Comptes Rendus Mathématique

Sami Douba

Junzhi Huang

Comptes Rendus
Mathématique
Sami Douba and Junzhi Huang
Density of systoles of hyperbolic manifolds
Volume 362 (2024), p. 1819-1824
Online since: 25 November 2024
https://doi.org/10.5802/crmath.689
This article is licensed under the
Creative Commons Attribution 4.0 International License.
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C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
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Comptes Rendus. Mathématique
2024, Vol. 362, p. 1819-1824
https://doi.org/10.5802/crmath.689
Research article / Article de recherche
Geometry and Topology, Group theory / Géométrie et Topologie, Théorie des
groupes
Density of systoles of hyperbolic manifolds
Densité de systoles de variétés hyperboliques
Sami Douba ∗,a and Junzhi Huang b
a Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440
Bures-sur-Yvette, France
b Department of Mathematics, Yale University, New Haven, CT 06511, USA
E-mails: douba@ihes.fr, junzhi.huang@yale.edu
Abstract. We show that for each n ≥ 2, the systoles of closed hyperbolic n-manifolds form a dense subset of
(0,+∞). We also show that for any n ≥ 2 and any Salem number λ, there is a closed arithmetic hyperbolic
n-manifold of systole log(λ). In particular, the Salem conjecture holds if and only if the systoles of closed
arithmetic hyperbolic manifolds in some (any) dimension fail to be dense in (0,+∞).
Résumé. Nous démontrons que, pour tout n ≥ 2, les systoles de variétés hyperboliques compactes sans bord
de dimension n constituent une partie dense de ]0,+∞[. Nous démontrons de plus que, pour tout n ≥ 2 et
tout nombre de Salem λ, il existe une variété hyperbolique arithmétique compacte sans bord de dimension
n et de systole log(λ). En particulier, la conjecture de Salem est vraie si et seulement si les systoles de variétés
hyperboliques arithmétiques compactes sans bord d’une certaine dimension (de manière équivalente, de
dimension quelconque) ne sont pas denses dans ]0,+∞[.
Keywords. Geometric topology, hyperbolic manifolds, systoles, arithmetic groups.
Mots-clés. Topologie géométrique, variétés hyperboliques, systoles, groupes arithmétiques.
2020 Mathematics Subject Classification. 22E40, 11F06.
Funding. S.D. was supported by the Huawei Young Talents Program. J.H. was partially supported by NSF
grant DMS-2005328.
Manuscript received 5 June 2024, accepted 17 September 2024.
The systole sys(M) of a closed nonpositively curved Riemannian manifold M is the length of a
shortest closed geodesic in M . It follows for instance from residual finiteness of surface groups
that there is a sequence of closed hyperbolic surfaces with systole going to infinity. Since for any
ϵ> 0 and any closed surface M of negative Euler characteristic there is a hyperbolic metric on M
of systole < ϵ, continuity of the systole function on moduli space then yields for any L > 0 a closed
hyperbolic surface of systole precisely L.
By Mostow rigidity [19], the set of systoles of closed hyperbolic n-manifolds is countable for
n ≥ 3. Nevertheless, we show the following.
Theorem 1. For any n ≥ 2, the set of systoles of closed hyperbolic n-manifolds is dense in (0,+∞).
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1820 Sami Douba and Junzhi Huang
That 0 is an accumulation point of systoles of closed hyperbolic 3-manifolds follows for in-
stance from Thurston’s hyperbolic Dehn filling theory (see [4, Sections E.5 and E.6]). The analo-
gous fact for 4-manifolds was established by Agol [1], whose strategy was to “inbreed” arithmetic
manifolds along totally geodesic hypersurfaces. This strategy was successfully extended to arbi-
trary dimensions independently by Belolipetsky–Thomson [3] and Bergeron–Haglund–Wise [6].
The present note is an elaboration on the utility of this inbreeding technique.
We also establish a variant of Theorem 1 for arithmetic hyperbolic manifolds. A Salem number
is a real algebraic integer λ > 1 of degree ≥ 4 that is Galois-conjugate to λ−1 and all of whose
remaining Galois conjugates lie on the unit circle.
Theorem 2. Let λ > 1 be a Salem number. Then for any n ≥ 2, there is a closed arithmetic
hyperbolic n-manifold of systole precisely log(λ).
The manifolds we exhibit in the proof of Theorem 2 are arithmetic of simplest type (the latter
being the only arithmetic construction that applies in every dimension) and are moreover classi-
cal in the language of Emery–Ratcliffe–Tschantz [11]. For n ≥ 4, it follows from Meyer’s theorem
on quadratic forms and [11, Theorem 5.2] that the exponential lengths of closed geodesics in any
classical simplest-type closed arithmetic hyperbolic n-manifold are Salem numbers.1
Lehmer’s conjecture for Salem numbers, or the Salem conjecture for short, asserts that the
Salem numbers are bounded away from 1; see [15, Section 13] and [21, p. 31]. In light of
the relationship between Salem numbers and lengths of geodesics in arithmetic hyperbolic
manifolds, Theorem 2 yields the following reformulation of the Salem conjecture.
Corollary 3. The Salem conjecture holds if and only if, for some, equivalently, any, n ≥ 2, the set
of systoles of closed arithmetic hyperbolic n-manifolds fails to be dense in (0,+∞).
Proof. It is known generally that, given any irreducible symmetric space X of noncompact
type, the Salem conjecture implies a uniform lower bound on systoles of closed arithmetic
locally symmetric manifolds modeled on X ; see Fra̧czyk–Pham [12]. On the other hand, since
positive powers of Salem numbers remain Salem numbers, failure of the Salem conjecture would
imply density of Salem numbers in (1,+∞), and hence density of systoles of closed arithmetic
hyperbolic n-manifolds in (0,+∞) for each n ≥ 2 by Theorem 2. □
We motivate the proofs of Theorems 1 and 2 with the following remark.
