4 because the
composition BZ/4 → BSpq2 (Z) → BSp2(R) ' BU(1) classifies the tautological line bundle.
Proof of the Theorem
Using work of Kreck [7], it is shown in [3, Equation 7.3 and Theorem 7.7] that for g ≥ 5 and n ≡
5 (mod 8) the mapping class group of orientation preserving diffeomorphisms of Wg = ]g Sn ×Sn
admits a splitting
π0 Diff
+(Wg ) ∼= Ẽ g ×coker(J )2n+1
into the (finite) cokernel coker(J )2n+1 of the stable J-homomorphism in dimension 2n + 1 and
the central extension
0 −→ Z/|bP2n+2| −→ Ẽ g −→ Spq2g (Z) −→ 0,
obtained from (6) by taking quotients by the subgroup |bP2n+2| ·Z ≤ Z generated by the order
of Kervaire–Milnor’s finite cyclic group bP2n+2 of homotopy (2n +1)-spheres bounding a paral-
lelisable manifold [5]. Since the finite residual of Eg is Z by the discussion above, that of Ẽ g is
Z/|bP2n+2|, which is nontrivial, since under our assumption that n ≡ 5 (mod 8), the group bP2n+2
is known not to vanish (see Section 7 loc. cit.). This implies that Ẽ g is not residually finite, and
hence neither is π0 Diff
+(Wg ) nor its non-orientation preserving variant π0 Diff(Wg ).
References
[1] K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer, 1982, x+306 pages.
[2] P. Deligne, “Extensions centrales non résiduellement finies de groupes arithmétiques”, C. R. Math. Acad. Sci. Paris
287 (1978), no. 4, p. A203-A208.
[3] S. Galatius, O. Randal-Williams, “Abelian quotients of mapping class groups of highly connected manifolds”, Math.
Ann. 365 (2016), no. 1-2, p. 857-879.
[4] D. Johnson, J. J. Millson, “Modular Lagrangians and the theta multiplier”, Invent. Math. 100 (1990), no. 1, p. 143-165.
[5] M. A. Kervaire, J. W. Milnor, “Groups of homotopy spheres. I”, Ann. Math. 77 (1963), p. 504-537.
C. R. Mathématique, 2020, 358, n 4, 469-473
Manuel Krannich and Oscar Randal-Williams 473
[6] M. Krannich, “Mapping class groups of highly connected (4k + 2)-manifolds”, https://arxiv.org/abs/1902.10097,
2019.
[7] M. Kreck, “Isotopy classes of diffeomorphisms of (k−1)-connected almost-parallelizable 2k-manifolds”, in Algebraic
topology, Aarhus 1978, Lecture Notes in Mathematics, vol. 763, Springer, 1979, p. 643-663.
[8] W. Meyer, “Die Signatur von lokalen Koeffizientensystemen und Faserbündeln”, Bonn. Math. Schr. (1972), no. 53,
p. viii+59.
[9] ——— , “Die Signatur von Flächenbündeln”, Math. Ann. 201 (1973), p. 239-264.
[10] D. W. Morris, “A lattice with no torsion-free subgroup of finite index (after P. Deligne)”, Available at http://people.
uleth.ca/~dave.morris/talks/deligne-torsion.pdf, 2009.
[11] J.-P. Serre, “Arithmetic groups”, in Homological group theory (Proc. Sympos., Durham, 1977), London Mathematical
Society Lecture Note Series, vol. 36, Cambridge University Press, 1979, p. 105-136.
[12] D. Sullivan, “Genetics of homotopy theory and the Adams conjecture”, Ann. Math. 100 (1974), p. 1-79.
[13] ——— , “Infinitesimal computations in topology”, Publ. Math., Inst. Hautes Étud. Sci. (1977), no. 47, p. 269-331.
[14] R. Thom, “Quelques propriétés globales des variétés différentiables”, Comment. Math. Helv. 28 (1954), p. 17-86.
C. R. Mathématique, 2020, 358, n 4, 469-473


Mapping class groups of simply connected high-dimensional manifolds need not be arithmetic

Songqi Wu

Xiaohui Ma

Hailin Sang

Xiequan Fan

A Berry–Esseen bound of order $\protect \frac{1}{\protect \sqrt{n}} $ for martingales

