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Comptes Rendus
Mathématique
Marco Aymone
Sign changes of the partial sums of a random multiplicative function II
Volume 362 (2024), p. 895-901
Online since: 7 October 2024
https://doi.org/10.5802/crmath.615
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Comptes Rendus. Mathématique
2024, Vol. 362, p. 895-901
https://doi.org/10.5802/crmath.615
Research article / Article de recherche
Number theory / Théorie des nombres
Sign changes of the partial sums of a random
multiplicative function II
Changements de signe des sommes partielles d’une
fonction multiplicative aléatoire II
Marco Aymone a
a Departamento de Matemática, Universidade Federal de Minas Gerais (UFMG),
Av. Antônio Carlos, 6627, CEP 31270-901, Belo Horizonte, MG, Brazil
E-mail: aymone.marco@gmail.com
Abstract. We study two models of random multiplicative functions: Rademacher random multiplicative
functions supported on the squarefree integers f , and Rademacher random completely multiplicative func-
tions f ∗. We prove that the partial sums ∑n≤x f ∗(n) and ∑n≤x f (n)pn change sign infinitely often as x →∞,
almost surely. The case
∑
n≤x
f ∗(n)p
n
is left as an open question and we stress the possibility of only a finite
number of sign changes, with positive probability.
Résumé. Nous étudions deux modèles de fonctions multiplicatives aléatoires : les fonctions multiplicatives
aléatoires de Rademacher supportées sur les entiers sans carrés f , et les fonctions multiplicatives aléatoires
complètement multiplicatives de Rademacher f ∗. Nous prouvons que les sommes partielles ∑n≤x f ∗(n) et∑
n≤x
f (n)p
n
changent de signe infiniment souvent comme x →∞, presque sûrement. Le cas ∑n≤x f ∗(n)pn reste
une question ouverte et nous soulignons la possibilité de seulement un nombre fini de changements de signe,
avec probabilité positive.
Keywords. Random multiplicative functions, Oscillation theorems.
Mots-clés. Fonctions multiplicatives aléatoires, théorèmes d’oscillation.
2020 Mathematics Subject Classification. 11K65, 11N99.
Funding. I am thankful the CNPq for supporting this visit at Aix-Marseille Université – grant PDE
no. 400010/2022-4 (200121/2022-7). This project also was funded by CNPq grant Universal no. 403037/2021-2
and by FAPEMIG grant Universal no. APQ-00256-23.
Manuscript received 19 October 2023, accepted 19 January 2024.
1. Introduction
1.1. Main results and motivation
A Rademacher random multiplicative function f is defined as follows: Over the primes, the
values f (p) are i.i.d. random variables taking ±1 values with probability 1/2 in each instance
(Rademacher random variables), and at the other values of n,
f (n) :=µ2(n) ∏
p|n
f (p),
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
896 Marco Aymone
where µ is the Möbius function.
By a Rademacher random completely multiplicative function f ∗, we mean that
f ∗(n) = ∏
pa∥n
f (p)a .
In a previous paper, Aymone, Heap and Zhao [4] proved that the partial sums
∑
n≤x f (n) of
a Rademacher random multiplicative function change sign infinitely often1 as x → ∞, almost
surely. The key idea was to show that, almost surely
lim
σ→1/2+
∫ ∞
1
( ∑
n≤x
f (n)
)
dx
x1+σ
= 0,
lim
σ→1/2+
∫ ∞
1
∣∣∣∣ ∑
n≤x
f (n)
∣∣∣∣ dxx1+σ =∞,
and these visibly capture an infinite number of sign changes.
Our task here is to study sign changes for weighted sums of f and f ∗:∑
n≤x
f (n)
nα
,
∑
n≤x
f ∗(n)
nα
, α≥ 0.
In both cases, we have that for α > 1/2, the partial sums converge to a non-vanishing Euler
product, and hence, it can be shown that these partial sums become positive for all x sufficiently
large, almost surely. Therefore, we restrict ourselves to the cases 0 ≤α≤ 1/2.
Our motivation comes from the study of the sign changes of the partial sums of the determin-
istic counterparts to f and f ∗: the Möbius µ and the Liouville λ. For 0 ≤α< 1/2, the partial sums∑
n≤x µ(n)n−α change sign infinitely often as x →∞, due to the analytic properties of 1/ζ(s) and
to the Landau’s oscillation theorem. Answering a question of Pólya, Haselgrove [19] proved that∑
n≤x λ(n) changes sign infinitely often as x → ∞. In the range 0 < α < 1/2, the case of an infi-
nite number of sign changes of
∑
n≤x λ(n)n−α recasts questions about the non-trivial Riemann
zeros, and at the present moment, unconditional results are unknown, see [28] by Mossinghoff
and Trudgian. In a paper by Humphries [22], it can be deduced from his results that under RH
and other hypothesis on Riemann zeros, the partial sums
∑
n≤x λ(n)n−α change sign infinitely
often as x →∞.
In both cases
∑
n≤x µ(n)n−1/2 and
∑
n≤x λ(n)n−1/2 the problem of an infinitude of sign changes
seems to be intricate and even assuming standard hypothesis, conditional results are unknown.