Remark 4. We provide an argument along the lines of [1] that, given any L > 0, there is a
closed hyperbolic surface of systole precisely L. Indeed, it suffices to find a closed hyperbolic
surface ML of systole ≥ L possessing two disjoint 2-sided simple closed geodesics Σ1 and Σ2
and an orthogeodesic segment ω joining the Σi of length L/2 and whose length is minimal
among all orthogeodesic segments with endpoints on Σ1 ∪Σ2. Cutting ML along the Σi and
then doubling the resulting surface along its boundary yields a (possibly disconnected) closed
hyperbolic surface of systole precisely L.
One can construct such a surface ML as follows. Let P ⊂ H2 be a right-angled pentagon with
an edge ω̃ of length precisely L/2, and let H1, H2 ⊂ H2 be the walls of P adjacent to ω̃. Let
ΓP < Isom(H2) be the group generated by the reflections in the walls of P . We can find a larger
right-angled convex polygon Q ⊂H2 that is a union of finitely many ΓP -translates of P such that
the walls of Q that enter the L2 -neighborhood of Hi are orthogonal to Hi for i = 1,2 (this idea
1Exponential geodesic lengths in closed arithmetic hyperbolic surfaces or 3-manifolds can be quadratic “Salem
numbers”; the latter are sometimes included in the definition of a Salem number. Exponential geodesic lengths in odd-
dimensional closed arithmetic hyperbolic orbifolds need not be Salem numbers even in this more general sense; see
Lemma 4.10 and Theorem 4.11 in [20], as well as [11, Theorem 7.7].
Sami Douba and Junzhi Huang 1821
originates in work of Scott [22]; see also [2, Section 3.1]). By residual finiteness and virtual torsion-
freeness of the group ΓQ < Isom(H2) generated by the reflections in the walls of Q, there is a finite-
index subgroup Λ< ΓQ such that Λ\H2 is a surface of systole ≥ L. We may now take ML =Λ\H2,
the Σi to be the projections to ML of the Hi , and ω to be the projection of ω̃.
Proof of Theorem 1. Given L,ϵ> 0, we exhibit a closed hyperbolic n-manifold M with |sys(M)−
L| < ϵ. As in Remark 4, it suffices to find a closed hyperbolic n-manifold ML,ϵ of systole ≥ L + ϵ
possessing two disjoint 2-sided closed embedded totally geodesic hypersurfaces Σ1 and Σ2, and
an orthogeodesic segment ω joining the Σi of length within ϵ/2 from L/2 and whose length is
minimal among all orthogeodesic segments with endpoints on Σ1 ∪Σ2. We can then cut ML,ϵ
along Σ1 ∪Σ2, select the component N containing ω of the resulting manifold with boundary,
and define M to be the double of N . The double of ω is indeed a shortest closed geodesic in M of
length within ϵ from L.
To that end, let k =Q(p2) and Ok =Z[
p
2]. Define f to be the quadratic form on Rn+1 given by
f (x1, . . . , xn+1) = x21 +·· ·+x2n −
p
2x2n+1.
We identify the level set {x ∈ Rn+1| f (x) = −1, xn+1 > 0} with n-dimensional hyperbolic space
Hn and O′( f ;R) with Isom(Hn), where O′( f ;R) is the index-2 subgroup of O( f ;R) preserving Hn .
By the Borel–Harish-Chandra theorem [7], we have that Γ := O′( f ;Ok ) is a uniform arithmetic
lattice of Isom(Hn).
Denote by H1 the hyperplane {x1 = 0} in Hn . Since O′( f ;k) is dense in O′( f ;R), there is an
element g ∈ O′( f ;k) such that ∣∣∣∣distHn (H1, g H1)− L2
∣∣∣∣< ϵ2 .
It follows again from the Borel–Harish-Chandra theorem that H1 and H2 := g H1 project to closed
immersed totally geodesic hypersurfaces in the orbifold Γ\Hn . Let ω̃ be the orthogeodesic
segment in Hn connecting H1 and H2. By the proof of Lemma 3.1 in Belolipetsky–Thomson [3],
we may now pass to a non-zero ideal I ⊂ Ok such that distHn (H1,γH2) ≥ distHn (H1, H2) for each
γ in the principal congruence subgroup Γ(I ) < Γ of level I . Up to diminishing the ideal I , we can
further assume that
(1) the principal congruence subgroup Γ(I ) is torsion-free, so that ML,ϵ := Γ(I )\Hn is a
manifold;
(2) we have sys(ML,ϵ) ≥ L+ϵ;
(3) for i = 1,2 and γ ∈ Γ(I ), either γHi = Hi , in which case γ furthermore preserves each side
of Hi , or distHn (Hi ,γHi ) ≥ distHn (H1, H2); in particular, the projections Σi of the Hi to
ML,ϵ are 2-sided closed embedded totally geodesic hypersurfaces in ML,ϵ.
Note that item (2) can be arranged by residual finiteness of the ring Ok , whereas item (3) can
be ensured by the fact that the stabilizer in Γ of either side of Hi is an intersection of congruence
subgroups of Γ for i = 1,2; see [16] and [5, Lemme principal]. We now have that the Σi are
disjoint and that any orthogeodesic segment in ML,ϵ with endpoints on Σ1 ∪Σ2 has length at
least that of the projection ω of ω̃ to ML,ϵ. Thus, the manifold ML,ϵ, the hypersurfaces Σi , and the
orthogeodesic segment ω are as desired. □
Remark 5. The proof of Theorem 1 demonstrates that we may take the closed hyperbolic
manifolds giving rise to a dense set of systoles in (0,+∞) to be quasi-arithmetic in the sense of
Vinberg and all share the same adjoint trace field (indeed, the same Vinberg ambient algebraic
group); see [23] and compare [9, Theorem 1.3]. In this case, infinitely many of these manifolds
would necessarily be nonarithmetic. This is because, for fixed D > 0 and µ > 1, there are
only finitely many monic integer polynomials of degree ≤ D and Mahler measure ≤ µ; see the
discussion immediately following Conjecture 10.2 in [13]. On the other hand, by varying the
1822 Sami Douba and Junzhi Huang
form f in the proof, one easily produces a family of pairwise incommensurable closed hyperbolic
manifolds in each dimension whose systoles remain dense in (0,+∞).