Jinsong Xu

Comptes Rendus
Mathématique
Jinsong Xu
A remark on the rank of finite p-groups of birational automorphisms
Volume 358, issue 7 (2020), p. 827-829.
<https://doi.org/10.5802/crmath.93>
© Académie des sciences, Paris and the authors, 2020.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2020, 358, n 7, p. 827-829
https://doi.org/10.5802/crmath.93
Birational Geometry / Géométrie birationnelle
A remark on the rank of finite p-groups of
birational automorphisms
Jinsong Xua
a Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, No.111,
Ren’ai Road, SIP, Suzhou, Jiangsu Province, China
E-mail: jinsong.xu@xjtlu.edu.cn
Abstract. In this note, we improve a result of Prokhorov and Shramov on the rank of finite p-subgroups of the
group of birational transformations of a rationally connected variety. Known examples show that the bounds
obtained are optimal in many cases.
Résumé. Dans cette note, nous améliorons un résultat de Prokhorov et Shramov sur le rang des p-sous-
groupes finis du groupe des transformations birationnelles d’une variété rationnellement connexe. Des
exemples connus montrent que les bornes obtenues sont optimales dans de nombreux cas.
Funding. This work is partially supported by XJTLU RDF-16-01-50.
Manuscript received 2nd March 2020, revised 16th June 2020, accepted 29th June 2020.
The goal of this note is to prove the following theorem on finite groups of birational transfor-
mations.
Theorem (Main Theorem). Let X be a rationally connected variety of dimension n over an
algebraically closed field k of characteristic 0. Let p be a prime number and let G be a finite p-
subgroup of the group of birational transformations Bir(X ). If p > n +1, then G is abelian and the
rank of G is at most n.
We shall see below in Example 1 that the inequality p > n +1 is optimal when n +1 is prime.
The same result, but with a non-optimal inequality p > L(n), was obtained in a series of papers
by Prokhorov and Shramov using – and in fact motivating – Birkar’s proof of the boundedness of
weak Fano varieties (see [2, 10]). Thus, our theorem provides a positive and optimal answer to
a question of Serre (see [13, Section 6] and [12, Question 1.1]). We refer to [12, Proposition 1.7]
and [11, 12] for surfaces and threefolds; for instance, in [12] it is shown that a finite p-subgroup
of Bir(P3k) is abelian and of rank ≤ 3 as soon as p ≥ 17; here we improve the inequality to p ≥ 5.
Our main tool is a remarkable fixed point theorem of Haution (see [3, Theorem 1.2.1]).
Theorem (Haution’s Theorem). Let k be an algebraically closed field, of arbitrary characteristic.
Let X be a projective variety over k, and G be a finite p-group acting by automorphisms on X .
Assume that one of the following conditions holds
(i) G is cyclic;
(ii) char(k) = p;
(iii) or dim X < p −1.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
828 Jinsong Xu
Then X (k)G = ∅ if and only if the Euler characteristic χ(X , F ) of every G-equivariant coherent
OX -module F is divisible by p.
Proof of the Main Theorem. Given a variety X , recall that it is rationally connected if for any
general points x, y ∈ X , we may find a rational curve C ⊂ X passing through x and y . This
property is a birational invariant. Therefore passing to a smooth regularization of G [9, Lemma-
Definition 3.1], we may assume that X is nonsingular and projective, and G acts regularly on X .
Observe that the structure sheaf OX is G-equivariant. Moreover, for a nonsingular projective
rationally connected variety X , one has χ(OX ) = 1, because dim H i (X , OX ) = dim H 0(X ,ΩiX ) = 0
for all i ≥ 1 (see [4, Chapter IV, 3.8]).
Therefore by case (iii) in Haution’s Theorem, G has a fixed point x ∈ X (k). The action of G on
the Zariski tangent space Tx,X is faithful because G is finite (see [6, Lemma 4]); so, G embeds
into the general linear group GL(n,k). From the assumption that p > n +1, we deduce that Tx,X