In [28] is left as an open problem to prove or disprove that
∑
n≤x λ(n)n−1/2 is negative for all x ≥ 17.
A remarkable consequence of this fact is that this implies that all non-trivial zeros of the Riemann
zeta function have multiplicity at most 2, see [28, Theorem 2.4]. In [22] it is showed that under
standard assumptions on the non-trivial zeros of the Riemann zeta function, the set of x for which∑
n≤x λ(n)pn is negative has total logarithmic asymptotic density.
Taking all of these facts into account, for the random counterparts f and f ∗ we were able to
proof the following results.
Theorem 1. Let f be a Rademacher random multiplicative function. Then for each 0 ≤ α ≤ 1/2,∑
n≤x
f (n)
nα changes sign infinitely often as x →∞, almost surely.
Theorem 2. Let f ∗ be a Rademacher random completely multiplicative function. Then for each
0 ≤α< 1/2 we have that ∑n≤x f ∗(n)nα changes sign infinitely often as x →∞, almost surely.
The case
∑
n≤x
f ∗(n)p
n
is left as an open question, and it is possible that these sums change sign
only a finite number of times with positive probability, here we give an heuristic reason. In a
recent paper, Angelo and Xu [1] (and later improvements by Kerr and Klurman [24]) studied the
1Here we say that a function M(x) changes sign infinitely often as x →∞ if at a certain point x0, M(x) is not always
non-negative or non-positive for all x ≥ x0.
Marco Aymone 897
weighted sums
∑
n≤x
f ∗(n)
n in the context of a “random Turán conjecture”. They proved that the
probability of
∑
n≤x
f ∗(n)
n being positive for all x ≥ 1 is at least 1− 10−45. This implies a certain
positivity for harmonic sums of f ∗ and it may be possible that such result may be extended for∑
n≤x
f ∗(n)
nσ , where σ> 1/2. Further, after a convolution argument, f ∗ = f ∗1PS , where 1PS (n) is
the indicator that n is a perfect square. Therefore,∑
n≤x
f ∗(n)p
n
≈ ∑
n≤x
f (n)p
n
log(x/n).
The later is known as a Riesz mean, and at some instances these averages stabilizes oscillatory
properties of an oscillating arithmetic function providing a finite number of sign changes for
these averages.
1.2. Proof method
Our proof follows closely the lines of [4]. The difference here is that, in contrast with unweighted
sums of a Rademacher random multiplicative function, we have the almost sure limits
lim
σ→1/2+
∫ ∞
1
( ∑
n≤x
f (n)p
n
)
dx
x1/2+σ
=∞,
lim
σ→1/2+
∫ ∞
1
∣∣∣∣ ∑
n≤x
f (n)p
n
∣∣∣∣ dxx1/2+σ =∞,
and this alone does not capture an infinite number of sign changes, unless we make the two
divergences above quantitative, and that these divergences differ in size. This can, indeed, be
achieved by using the following quantitative statement by Harper:
Theorem 3 (Harper [13, p. 25]). Let p run over the primes and
(
f (p)
)
p be a sequence of inde-
pendent Rademacher random variables. For fixed A > 3, almost surely, there exists a sequence
σk → 1/2+ such that
sup
1≤t≤2
(
log
(
1
σk−1/2
))2
(∑
p
f (p)cos(t log p)
pσk
−2loglog
(
1
σk−1/2
))
≥ log
(
1
σk−1/2
)
− A loglog
(
1
σk−1/2
)
.
1.3. Background
Here we are intended to give a list of results in the literature (perhaps non-complete) on random
multiplicative functions. The first result on partial sums of random multiplicative functions is
due to Wintner [31], where he considered a question by Levy [27] and showed that
∑
n≤x f (n) is
almost surelyΩ(x1/2−ϵ) and ≪ x1/2+ϵ, for any ϵ> 0.
Later, these results have been improved by Erdős [10] and subsequently by Halász [11]. Later
Basquin [5] and Lau, Tenenbaum and Wu [26] made a significant improvement by proving that,
almost surely, ∑
n≤x
f (n) ≪px(loglog x)2+ϵ,
for any ϵ> 0.
In this year of 2023, Caich [8] proved that we can replace (loglog x)2+ϵ above by (loglog x)1/4+ϵ.
This combined with Harper’s Omega bound [17] gives a sharp description of the size of the
fluctuations of the partial sums of f :∑
n≤x
f (n) =Ω(px(loglog x)1/4−ϵ),
for any ϵ> 0, almost surely.
898 Marco Aymone
Central limit Theorems also have been studied in various settings: [9, 14, 21, 25, 30].
The problem of moments also have been extensively studied, with the remarkable result of
Harper that shows that the first moment exhibits better than squareroot cancellation [16]. Other
related works (including related models): [3, 6, 7, 12, 15, 18, 20, 32]. And another viewpoints of
investigation [1, 2, 4, 23, 24, 29].