In fact, it is possible to achieve the latter without varying f ; one can select from any family
of closed hyperbolic manifolds whose systoles are dense in (0,+∞) pairwise incommensurable
manifolds whose systoles remain dense in (0,+∞). Indeed, given a closed hyperbolic manifold
M , the set of all geodesic lengths of manifolds commensurable to M is closed and discrete in
[0,+∞). When M is nonarithmetic, the latter follows from Margulis’s arithmeticity criterion [18,
Theorem IX.6.5], and when M is arithmetic, from the fact about Mahler measures of bounded-
degree monic integer polynomials mentioned in the previous paragraph.
Remark 6. Instead of appealing to the work of Belolipetsky–Thomson (loc. cit.) to find the
manifold ML,ϵ in the proof of Theorem 1, we could have instead used [6, Corollary 1.12] as in
the proof of Theorem 4 in [10]. An interesting feature of the former approach is that it afforded us
a manifold ML,ϵ that is congruence arithmetic.
Proof of Theorem 2. Let µ=λ+λ−1. Let k =Q(µ), and Ok be the ring of integers of k. Following
the proof of [11, Theorem 6.3], we define the form f in n +1 variables over k to be that given by
the symmetric matrix  1 µ/2In−1
µ/2 1
 .
The form f is of signature (n,1), so that we may again identify Hn with one of the two sheets of
the hyperboloid {x ∈ Rn+1| f (x) = −1} and Isom(Hn) with O′( f ;R), where O′( f ;R) is the index-2
subgroup of O( f ;R) preserving Hn . Moreover, the field k is totally real, and for any embedding
σ : k → R other than the identity (of which at least one exists), the form f σ is positive definite. It
follows that Γ := O′( f ;Ok ) is a uniform arithmetic lattice of O′( f ;R) = Isom(Hn), which contains
the reflections
τ1 :=
−1 −µIn−1
1
 , τ2 :=
−µ 1−µ2In−1
1 µ

whose respective fixed hyperplanes H1, H2 ⊂Hn satisfy
distHn (H1, H2) = log(λ)/2.
As in the proof of Theorem 1, by the aforementioned work of Belolipetsky–Thomson (loc. cit.),
we may pass to a non-zero ideal I ⊂Ok such that
• the principal congruence subgroup Γ(I ) is torsion-free, so that Mlog(λ) := Γ(I )\Hn is a
manifold;
• we have sys(Mlog(λ)) ≥ log(λ);
• the Hi project to disjoint closed embedded 2-sided totally geodesic hypersurfacesΣ1 and
Σ2 in Mlog(λ);
• any orthogeodesic segment in Mlog(λ) with endpoints on Σ1 ∪Σ2 has length ≥ log(λ)/2.
Now cutting Mlog(λ) along the Σi and then doubling the resulting manifold along its boundary
yields a (possibly disconnected) closed manifold M of systole precisely log(λ) each of whose
components is arithmetic. □
Remark 7. Note that each component of the output manifold M in the proof of Theorem 2 is
arithmetic because the reflections τi in the hyperplanes Hi both belong to the input arithmetic
lattice Γ, whereas there is no such guarantee in the proof of Theorem 1.
Sami Douba and Junzhi Huang 1823
We conclude by remarking that there are natural spectral analogues of the questions consid-
ered in this paper, namely, what are the possible values of the smallest nonzero eigenvalue of
the Laplacian of a closed hyperbolic manifold (respectively, an arithmetic hyperbolic manifold)?
For recent progress in dimension 2, where these questions are already nontrivial, see Kravchuk–
Mazáč–Pal [14] and Magee [17]. See also Breuillard–Deroin [8] for a spectral reformulation of the
Salem conjecture.
Acknowledgements
We thank Misha Belolipetsky, Bram Petri, Alan Reid, and the anonymous referee for helpful
comments. This work was initiated during a visit of S.D. to Yale University in March 2024; S.D.
thanks Sebastian Hurtado and the entire geometry/topology/dynamics team at Yale for their
hospitality.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
[1] I. Agol, “Systoles of hyperbolic 4-manifolds”, 2006. https://arxiv.org/abs/math/0612290.
[2] I. Agol, D. D. Long and A. W. Reid, “The Bianchi groups are separable on geometrically finite
subgroups”, Ann. Math. 153 (2001), no. 3, pp. 599–621.
[3] M. V. Belolipetsky and S. A. Thomson, “Systoles of hyperbolic manifolds”, Algebr. Geom.
Topol. 11 (2011), no. 3, pp. 1455–1469.
[4] R. Benedetti and C. Petronio, Lectures on hyperbolic geometry, Springer, 1992, pp. xiv+330.
[5] N. Bergeron, “Premier nombre de Betti et spectre du laplacien de certaines variétés hyper-
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[6] N. Bergeron, F. Haglund and D. T. Wise, “Hyperplane sections in arithmetic hyperbolic
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[7] A. Borel and Harish-Chandra, “Arithmetic subgroups of algebraic groups”, Ann. Math. 75
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[15] D. H. Lehmer, “Factorization of certain cyclotomic functions”, Ann. Math. 34 (1933), no. 3,
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1824 Sami Douba and Junzhi Huang
[16] D. D. Long, “Immersions and embeddings of totally geodesic surfaces”, Bull. Lond. Math.
Soc. 19 (1987), no. 5, pp. 481–484.
[17] M. Magee, “The limit points of the bass notes of arithmetic hyperbolic surfaces”, 2024.