contains no irreducible G-submodule of dimension > 1 because any such G-submodule has
dimension divisible by p. Hence the action of G on Tx,X is diagonalizable, and we conclude that
G is abelian of rank ≤ n. 
Example 1. The lower bound p > n +1 in the Main Theorem is sharp when n +1 is prime: the
Fermat hypersurface X ⊂Pn+1k of degree n +1, defined by the equation
xn+10 +xn+11 +·· ·+xn+1n+1 = 0,
is a smooth Fano variety, hence rationally connected, and it admits a faithful action of
(Z/(n +1)Z)n+1, an abelian group of rank n +1.
Example 2. The rank r of an elementary p-subgroup of Bir(X ) can be greater than n if p ≤ n+1.
(1) For rational surfaces, it is known that r ≤ 4 if p = 2, and r ≤ 3 if p = 3; both of these upper
bounds are sharp, see [1].
(2) Now, suppose that X is a rationally connected threefold. If p = 2, then r ≤ 6 [8, Theo-
rem 1.2], and this is sharp. If p = 3, the product of the Fermat cubic surface with P1k gives
an example showing that (Z/3Z)4 embeds into Bir(P3k). Therefore, our main theorem is
sharp in dimension 3. Note that we also have r ≤ 5 when p = 3 [7, Theorem 1.2], but it
is unclear whether r = 5 can be reached. In a recent manuscript, Kuznetsova obtained
some new results on the rank of 3-groups. We refer the interested readers to [5].
Acknowledgements
This note was written during the author’s visit to Laboratory of Algebraic Geometry, HSE Uni-
versity in Dec 2019. The author would like to thank Constantin Shramov for his invitation and
valuable suggestions on examples. He also thanks Yuri Prokhorov for useful discussion and the
anonymous referee for many suggestions on improving the manuscript.
References
[1] A. Beauville, “p-Elementary subgroups of the Cremona group”, J. Algebra 314 (2007), no. 2, p. 553-564.
[2] C. Birkar, “Singularities of linear systems and boundedness of Fano varieties”, https://arxiv.org/abs/1609.05543,
2016.
[3] O. Haution, “Fixed point theorems involving numerical invariants”, Compos. Math. 155 (2019), no. 2, p. 260-288.
[4] J. Kollár, Rational Curves on Algebraic Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 32,
Springer, 1995.
[5] A. Kuznetsova, “Finite 3-subgroups in Cremona group of rank 3”, https://arxiv.org/abs/2001.01308, 2020.
C. R. Mathématique, 2020, 358, n 7, 827-829
Jinsong Xu 829
[6] V. L. Popov, “Jordan Groups and Automorphism Groups of Algebraic Varieties”, in Automorphisms in Birational and
Affine Geometry. Papers based on the presentations at the conference, Levico Terme, Italy, October 29 – November 3,
2012, Springer Proceedings in Mathematics & Statistics, vol. 79, Springer, 2014, p. 185-213.
[7] Y. G. Prokhorov, “p-elementary subgroups of the Cremona group of rank 3”, in Classification of algebraic varieties.
Based on the conference on classification of varieties, Schiermonnikoog, Netherlands, May 2009, EMS Series of
Congress Reports, European Mathematical Society, 2011, p. 327-338.
[8] ——— , “2-Elementary Subgroups of the Space Cremona Group”, in Automorphisms in Birational and Affine Geom-
etry. Papers based on the presentations at the conference, Levico Terme, Italy, October 29 – November 3, 2012, Springer
Proceedings in Mathematics & Statistics, vol. 79, Springer, 2014, p. 215-229.
[9] Y. G. Prokhorov, C. Shramov, “Jordan property for groups of birational selfmaps”, Compos. Math. 150 (2014), no. 12,
p. 2054-2072.
[10] ——— , “Jordan property for Cremona groups”, Am. J. Math. 138 (2016), no. 2, p. 403-418.
[11] Y. G. Prokhorov, C. Shramov, “Jordan constant for Cremona group of rank 3”, Mosc. Math. J. 17 (2017), no. 3, p. 457-
509.
[12] ——— , “p-subgroups in the space Cremona group”, Math. Nachr. 291 (2017), no. 8-9, p. 1374-1389.
[13] J.-P. Serre, “A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an
arbitrary field”, Mosc. Math. J. 9 (2009), no. 1, p. 193-208.
C. R. Mathématique, 2020, 358, n 7, 827-829