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[18] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Springer, 1991, pp. x+388.
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Comptes Rendus
Mathématique
Ilias Laib
New proof and generalization of some results on translated sums over k-almost
primes
Volume 362 (2024), p. 481-486
Online since: 31 May 2024
https://doi.org/10.5802/crmath.552
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2024, Vol. 362, p. 481-486
https://doi.org/10.5802/crmath.552
Research article / Article de recherche
Number theory / Théorie des nombres
New proof and generalization of some results
on translated sums over k-almost primes
Nouvelle preuve et généralisation de certains résultats
sur des sommes translatées sur k-presque premiers
Ilias Laib a
a ENSTP and Laboratory of Equations with Partial Non-Linear Derivatives ENS Vieux
Kouba, Algiers, Algeria.
E-mail: i.laib@enstp.edu.dz
Abstract. A sequence A of strictly positive integers is said to be primitive if none of its terms divides the
others, Erdős conjectured that the sum f (A ,0) ≤ f (N1,0), where N1 is the sequence of prime numbers
and f (A ,h) = ∑a∈A 1a(log a+h) . In 2019, Laib et al. proved that the analogous conjecture of Erdős
f (A ,h) ≤ f (N1,h) is false for h ≥ 81 on a sequence of semiprimes. Recently, Lichtman gave the best lower
bound h = 1.04 · · · on semiprimes and he obtained other results for translated sums on k-almost primes
with 2 < k ≤ 20 and when k sufficiently large. In this note, we propose a new proof of the same result on
semiprimes, and we generalize the result on k -almost primes for any k ≥ 2.
Résumé. Une suite A d’entiers strictement positifs est dite primitive si aucun de ses termes ne divise les
autres, Erdős a conjecturé que la somme f (A ,0) ≤ f (N1,0), où N1 est la suite des nombres premiers et
f (A ,h) = ∑a∈A 1a(log a+h) . En 2019, Laib et al. ont prouvé que la conjecture analogue d’Erdős f (A ,h) ≤
f (N1,h) est fausse pour h ≥ 81 sur la suite de nombres semi-premiers. Récemment, Lichtman a donné
le meilleur minorant h = 1.04 · · · sur les nombres semi-premiers et il a obtenu d’autres résultats pour des
sommes translatées sur k-presque premiers avec 2 < k ≤ 20 et lorsque k est suffisamment grand. Dans cette
note, nous proposons une nouvelle démonstration du même résultat sur les nombres semi-premiers, et nous
généralisons le résultat sur les k-presque premiers pour tout k ≥ 2.
2020 Mathematics Subject Classification. 11B05, 11Y55, 11L20.
Manuscript received 14 January 2023, revised 14 March 2023, accepted 26 July 2023.
1. Introduction
A sequence A of strictly positive integers is said to be primitive if none of its terms divides the
others. It is obvious that the sequenceN1 of prime numbers and the following examples
A dk =
{
pα11 p
α2
2 . . . p
αd
d
∣∣α1, . . . , αk ,d ∈N, α1 +·· ·+αd = k}
Cd =
{
pn ∈N1
∣∣n > d} ,
Bdk =A dk ∪Cd ,
Nk = {n ∈N |Ω (n) = k} ,
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
482 Ilias Laib
are primitive sequences, whereΩ(n) is the numbers of prime factors of n counted with multiplic-
ity. Note that the sequence (Nk )k >2 is the sequence of k-almost primes and N2 is the sequence
of semiprimes. In [2], Erdős proved that the series f (A ,0) converges uniformly for any primitive
sequence A ̸= {1}. In [3], Erdős and Zhang conjectured, for any primitive sequence A ̸= {1}, that
f (A ,0) ≤ f (N1,0). In [1], Cohen compute f (N1,0) and he obtained
f (N1,0) = 1.63661632335126086856965800392186367118159707613129 · · · .
In [3], Erdős and Zhang proved that f (A ,0) ≤ 1.84 and in [9], Lichtman and Pomerance proved
that f (A ,0) < 1.781 · · · . In [10], Zhang proved that f (Nk ,0) < f (N1,0) for all k > 1. Recently, in [5],
recently, Laib et al. proved that f (A ,h) > f (N1,h) for h ≥ 81 and the author improved it later
in [4] for h ≥ 60. Also, in [6], Laib and Rezzoug proved that for all k > 1 and
h ≥ kk !e
k+1
(k +1)k−1 −k ! , f (Nk ,h) > f
(
B
dk
k ,h
)
> f (N1,h).
For the purposes of this note, we define hk as follows
Definition 1. For any integer k > 1, the value hk is defined as the greatest lower bound of the set of
all positive real numbers, where for any h ≥ hk , we have f (Nk ,h) ≥ f (N1,h).
Note that for k ≤ 10, we have h2 = 1.04466, h3 = 0.98214, h4 = 0.93018, h5 = 0.89038, h6 =
0.86146, h7 = 0.84126, h8 = 0.82759, h9 = 0.81859 and h10 = 0.81280.
In [8], Lichtman gave the best lower bound h2 = 1.04 · · · for semiprimes, and other best result hk
for k-almost primes when 2 < k ≤ 20.
Motivated by the above works, we show in this paper by different methods the general case k > 1.
Our main results are given by the following two theorems.
Theorem 2. For all k > 1 and any h > hk , we have f (Nk ,h) > f (N1,h), where hk is the common
limit of the decreasing sequence (x(k)n )n≥0, and the increasing sequence (y
(k)
n )n≥0 defined asx
(k)
0 = kk !e
k+1
(k+1)k−1−k ! ,
f
(
Nk , x
(k)
n
)
= f
(
N1, x
(k)
n+1
) and {y (k)0 = 0,
f
(
Nk , y
(k)
n
)
= f
(
N1, y
(k)
n+1
)
.
Moreover f (Nk ,h) > f (N1,h∞), for h > h∞, where h∞ = 0.8035236546387282 · · · is the unique real
solution of the equation f (N1,h) = 1, for all k sufficiently large.
The case when k = 2 becomes a special case of the previous theorem. A computer calculation
gives h2 = 1.0446 · · · , so we show the next result as the general case of [8, Theorem 2], and which
confirms that all hk ≤ h2.
Theorem 3. For h > h2 = 1.0446 · · · and for all k > 1, we have
f (Nk ,h) > f (N1,h) .
The proof strategy used by Lichtman in [8] is based on the introduction of the zeta function for
k-almost primes as
Pk (s) =
∑
n∈Nk
1
ns
,
where P1(s) := P (s) denotes the prime zeta function. He proved the decreasing of the function
s 7→ Pk (s)−P (s) over the interval (1, s′k ) with the condition tk < s′k , for 2 ≤ k ≤ 20 where tk , s′k are,
respectively, the solutions of the equations
Pk (s)/
(
2−s +3−s)= 1 and Pk−1 (s) = 1.
We present improvements and other proofs of [8, Theorems 1 and 2]. Our proofs are based on the
following lemmas.
Ilias Laib 483
2. Lemmas
Lemma 4. Let f , g ∈C 1[a,b], be two strictly decreasing functions on the interval [a,b], such that
g (b) > f (b) and g (a) < f (a) and let (xn)n≥0 and (yn)n≥0 be two sequences defined as{
x0 = b, g (xi ) = f (xi+1) ,
y0 = a, g
(
yi
)= f (yi+1) , i ≥ 0.
Then, the sequence (xn)n≥0 is strictly decreasing and the sequence (yn)n≥0 is strictly increasing.
Furthermore, if there exist i , j ∈ N such that the function x 7→ g (x) − f (x) is monotonic on the
interval [y j , xi ], then these sequences are adjacent i.e. share a common limit.
Proof. Let (hn(x))n≥0 and (kn(x))n≥0 be two sequences of continuous and strictly increasing
functions defined on the interval [yn , xn] by hn(x) = g (xn)− f (x) and kn(x) = g (yn)− f (x). To
prove this lemma we proceed by induction on n ≥ 0. Indeed, for n = 0, since h0 is strictly
increasing with h0(b) > 0 > h0(a), by the intermediate value theorem, there exists a unique x1 ∈
(a,b) such that h0(x1) = 0. That is, g (b) = f (x1). Similarly, there exists a unique y1 ∈ (a,b) such
that g (a) = f (y1). And since g is strictly decreasing, 0 < g (a)− g (b) = f (y1)− f (x1) so f strictly
decreasing gives x1 > y1. Assume that b > x1 > ·· · > xn > yn > ·· · > y1 > a, g (xn) > f (xn) and
g (yn) < f (yn), since hn is strictly increasing with hn(xn) > 0 > hn(yn), by the intermediate value
theorem, there exists a unique xn+1 ∈ (yn , xn) such that hn(xn+1) = 0. That is, g (xn) = f (xn+1).
Similarly, there exists a unique yn+1 ∈ (yn , xn) such that g (yn) = f (yn+1). And since g is strictly
decreasing, 0 < g (yn)− g (xn) = f (yn+1)− f (xn+1) so f strictly decreasing gives xn+1 > yn+1.
For the second part of lemma, we have the two sequences (xn)n≥0, (yn)n≥0 are bounded and
monotonic. So, they converge, respectively, to some values l1 and l2 with xn > l1 ≥ l2 > yn for
any n ≥ 0, where g (l1) = f (l1) and g (l2) = f (l2). By the fact that the function x 7→ g (x)− f (x) is
monotonic on the interval [y j , xi ] for some indices i and j , it follows that g (x) = f (x) has a unique
root l ∈ [y j , xi ]. So necessarily l = l1 = l2 which completes the proof. □
Lemma 5 (Lichtman, 2022). For k ≥ 1, we have
Pk (s) =
k∑
j=1
Pk− j (s)P
(
j s
)
k
and f (Nk ,h) =
∫ ∞
1
Pk (s)e
(1−s)hd s.
Lemma 6. For k > 13 and h ≥ 0 the series
Dk (h) :=
∫ ∞
1
(
(P (s))k
k !
−P (s)
)
e(1−s)hd s
has a unique real zero h = lk where lk < 1.9.
Proof. Let k > 13. According to Lemma 5 and from Theorem of Zhang [10], we get∫ ∞
1
(P (s))k
k !
e(1−s)0d s <
∫ ∞
1
Pk (s)e
(1−s)0d s <
∫ ∞
1
P (s)e(1−s)0d s. (1)
Then, by [7, equation (5.10)], we have
0 < P (s)− log
( α
s −1
)
< 1.4(s −1) , s ∈ [1,2] ,
where α= exp(−∑m≥2 P (m)/m) = 0.72926 · · · .
Thus, for every k > 13, since log(α/0.0134) < 4, we get∫ 1.0134
1
(P (s))k d s >
∫ 0.0134
0
(
log
(α
s
))k
d s
=α
∫ ∞
log(α/0.0134)
uk e−udu
>αΓ (k +1,4) > 0.72926k !,
484 Ilias Laib
where Γ(k +1,4) is the incomplete Gamma function. It follows that∫ ∞
1
(P (s))k
k !
e(1−s)1.9d s ≥
∫ 1.0134
1
(P (s))k
k !
e(1−s)1.9d s
> e(−0.0134)1.9
∫ 1.0134
1
(P (s))k
k !
d s
> 0.72926e(−0.0134)×1.9
≥ 0.71 > 0.702
≥
∫ ∞
1
P (s)e(1−s)×1.9d s.
The series (P (s))
k
k ! −P (s) changes sign from positive to negative around its unique real root σk > 1,
where P (σk ) = (k !)
1
k−1 . So we have
Dk (h)e(
σk−1)h =
∫ σk
1
(
(P (s))k
k !