A remark on the rank of finite $p$-groups of birational automorphisms

Jianjun Xu

New asymptotic expansions on hyperfactorial functions

J.L. Ramírez Alfonsín

M. Skałba

Comptes Rendus
Mathématique
J.L. Ramírez Alfonsín and M. Skałba
Primes in numerical semigroups
Volume 358, issue 9-10 (2020), p. 1001-1004.
<https://doi.org/10.5802/crmath.104>
© Académie des sciences, Paris and the authors, 2020.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2020, 358, n 9-10, p. 1001-1004
https://doi.org/10.5802/crmath.104
Number Theory / Théorie des nombres
Primes in numerical semigroups
J.L. Ramírez Alfonsín∗, a and M. Skałbab
a UMI2924 - Jean- Christophe Yoccoz, CNRS-IMPA, Brazil and Univ. Montpellier,
CNRS, Montpellier, France
b Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa,
Poland
E-mails: jorge.ramirez-alfonsin@umontpellier.fr (J.L. Ramírez Alfonsín),
skalba@mimuw.edu.pl (M. Skałba)
Abstract. Let 0 < a < b be two relatively prime integers and let 〈a,b〉 be the numerical semigroup generated
by a and b with Frobenius number g (a,b) = ab−a−b. In this note, we prove that there exists a prime number
p ∈ 〈a,b〉 with p < g (a,b) when the product ab is sufficiently large. Two related conjectures are posed and
discussed as well.
2020 Mathematics Subject Classification. 11D07, 11N13.
Funding. J.L. Ramírez Alfonsín was partially supported by Grant MATHAM-SUD 18-MATH-01, Project
FLaNASAGraTA and INSMI-CNRS..
Manuscript received 22nd July 2020, revised and accepted 30th July 2020.
Let 0 < a < b be two relatively prime integers. Let S = 〈a,b〉 = {n |n = ax +by, x, y ∈Z, x, y ≥ 0}
be the numerical semigroup generated by a and b. A well-known result due to Sylvester [5] states
that the largest integer not belonging to S, denoted by g (a,b), is given by ab − a − b. g (a,b) is
called the Frobenius number (we refer the reader to [3] for an extensive literature on the Frobenius
number).
We clearly have that any prime p larger than g (a,b) belongs to 〈a,b〉. A less obvious and more
intriguing question is whether there is a prime p ≤ g (a,b) belonging to 〈a,b〉.
In this note, we show that there always exists a prime p ∈ 〈a,b〉, p < g (a,b) when the product
ab is sufficiently large. The latter is a straight forward consequence of the below Theorem.
Let 0 < u < v be integers. We define
πS [u, v] = |{p prime | p ∈ S, u ≤ p ≤ v}|.
For short, we may write πS instead of πS [0, g (a,b)].
Theorem 1. Let 3 ≤ a < b be two relatively prime integers and let S = 〈a,b〉 be the numerical
semigroup generated by a and b. Then, for any fixed ε> 0 there exists C (ε) > 0 such that
πS >C (ε) g (a,b)
log(g (a,b))2+ε
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1002 J.L. Ramírez Alfonsín and M. Skałba
for ab sufficiently large.
Let us quickly introduce some notation and recall some facts needed for the proof of Theo-
rem 1.
Let S = 〈a,b〉 and let 0 < u < v be integers. We define
nS [u, v] = |{n ∈N | u ≤ n ≤ v, n ∈ S}|
and
ncS [u, v] = |{n ∈N | u ≤ n ≤ v, n 6∈ S}|.
For short, we may write nS instead of nS [0, g (a,b)] and ncS instead of n
c
S [0, g (a,b)]. The set of
elements in ncS =N\ S are usually called the gaps of S.
It is known [3] that S is always symmetric, that is, for any integer 0 ≤ s ≤ g (a,b)
s ∈ S if and only if g (a,b)− s 6∈ S.
It follows that
nS = g (a,b)+1
2
.
We may now prove Theorem 1.
Proof of Theorem 1. Let ε> 0 be fixed. We distinguish two cases.
Case 1. Suppose that a > (log(ab))1+ε. Let us take c = ab/(log(ab))1+ε. It is known [1] that if
k ∈ [0, . . . , g (a,b)] then
nS [0,k] =
⌊
k
b
⌋∑
i=0
(⌊
k − i b
a
⌋
+1
)
.
In our case, we obtain that
nS [0,c] ≤
⌊ c
a
⌋
+
⌊ c
b
⌋(⌊c −b
a
⌋
+1
)
+1 ≤
⌊ c
a
⌋
+
⌊ c
b
⌋(⌊ c
a
⌋
+1
)
+1
≤ c
a
+ c
b
+ c
2
ab
+1 = bc +ac + c
2 +ab
ab
< 2c
2 + c2 + c2
ab
= 4c
2
ab
= 4ab
(log(ab))2+2ε
where the last inequality holds since c > b > a.
Due to the symmetry of S, we have
ncS [g (a,b)− c, g (a,b)] = nS [0,c] <
4ab
(log(ab))2+2ε
. (1)
Let π(x) be the number of primes integers less or equals to x. We have
π(g (a,b))−π(g (a,b)− c) >> c
log(ab)
= ab
(log(ab))2+ε
(2)
when ab is large enough. The latter follows from Prime Number Theorem for short intervals
(when c = ab/(log(ab))1+ε is large enough in comparison to g (a,b) = ab −a −b).
Finally, by combining equations (1) and (2), we obtain
πS ≥πS [g (a,b)− c, g (a,b)] ≥π(g (a,b))−π(g (a,b)− c)−ncS [g (a,b)− c, g (a,b)]
> ab
(log(ab))2+ε
− 4ab
(log(ab))2+2ε
> 0
where the last inequality holds since (log(ab))ε > 4 for ab large enough for the fixed ε. The above
leads to the desired estimate of πS .
C. R. Mathématique, 2020, 358, n 9-10, 1001-1004
J.L. Ramírez Alfonsín and M. Skałba 1003
Case 2. Suppose that 3 ≤ a ≤ (log(ab))1+ε.
If p ∈ [b, . . . , g (a,b)] is a prime and p ≡ b (mod a) then p is clearly representable as
p = b + p−ba a. By Siegel–Walfisz theorem [2, 7], the number of such primes p, denoted by N , is
N = 1
ϕ(a)
∫ g (a,b)
b
du
logu
+R
where ϕ is the Euler totient function and |R| < D ′(ε) g (a,b)
(log(g (a,b)))2+2ε uniformly in a and g (a,b).
Since the function 1/logu is decreasing on the interval [b, g (a,b)] then∫ g (a,b)
b
du
logu
> (g (a,b)−b) · 1
log g (a,b)
and therefore
N > 1
ϕ(a)
· g (a, b)−b
log(g (a, b))
−D ′(ε) g (a, b)
(log(g (a, b)))2+2ε
. (3)
Now, we have that
1
ϕ(a)
· g (a,b)−b
log(g (a,b))
· (log(g (a,b)))
2+ε
g (a,b)
= 1
ϕ(a)
log(g (a,b))1+ε
(
1− b
g (a,b)
)
> 1
log(ab)1+ε
log(g (a,b))1+ε
(
1− b
g (a,b)
)
(since(log(ab))1+ε ≥ a >ϕ(a))
>
(
log(ab)− log(3)
log(ab)
)1+ε 1
5
> F > 0
(
since g (a,b) > ab/3 and b
g (a,b)
≤ 4
5
)
for some absolute F > 0, uniformly for ab ≥ D ′′(ε) with a ≥ 3.
It yields to
1
ϕ(a)
· g (a,b)−b
log(g (a,b))
≥ F g (a,b)
log(g (a,b))2+ε
(4)
and combining equations (3) and (4) we obtain
N > F ′ g (a,b)
log(g (a,b))2+ε
for ab large enough for the fixed ε. The latter leads to the desired estimate of πS also in this case.

1. Concluding remarks
A number of computer experiments lead us to the following.
Conjecture 2. Let 2 ≤ a < b be two relatively prime integers and let S be the numerical semigroup
generated by a and b. Then,
πS > 0.
In analogy with the symmetry of 〈a,b〉mentioned above, our task of looking for primes in 〈a,b〉
is related with the task of finding primes in [g (a,b)− 1)/2, . . . , g (a,b)]. From this point of view,
Conjecture 2 can be thought of as a counterpart of the famous Chebyshev theorem stating that
there is always a prime in [n, . . . , 2n] for any n ≥ 2, see [4, Chapter 3]. A way to attack Conjecture 2
could be by applying effective versions of Siegel–Walfisz theorem. For instance, one may try to
use [6, Corollary 8.31] in order to get computable constants in our estimates. However, it is not
an easy task to trace all constants appearing in the relevant estimates of L(x,χ) (but in principle
possible). The remaining cases for small values ab must to be treated by computer.
C. R. Mathématique, 2020, 358, n 9-10, 1001-1004
1004 J.L. Ramírez Alfonsín and M. Skałba
Conjecture 3. Let 2 ≤ a < b be two relatively prime integers and let S be the numerical semigroup
generated by a and b. Then,
πS ∼ π(g (a,b))
2
for a →∞.
In the same spirit as the prime number theorem, this conjecture seems to be out of reach.
The famous Linnik’s theorem asserts that there exist absolute constants C and L such that: for
given relatively prime integers a,b the least prime p satisfying p ≡ b (mod a) is less than C aL . It
is conjectured that one can take L = 2, but the current record is only that L ≤ 5 is allowed, see [8].
On the same flavor of Linnik’s theorem that concerns the existence of primes of the form ax+b,
Theorem 1 is concerning the existence of primes of the form ax+by with x, y ≥ 1 less than ab for
sufficiently large ab. This relation could shed light on in either direction.
References
[1] G. Márquez-Campos, J. L. Ramírez Alfonsín, J. M. Tornero, “Integral points in rational polygons: a numerical semi-
group approach”, Semigroup Forum 94 (2017), no. 1, p. 123-138.
[2] K. Prachar, Primzahlverteilung, Die Grundlehren der Mathematischen Wissenschaften, vol. 91, Springer, 1957.
[3] J. L. Ramírez Alfonsín, The Diophantine Frobenius Problem, Oxford Lecture Series in Mathematics and its Applications,
vol. 30, Oxford University Press, 2005.
[4] W. Sierpiński, Elementary Theory of Numbers, 2nd ed., North-Holland Mathematical Library, vol. 31, PWN-Polish
Scientific Publisher; North-Holland, 1988.
[5] J. J. Sylvester, “Question 7382”, Mathematical Questions from Educational Times 41 (1884), p. 21.
[6] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3rd expanded ed., Graduate Studies in
Mathematics, vol. 163, American Mathematical Society, 2015.
[7] A. Walfisz, “Zur additiven Zahlentheorie. II”, Math. Z. 40 (1936), no. 1, p. 592-607.
[8] T. Xylouris, Über die Nullstellen der Dirichletschen L-Funktionen und die kleinste Primzahl in einer arithmetischen
Progression, Bonner Mathematische Schriften, vol. 404, University Bonn; Mathematisches Institut and Mathematisch-
Naturwissenschaftliche Fakultät, 2011.
C. R. Mathématique, 2020, 358, n 9-10, 1001-1004


Primes in numerical semigroups

Théophile Chaumont-Frelet

Alexandre Ern

Martin Vohralík

Polynomial-degree-robust $\protect H({\protect \bf curl})$-stability of discrete minimization in a tetrahedron