−P (s)
)
e(σk−s)hd s −
∫ ∞
σk
(
P (s)− (P (s))
k
k !
)
e(σk−s)hd s.
The expression given represents the difference between two integrals, both of which have positive
integrands. The first integral is monotonically increasing, whereas the second is monotonically
decreasing with respect to the variable h ≥ 0. Consequently, the difference between the two
integrals is monotonically increasing with respect to h ≥ 0. Moreover, since Dk (1.9) > 0, it
follows that for all h ≥ 1.9, we have ∫ ∞1 (P (s))kk ! e(1−s)hd s > ∫ ∞1 P (s)e(1−s)hd s. Combining this with
equation (1), we can conclude that the real zero h = lk of the series Dk (h) is unique and lies in the
interval [0,1.9]. This completes the proof. □
3. Proofs of Theorems 2 and 3
Proof of Theorem 2. For 2 ≤ k ≤ 13, the two series of functions h 7→ f (Nk ,h) and h 7→ f (N1,h),
satisfy the conditions of Lemma 4 on the interval [y (k)0 , x
(k)
0 ]. So by Lemma 4 the sequence
(x(k)n )n≥0 is strictly decreasing and the sequence (y
(k)
n )n≥0 is strictly increasing, then they con-
verge. The series of functions h 7→ f (Nk ,h) converges uniformly according to Erdős [2] and the
series of derivatives functions
h 7→ d
dh
f (Nk ,h) := f ′ (Nk ,h) =−
∑
a∈Nk
1
a
(
log(a)+h)2
converges uniformly on the interval [y (k)0 , x
(k)
0 ]. Also, for any h ∈ [y (k)1 , x(k)5 ], we have
f ′ (Nk ,h)− f ′ (N1,h) > f ′
(
Nk , y
(k)
1
)
− f ′
(
N1, x
(k)
5
)
=∆k
≥ f ′
(
N2, y
(2)
1
)
− f ′
(
N1, x
(2)
5
)
= f ′ (N2,0.5183)− f ′ (N1,1.56114)
=−0.2173 · · ·+0.2209 · · ·
≥ 0.0036 > 0.
So the two sequences (xn)n≥0, (yn)n≥0 converge to the common limit hk which is the unique
solution of f (Nk ,h) = f (N1,h) and f (Nk ,h) > f (N1,h) for any h > hk .
For k > 13, we have Pk (s) > (P (s))
k
k ! , so from Lemma 5 and Lemma 6, we get
f (Nk ,h) >
∫ ∞
1
(P (s))k
k !
e(1−s)hd s > f (N1,h) , for any h ≥ 1.9.
Ilias Laib 485
As f (Nk ,0) < f (N1,0), so there exists hk ∈ [0,1.9] such that f (Nk ,hk ) = f (N1,hk ) and f (Nk ,h) >
f (N1,h) for h > hk . However, for any h ∈ [0,1.9], we have
f
′
(Nk ,h)− f
′
(N1,h) > f
′
(Nk ,0)− f
′
(N1,1.9)
>− f (N1,0)
k log(2)
− f ′ (N1,1.9)
≥− 1.63 · · ·
14log(2)
+0.1764 · · ·
≥ 0.00778 > 0,
this implies that the solution hk is unique, which is the common limit of the two sequences
(xn)n≥0, (yn)n≥0. For the second part of Theorem 1, we use the same method used in [8],
which gives us the unique root of f (N1,h)− 1 is h∞ = 0.8035236546387282 · · · . This completes
the proof. □
Proof of Theorem 3. The proof of the theorem comes directly from the inequality f (Nk ,h2) >
f (N1,h2) for all k > 1 of [8] and the inequality f (Nk ,0) < f (N1,0) for all k > 1 of Zhang [10], so
the unique solution hk which is in the interval [0,1.9], becomes in the interval [0,h2]. Then for
h < hk , we have f (Nk ,h)− f (N1,h) < 0 and for h > hk we have f (Nk ,h)− f (N1,h) > 0, and since
h2 > hk , this completes the proof. □
4. Calculations table
For 2 ≤ k ≤ 13, by using Mathematica1 we compute (x(k)i )1≤i≤5 and (y (k)1 ) as the unique roots of
the equations f (Nk , x
(k)
n ) = f (N1, x(k)n+1) and f (Nk , y (k)n ) = f (N1, y (k)n+1), respectively, where x(k)0 =
kk !ek+1
(k+1)k−1−k ! and y
(k)
0 = 0. We remark that, for k ≥ 5 the sequences (x(k)n )n≥0 and (y (k)n )n≥0 converge
rapidly to hk which is the unique solution of f (Nk ,h) = f (N1,h), and for k ≥ 11 the two digits after
the point of hk are fixed with that of h∞, then for k ≥ 16 the three digits after the point of hk are
fixed with that of h∞, . . . etc.
So we can ask the question:
Open question: The sequence (hk )k ≥2, decrease and converges rapidly to
h∞ = 0.8035236546387282 · · · .
For 13 < k ≤ 25, we compute the unique root hk of the equation f (Nk ,h) = f (N1,h) by using
Mathematica. 2
1P [k −integer, s−] := If[k == 1,PrimeZetaP[s],Expand[(Sum[P [1, j ∗ s]∗P [k − j , s], j ,1,k −1]+P [1,k ∗ s])/k]]
FindRoot[NIntegrate[P [k, s]/e[(s−1)x
(k)
n ], s,1,∞] == NIntegrate[PrimeZetaP[s]/e[(s−1)x
(k)
n+1], s,1,∞], x(k)n+1,0.1],
FindRoot[NIntegrate[P [k, s]/e[(s−1)y
(k)
n ], s,1,∞] == NIntegrate[PrimeZetaP[s]/e[(s−1)y
(k)
n+1], s,1,∞]y (k)n+1,0.1],
∆k = NIntegrate[((1− s)P [k, s])/e[(s−1)y
(k)
1 ], s,1,∞]−NIntegrate[((1− s)PrimeZetaP[s])/e[(s−1)x
(k)
5 ], s,1,∞]
2FindRoot[NIntegrate[P [k, s]/e[(s−1)X ], s,1,∞] == NIntegrate[PrimeZetaP[s]/e[(s−1)X ], s,1,∞], X ,0.1]
486 Ilias Laib
Table 1. The values of (x(k)i )1≤ i ≤5, y
(k)
1 ,hk and ∆k , for 2 ≤ k ≤ 13.
k x(k)1 x
(k)
2 x
(k)
3 x
(k)
4 x
(k)
5 y
(k)
1 hk ∆k
2 23.4028 8.47027 3.86317 2.21842 1.56114 0.5183 1.04466 0.00361
3 11.0425 2.65532 1.33427 1.06189 1.00053 0.73419 0.98214 0.25936
4 6.84739 1.4812 0.99408 0.93783 0.93110 0.80975 0.93018 0.32833
5 4.70836 1.09508 0.90330 0.89120 0.89043 0.82987 0.89038 0.36537
6 3.43161 0.94071 0.86419 0.86155 0.86146 0.83018 0.86146 0.38850
7 2.61004 0.87232 0.84184 0.84127 0.84126 0.82474 0.84126 0.40332
8 2.05938 0.83979 0.84167 0.82772 0.82759 0.81875 0.82759 0.41277
9 1.68123 0.82336 0.81862 0.81860 0.81860 0.81381 0.81859 0.41872
10 1.41757 0.81465 0.81280 0.81280 0.81280 0.81021 0.81280 0.42242
11 1.23208 0.80986 0.80915 0.80915 0.80915 0.80775 0.80915 0.42469
12 1.10107 0.80716 0.80689 0.80689 0.80689 0.80613 0.80689 0.42605
13 1.00855 0.80562 0.80551 0.80551 0.80551 0.80551 0.80551 0.42687
Table 2. The values of hk , for 13 < k ≤ 25.
k hk k hk k hk k hk
14 0.80469 17 0.80374 20 0.80356 23 0.80353
15 0.80420 18 0.80365 21 0.80355 24 0.80352
16 0.80391 19 0.80359 22 0.80355 25 0.80352
Acknowledgments
The author would like to thank editor and the anonymous referee for his careful reading and their
comments. Also, the author is thankful to Miloud Mihoubi for his help to proofread this paper.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
[1] H. Cohen, “High precision computation of Hardy-Littlewood constants”, http://www.math.u-bordeaux1.fr/cohen/.
[2] P. Erdős, “Note on sequences of integers no one of which is divisible by any other”, J. Lond. Math. Soc. 10 (1935),
p. 126-128.
[3] P. Erdős, Z. Zhang, “Upper bound of
∑
1/(ai log ai ) for primitive sequences”, Proc. Am. Math. Soc. 117 (1993), no. 4,
p. 891-895.
[4] I. Laib, “Note on translated sum on primitive sequences”, Notes Number Theory Discrete Math. 27 (2021), no. 3, p. 39-
43.
[5] I. Laib, A. Derbal, R. Mechik, “Somme translatée sur des suites primitives et la conjecture d’Erdő s”, C. R. Math. 357
(2019), no. 5, p. 413-417.
[6] I. Laib, N. Rezzoug, “On a sum over primitive sequences of finite degree”, Math. Montisnigri 53 (2022), p. 26-32.
[7] J. D. Lichtman, “Almost primes and the Banks–Martin conjecture”, J. Number Theory 211 (2020), p. 513-529.
[8] ——— , “Translated sums of primitive sets”, C. R. Math. 360 (2022), p. 409-414.
[9] J. D. Lichtman, C. Pomerance, “The Erdős conjecture for primitive sets”, Proc. Am. Math. Soc., Ser. B 6 (2019), p. 1-14.
[10] Z. Zhang, “On a problem of Erdős concerning primitive sequences”, Math. Comput. 60 (1993), no. 202, p. 827-834.


New proof and generalization of some results on translated sums over k-almost primes

Nguyen Van Phu

Comptes Rendus
Mathématique
Nguyen Van Phu
A note on the weighted log canonical threshold
Volume 361 (2023), p. 969-971
Published online: 7 September 2023
https://doi.org/10.5802/crmath.456
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
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e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 969-971
https://doi.org/10.5802/crmath.456
Complex analysis and geometry / Analyse et géométrie complexes
A note on the weighted log canonical
threshold
Nguyen Van Phua
a Faculty of Natural Sciences, Electric Power University, Hanoi, Vietnam
E-mail: ltphudk@gmail.com
Abstract. In this paper, we introduce and study a set relative to singularities of plurisubharmonic functions.
We prove that this set is countable under the condition h > 0 on B\ {0}.
Mathematical subject classification (2010). 32U05, 32U15, 32U40, 32W20.
Manuscript received 28 October 2022, revised 11 December 2022, accepted 18 December 2022.
1. Introduction
Let Ω be a domain in Cn , z0 ∈ Ω and ϕ be a plurisubharmonic function on Ω (briefly, psh).
Following Demailly and Kollár [3], we introduce the log canonical threshold of ϕ at z0:
cϕ(z0) = sup{c > 0 : e−2cϕ is L1(dV2n) on a neighborhood of z0}, (1)
where dV2n denotes the Lebesgue measure of Cn .
It is an invariant of the singularity of ϕ at z0. We refer the readers to [1, 2, 4–7, 9, 10] for further
information and applications to this number.