Dragan Stankov

The number of unimodular roots of some reciprocal polynomials

Nathalie Ayi

Maxime Herda

Hélène Hivert

Isabelle Tristani

A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium

Igor Pak

Greta Panova

Comptes Rendus
Mathématique
Igor Pak and Greta Panova
Breaking down the reduced Kronecker coefficients
Volume 358, issue 4 (2020), p. 463-468.
<https://doi.org/10.5802/crmath.60>
© Académie des sciences, Paris and the authors, 2020.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2020, 358, n 4, p. 463-468
https://doi.org/10.5802/crmath.60
Theory of Representations, Algebra, Combinatorics / Théorie des représentations,
Algèbre, Combinatoire
Breaking down the reduced Kronecker
coefficients
Analyse fine des coefficients de Kronecker réduits
Igor Paka and Greta Panova∗, b
a Department of Mathematics, UCLA, Los Angeles, CA 90095, USA
b Department of Mathematics, USC, Los Angeles, CA 90089, USA.
E-mails: pak@math.ucla.edu, gpanova@usc.edu.
Abstract. We resolve three interrelated problems on reduced Kronecker coefficients g (α,β,γ). First, we dis-
prove the saturation property which states that g (Nα, Nβ, Nγ) > 0 implies g (α,β,γ) > 0 for all N > 1. Sec-
ond, we esimate the maximal g (α,β,γ), over all |α|+|β|+|γ| = n. Finally, we show that computing g (λ,µ,ν) is
strongly #P-hard, i.e. #P-hard when the input (λ,µ,ν) is in unary.
Funding. Both authors were partially supported by the NSF.
Manuscript received 6th April 2020, revised 3rd May 2020, accepted 25th April 2020.
1. Introduction
The reduced Kronecker coefficients were introduced by Murnaghan in 1938 as the stable limit of
Kronecker coefficients, when a long first row is added:
g (α,β,γ) := lim
n→∞g
(
α[n],β[n],γ[n]
)
, where α[n] := (n −|α|,α1,α2, . . .), n ≥ |α|+α1, (1)
see [15, 16]. They generalize the classical Littlewood–Richardson (LR–) coefficients:
g (α,β,γ) = cαβγ for |α| = |β|+ |γ|,
see [12]. As such, they occupy the middle ground between the Kronecker and the LR–coefficients.
While the latter are well understood and have a number of combinatorial interpretations, the
former are notorious for their difficulty. It is generally believed that the reduced Kronecker
coefficients are simpler and more accessible than the (usual) Kronecker coefficients, cf. [9, 18].
The results of this paper suggest otherwise, see Remark 12.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
464 Igor Pak and Greta Panova
1.1. Saturation property
The Kronecker coefficients g (λ,µ,ν), are defined as
g (λ,µ,ν) := 〈χλχµ,χν〉 = 1
n!
∑
σ∈Sn
χλ(σ)χµ(σ)χν(σ),
where λ,µ,ν` n, and χλ is the irreducible character of Sn corresponding to partition λ. Similarly,
the Littlewood–Richardson coefficients are defined as
cλµν :=
〈
χλ,χµ⊗χν ↑SnSk×Sn−k
〉
, where λ` n,µ` k,ν` n −k.
It is easy to see that cNλNµ,Nν ≥ cλµν for all N ≥ 1, where Nλ= (Nλ1, Nλ2, . . .). The saturation property
is the fundamental result by Knutson and Tao [11], giving a converse:
cNλNµ,Nν > 0 for some N ≥ 1 =⇒ cλµν > 0.
For a partition α ` k and n ≥ k + α1, we have α[n] = (n − k,α1,α2, . . .) ` n. It is known
that g (α[n + 1],β[n + 1],γ[n + 1]) ≥ g (α[n],β[n],γ[n]) for all n, whenever the right hand side is
defined. In this notation, Murnaghan’s result (1) states that g (α,β,γ) = g (α[n],β[n],γ[n]) for n
large enough.
The saturation property fails for the Kronecker coefficients, i.e. g (22,22,22) = 1 but
g (12,12,12) = 0. It is a long-standing open problem whether it holds for the reduced Kro-
necker coefficients. This was independently conjectured in 2004 by Kirillov [9, Conj. 2.33] and
Klyachko [10, Conj. 6.2.4] :
Conjecture 1 (Kirillov, Klyachko). The reduced Kronecker coefficients satisfy the saturation prop-
erty:
g (Nα, Nβ, Nγ) > 0 for some N ≥ 1 =⇒ g (α,β,γ) > 0.
This conjecture was motivated by the known converse:
g (α,β,γ) > 0 =⇒ g (Nα, Nβ, Nγ) > 0 for all N ≥ 1,
see below. Here is the first result of this paper.
Theorem 2. For all k ≥ 3, the triple of partitions (1k2−1,1k2−1,kk−1) is a counterexample to
Conjecture 1. Moreover, for every partition γ s.t. γ2 ≥ 3, there are infinitely many pairs (a,b) ∈ N2
for which the triple of partitions (ab , ab ,γ) is a counterexample to Conjecture 1.
These results both contrast and complement [5, Cor. 6], which confirms the saturation prop-
erty for triples of the form (ab , ab , a).
1.2. Maximal values
Our second result is a variation on Stanley’s recent bounds on the maximal Kronecker and LR–
coefficients:
Theorem 3 (cf. [25, 26], and also [21]). We have:
max
λ`n
max
µ`n
max
ν`n
g (λ,µ,ν) =
p
n!e−O(
p
n), (2)
max
0≤k≤n
max
λ`n
max
µ`k
max
ν`n−k
cλµ,ν = 2n/2−O(
p
n). (3)