For every non-negative Radon measures µ on a neighbourhood of z0 ∈ Cn . Following Pham
in [8], we introduce the weighted log canonical threshold of ϕ with weight µ at z0 to be:
cµ,ϕ(z0) = sup
{
c > 0 : ∃ r > 0 ,
∫
B(0,r )
e−2cϕ(z+z0)dµ(z) <+∞
}
. (2)
In the case if µ = hdV2n where h ∈ L1(dV2n),h > 0 on B \ {0},h ∈ L∞(B) then we introduce the
weighted log canonical threshold of ϕ with weight µ at z0 to be:
chdV2n ,ϕ(z0) = sup
{
c > 0 :∃ r > 0 ,
∫
B(z0,r )
e−2cϕ(z)h(z − z0)dV2n(z) <+∞
}
. (3)
From the definition of chdV2n ,ϕ(z0) and cϕ(z0), we have chdV2n ,ϕ(z0) ≥ cϕ(z0). In the paper, we
study properties of the set Eh,ϕ = {z ∈Ω : chdV2n ,ϕ(z) > cϕ(z)}. The main result of the paper prove
that the set Eh,ϕ is a countable set.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
970 Nguyen Van Phu
2. Main Theorem
Theorem 1. If h ∈ L1(dV2n),h > 0 on B\ {0},h ∈ L∞(B) then
Eh,ϕ = {z ∈Ω : chdV2n ,ϕ(z) > cϕ(z)}
is a countable set.
Proof. We have
Eh,ϕ =∪c∈Q{z ∈Ω : cϕ(z) < c < chdV2n ,ϕ(z)}.
It means that we need to prove the following set
Ec,h,ϕ = {z ∈Ω : cϕ(z) < c < chdV2n ,ϕ(z)}
is a countable set. Indeed, let z0 ∈ Ec,h,ϕ. We have
cϕ(z0) < c < chdV2n ,ϕ(z0).
Since c < chdV2n ,ϕ(z0) we can find r > 0 such that∫
B(z0,r )
e−2cϕ(z)h(z − z0)dV =
∫
B(0,r )
e−2cϕ(z+z0)h(z)dV2n <+∞.
Since h > 0 on B(0,r ) \ {0} (h(0) = 0), we have ∀ w ∈ B(z0,r ) \ {z0} there exists δ > 0 such that
B(w,δ) ⊂B(z0,r ) \ {z0} and ∫
B(0,δ)
e−2cϕ(z+w)dV2n(z) <+∞.
We obtain that cϕ(z) > c, ∀ w ∈B(z0,r ) \ {z0}. Thus, if z0 ∈ Ec,h,ϕ then
Ec,h,ϕ∩B(z0,r ) \ {z0} =∅.
So we have Ec,h,ϕ is a countable set. From
Eh,ϕ =
⋃
c∈Q
Ec,h,ϕ
we have Eh,ϕ is a countable set. □
The following proposition shows a corollary of main theorem.
Corollary 2. LetΩ be a domain of Cn and f :Ω−→C be a holomorphic function. Assuming that
h ∈ L1(dV2n),h > 0 on B\ {0},h ∈ L∞(B). Then
Eh,log | f | ⊂ {z ∈Ω : f = 0}sing,
where {z ∈Ω : f = 0}sing is the singularities of the hypersurface {z ∈Ω : f = 0}.
Proof. By Theorem 1, we have Eh,log | f | is a countable subset of {z ∈Ω : f = 0}. Take z0 ∈ {z ∈Ω :
f = 0}reg. We only need to prove that z0 ̸∈ Eh,log | f |. Indeed, we have f (z) = hm in a neighborhood
of the point z0, where {z ∈Ω : f = 0} defined locally at the point z0 by h. On the other hand, from
the proof of Theorem 1 we have
cϕ(z0) ≤ chdV2n ,ϕ(z0) ≤ limr→0 min{cϕ(z) : z ∈B(z0,r )\{z0}}.
Therefore
clog | f |(z0) = chdV2n ,log | f |(z0) =
1
m
.
This implies that z0 ̸∈ Eh,log | f |. □
Example. We choose f (z) = zm11 +·· ·+ zmnn and h(z) = ∥z∥2t (t > 0). Then
(1) Eh,log | f | = {0} if
∑n
j=1
1
m j
< 1.
(2) Eh,log | f | =; if
∑n
j=1
1
m j
≥ 1.
Nguyen Van Phu 971
References
[1] J.-P. Demailly, “Monge-Ampère operators, Lelong Numbers and Intersection theory”, in Complex analysis and
geometry, The University Series in Mathematics, Plenum Press, 1993, p. 115-193.
[2] J.-P. Demailly, P. H. Hiê. p, “A sharp lower bound for the log canonical threshold”, Acta Math. 212 (2014), no. 1, p. 1-9.
[3] J.-P. Demailly, J. Kollár, “Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano
orbifolds”, Ann. Sci. Éc. Norm. Supér. 34 (2001), no. 4, p. 525-556.
[4] T. de Fernex, L. Ein, M. Mustaţă, “Bounds for log canonical thresholds with applications to birational rigidity”, Math.
Res. Lett. 10 (2003), no. 2-3, p. 219-236.
[5] ——— , “Shokurov’s ACC Conjecture for log canonical thresholds on smooth varieties”, Duke Math. J. 152 (2010),
no. 1, p. 93-114.
[6] L. M. Hai, P. H. Hiê. p, T. Tung, “Estimates of level sets of holomorphic functions and applications to the weighted log
canonical thresholds”, J. Geom. Anal. (2021), p. 3783-3819.
[7] P. H. Hiê. p, “A comparison principle for log canonical threshold”, C. R. Acad. Sci. Paris 351 (2013), no. 11-12, p. 441-
443.
[8] ——— , “The weighted log canonical threshold”, C. R. Acad. Sci. Paris 352 (2014), no. 4, p. 283-288.
[9] ——— , “Continuity properties of certain weighted log canonical thresholds”, C. R. Acad. Sci. Paris 355 (2017), no. 1,
p. 34-39.
[10] ——— , “Log canonical thresholds and Monge–Ampère masses”, Math. Ann. 370 (2018), no. 1-2, p. 556-566.


A note on the weighted log canonical threshold

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