In [21], we refine (3) and prove that the maximal Kronecker and LR–coefficients appear when
all three partitions have near-maximal dimension, which in turn implies that they have a Vershik–
Kerov–Logan–Shepp shape. See also [20] for refined upper bounds on (reduced) Kronecker coef-
ficients with few rows. Here we obtain the following analogue of Stanley’s Theorem 3.
C. R. Mathématique, 2020, 358, n 4, 463-468
Igor Pak and Greta Panova 465
Theorem 4. We have:
max
a+b+c≤3n
max
α`a
max
β`b
max
γ`c
g (α,β,γ) =
p
n!eO(n).
1.3. Complexity
Our final result is on complexity of computing the reduced Kronecker coefficients. Via reduction
to LR–coefficients, computing the reduced Kronecker coefficients is classically #P-hard, see [17].
The following recent result by Ikenmeyer, Mulmuley and Walter is a far-reaching extension:
Theorem 5 (cf. [7] and Remark 14). Computing the Kronecker coefficients g (λ,µ,ν) is strongly
#P-hard.
Here by strongly #P-hard we mean #P-hard when the input (λ,µ,ν) is given in unary. In
other words, the input size of the problem is the total number of squares in the three Young
diagrams. The theorem is in sharp contrast with computing χ(n−k,k)[λ] which is #P-complete but
not strongly #P-complete, see [19, §7].
Theorem 6. Computing the reduced Kronecker coefficients g (α,β,γ) is strongly #P-hard.
Let us mention that the problem of computing the (reduced) Kronecker coefficients is not
known to be in #P, see [19]. In fact, finding a combinatorial interpretation for (reduced) Kronecker
coefficients is a classical open problem [24, Prob. 10]. Note also that Theorem 6 is stronger than
Theorem 5, since in the limit (1) it suffices to take n ≥ |α|+|β|+|γ|, see [3,27]. Indeed, this implies
that the reduced Kronecker coefficient problem is a subset of instances of the usual Kronecker
coefficient problem (cf., however, Remark 16).
2. Disproof of the saturation property
We assume the reader is familiar with basic results and standard notations in Algebraic Combi-
natorics, see [23]. We also need the following two results on Kronecker coefficients.
Lemma 7 (Symmetries). For every λ,µ,ν` n, we have:
g (λ,µ,ν) = g (λ′,µ′,ν) = g (µ,λ,ν) = g (λ,ν,µ).
Lemma 8 (Semigroup property [4, 13]). Suppose α,β,γ ` m, such that g (α,β,γ) > 0. Then, for
all partitions λ,µ,ν` n, we have:
g (λ+α,µ+β,ν+γ) ≥ g (λ,µ,ν).
This result is crucial for understanding of reduced Kronecker coefficients. First, since
g (1,1,1) = 1, we conclude that the sequence {g (α[n],β[n],γ[n])} is weakly increasing with n. Sim-
ilarly, the sequence
{
g (Nλ, Nµ, Nν)
}
is weakly increasing with N if g (λ,µ,ν) > 0.
Let `(λ) be the number of parts of the partition λ, and d(λ) := max{k : λk ≥ k} be the Durfee
size. For the proof of Theorem 2, we need several known technical results which we state below.
Lemma 9 (cf. [6]). Let λ,µ,ν` n be such that d(λ) > 2d(µ)d(ν). Then g (λ,µ,ν) = 0.
Lemma 10 (cf. [1, Cor. 3.2]). Let λ=λ′ be a self-conjugate partition. Then g (λ,λ,λ) > 0.
Lemma 11 (cf. [8, Thm. 1.10]). Let X := {1,12,14,16,21,31}, and let partition ν ∉ X . Denote
` := max{`(ν)+1,9}, and suppose r > 3`3/2, s ≥ 3`2, and |ν| ≤ r s/6. Then g (sr , sr ,ν[r s]) > 0.
C. R. Mathématique, 2020, 358, n 4, 463-468
466 Igor Pak and Greta Panova
Proof of Theorem 2. We prove the first statement of the theorem. Let k ≥ 3, and let α = (1k2−1),
γ = (kk−1) be as in the theorem. Since d(α[n]) = 1 and d(γ[n]) = k for all n ≥ k2, we have
2d(α[n])2 = 2 < d(γ[n]) = k. Thus, we have g (α,α,γ) = 0 by Lemma 9.
By Lemma 10, the symmetry and semigroup properties (Lemma 7 and 8), we have:
g (kα,kα,kγ) = g (kk2−1,kk2−1, (k2)k−1)≥ g (kk2−1[k3],kk2−1[k3], (k2)k−1[k3])
≥ g (kk2 ,kk2 , (k2)k)= g ((k2)k , (k2)k , (k2)k)≥ g (kk ,kk ,kk)> 0.
This contradicts the saturation property in Conjecture 1 for N = k, and proves the first part of the
theorem.
For the second part, we construct the counterexample based on Lemma 11. For a partition γ,
let ` := max{`(γ)+1,9} as in the lemma. Let b ≥ max{3`3/2, |γ|/(6√d(γ[n])/2−6)}. Since γ2 ≥ 3,
we have d(γ[n]) ≥ 3. Thus, there exists at least one a ≥ 1, such that |γ|/(6b) ≤ a <√d(γ[n])/2. Let
us show now that (a,b) is a pair as in the theorem.
Takeα := (ab). Since d(α[n]) ≤ a, we have 2d(α[n])2 ≤ 2a2 < d(γ[n]). Thus, we have g (α,α,γ) =
0 by Lemma 9. On the other hand, let N ≥ 3`2/a, ν := Nγ, r := b + 1, and s := N a. Then
|ν| ≤ N ab/6 < r s/6, r > 3`3/2, and s = N a ≥ 3`2, by construction. Since ν 6∈ X for all N > 1,
the conditions of Lemma 11 are satisfied. We conclude:
g (Nα, Nα, Nγ) = g (N ab , N ab , Nγ)≥ g (sb+1, sb+1, Nγ[r s])= g (sr , sr ,ν[r s])> 0,
which implies that (α,α,γ) is a counterexample to the saturation property. Since the construction
works for all b large enough as above, this proves the second part of the theorem. 
3. Bounds and complexity via identities
Proof of Theorem 4. We follow [21] in our exposition. We start with the following identity [2,
Cor. 4.5]:
g (α,β,γ) =
bk/2c∑
m=0
∑
π`q+m−b
∑
ρ`q+m−a
∑
σ`m
∑
λ,µ,ν`k−2m
cανπρc
β
µπσc
γ
λρσ
g (λ,µ,ν), (4)
where a = |α|, b = |β|, q = |γ|, k = a +b −q , and
cλαβγ =
∑
τ
cλατc
τ
βγ.
For the upper bound, by [21, Thm. 1.5] which extends (3) in Theorem 3, we have:
cλαβ ≤
(
N
a
)1/2
for all λ` N ,α` a,β` N −a.
Using the Vandermonde identity for the sums of binomial coefficients, we have:
cλαβγ ≤
(
N
a,b, N −a −b
)1/2
≤ 3N /2 for all λ` N ,α` a,β` b,γ` N −a −b.
In this notation, the theorem is a maximum over a + b + q ≤ 3n. Combining these with (2) in
Theorem 3, we have:
g (α,β,γ) ≤ (3n/2) ·p (3n)6 ·33n/2 ·
p
n! =
p
n!eO(n).
For the lower bound, let α,β,γ ` n, and note that g (α,β,γ) ≥ g (α,β,γ), which is achieved in (4)
for m = 0. The result now follows from part (2) of Theorem 3. 
C. R. Mathématique, 2020, 358, n 4, 463-468
Igor Pak and Greta Panova 467
Proof of Theorem 6. Let λ = (λ1,λ2, . . .) ` n, which can be viewed as an infinite nonincreasing
sequence by appending zeros at the end. Denote λ̃ := (λ2,λ3, . . .). For all i ≥ 1, define
λ〈i 〉 := (λ1 +1,λ2 +1, . . . ,λi−1 +1,λi+1,λi+2, . . .),
so in particular λ〈1〉 = λ̃. The result is a direct consequence of the following identity:
g (λ,µ,ν) =
`(µ)`(ν)∑
i=1
(−1)i g (λ〈i 〉, µ̃, ν̃), (5)
see [3, Thm. 1.1]. From Theorem 5, computing g (λ,µ,ν) is #P-hard in unary. The identity (5) has
polynomially many terms, and thus gives a polynomial reduction. 
4. Final remarks and open problems
Remark 12. All three results in this paper are centered around the same (philosophical) claim,
that the reduced Kronecker coefficients are closer in nature to the (usual) Kronecker coefficients
than to the LR–coefficients. This is manifestly evident from both the statements and the proofs of
the theorems. However, this should not be taken as a suggestion that the LR–coefficients are not
strongly #P-hard. We do, in fact, conjecture that computing cλµν is strongly #P-hard [19, Conj. 8.1],
but this remains beyond the reach of existing technology.
Remark 13. There is a general setting which extends the stability of Kronecker coefficients to
other families of stable limits, see [22]. Manivel asks if the saturation property holds for all these
families, but notes that “we actually have only very limited evidence for that” [13]. In view of our
results, it would be interesting to see if the saturation property holds for any of these families of
stable coefficients.
Remark 14. Theorem 5 is not stated in [7] in this form. It does however follow directly from the
proof, which is essentially a parsimonious reduction from the 3-Partition problem classically
known to be (strongly) NP-complete, and thus the counting is (strongly) #P-complete.
Remark 15. Among other consequences, the saturation property implies that the vanishing
problem cλµν >? 0 is in P, see [14]. The main result of [7] proved that the vanishing problem
g (λ,µ,ν) >? 0 is NP-hard, refuting Mulmuley’s conjecture (cf. e.g. [19, §2]). Following the pat-
tern in Remark 12 above, we conjecture that the vanishing problem g (α,β,γ) >? 0 for reduced
Kronecker coefficients is also NP-hard.
Remark 16. There is a subtle but important technical differences between the way we state
Theorem 5 and the way it is stated in [7]. While we use the (standard) Turing reduction to
derive Theorem 6 from Theorem 5, the original proof in [7] uses a more restrictive many-to-
one reduction. Such a reduction for Theorem 6 would also resolve our conjecture above on the
vanishing problem.
Acknowledgements
The authors are grateful to Christine Bessenrodt, Chris Bowman and Rosa Orellana for interesting
conversations and helpful remarks. Special thanks to Christian Ikenmeyer for the explanation of
the inner working of [7], and to Mike Zabrocki for help with the computer algebra. The results of
the paper were obtained during authors’ back to back visits at the Oberwolfach Research Institute
for Mathematics and the Mittag-Leffler Institute; we are grateful for their hospitality.
C. R. Mathématique, 2020, 358, n 4, 463-468
468 Igor Pak and Greta Panova
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C. R. Mathématique, 2020, 358, n 4, 463-468


Breaking down the reduced Kronecker coefficients

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Breaking down the reduced Kronecker coefficients

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