2. Preliminaries
2.1. Notation
Here p stands for a generic prime number. We use the standard Vinogradov notation f (x) ≪ g (x)
or Landau’s f (x) =O(g (x)) whenever there exists a constant c > 0 such that | f (x)| ≤ c|g (x)|, for all
x in a set of parameters. When not specified, this set of parameters is an infinite interval (a,∞)
for sufficiently large a > 0. The standard f (x) = o(g (x)) means that f (x)/g (x) → 0 when x → a,
where a could be a complex number or ±∞. For g (x) > 0 for all x, we say that f (x) =Ω(g (x)) if
limsupx→∞
| f (x)|
g (x) > 0.
2.2. Some Lemmas
By the pioneering work of Wintner [31], we have that F (s) := ∑∞n=1 f (n)ns converges for all Re(s) >
1/2, almost surely. His proof can be modified to show that the same is true for F∗(s) =∑∞n=1 f ∗(n)ns .
Moreover we have:
Lemma 4. For all Re(s) > 1/2, almost surely we have
F (s) =∏
p
(
1+ f (p)
p s
)
,
F∗(s) =∏
p
(
1− f (p)
p s
)−1
.
This leads to the almost sure identities
F (s) = exp
(∑
p
f (p)
p s
− 1
2
logζ(2s)+O(1)
)
, (1)
F∗(s) = exp
(∑
p
f (p)
p s
+ 1
2
logζ(2s)+O(1)
)
, (2)
where the O(1) term above is actually an analytic function that is uniformly bounded in Re(s) ≥
1/3+ϵ, for small fixed ϵ> 0. For a proof of these results we refer reader to [4, Lemma 2.4] and the
references therein. To conclude this section, we have
Lemma 5. Let ( f (p))p be independent Rademacher random variables. Then, for all ϵ> 0, we have
almost surely as σ→ 1/2+ ∑
p
f (p)
pσ
≪
(
log
(
1
σ−1/2
))1/2+ϵ
.
For a proof we refer reader to [4, Lemma 2.1 and Remark 2.1].
Marco Aymone 899
3. Proof of the main results
3.1. The case
∑
n≤x
f (n)p
n
By the upper bound for the partial sums of a Rademacher random multiplicative function [5]
and [26], ∑
n≤x
f (n) ≪px(loglog x)2+ϵ,
we have that by partial summation ∑
n≤x
f (n)p
n
≪ (log x)2, a.s.
Further refinements of this last upper bound could, perhaps, be achieved, by following the lines
of [3] and more recently of [12].
Now, for Re(s) =σ> 1/2,
F (s) :=
∞∑
n=1
f (n)
ns
=
∞∑
n=1
f (n)p
n
1
ns−1/2
= (s −1/2)
∫ ∞
1
( ∑
n≤x
f (n)p
n
)
dx
xs+1/2
.
By equation (1) and Lemma 5 combined with the estimate logζ(2σ) = log(1/(2σ− 1))+O(1)
valid for σ → 1/2+, we obtain that as σ → 1/2+, almost surely we have F (σ) = (2σ− 1)1/2+o(1).
Hence, we almost surely have as σ→ 1/2+∫ ∞
1
( ∑
n≤x
f (n)p
n
)
dx
xσ+1/2
= 1
(2σ−1)1/2+o(1) . (3)
On the other hand, for t ≥ 1∣∣∣∣F (σ+ i t )t
∣∣∣∣≪ ∫ ∞
1
∣∣∣∣ ∑
n≤x
f (n)p
n
∣∣∣∣ dxxσ+1/2 .
Now, by equation (1),
|F (s)| = exp
(∑
p
f (p)cos(t log p)
pσ
− 1
2
Re(logζ(2s))+O(1)
)
.
We recall that | logζ(σ+ i t )| ≤ loglog |t |+O(1), for σ≥ 1 and t ≥ 2. By Theorem 3, almost surely
we have an infinite sequence of points σk → 1/2+ and tk ≥ 1 such that∣∣∣∣F (σk + i tk )tk
∣∣∣∣≥ exp((1+o(1)) log( 1σk −1/2
))
≫ 1
(σk −1/2)0.9
.
Therefore, almost surely, there exists a sequence σk → 1/2+ such that∫ ∞
1
∣∣∣∣ ∑
n≤x
f (n)p
n
∣∣∣∣ dxxσk+1/2 ≫ 1(σk −1/2)0.9 . (4)
To complete the argument, we see that the quantities at the left of (3) and (4) have different
size, and so the partial sums
∑
n≤x
f (n)p
n
cannot be always non-negative or non-positive, there
always must be infinite sign changes, almost surely.
3.2. The case
∑
n≤x
f ∗(n)
nα
Let then 0 ≤ α< 1/2. Let F∗(s) be the Dirichlet series of f ∗. Similarly as in the previous case, by
equation (2) and Lemma 5, we have that
F∗(σ) = 1
(2σ−1)1/2+o(1) ,
almost surely as σ→ 1/2+.
900 Marco Aymone
By partial summation and integration,
F∗(s) = (s −α)
∫ ∞
1
( ∑
n≤x
f ∗(n)
nα
)
dx
xs+1−α
.
Therefore, almost surely as σ→ 1/2+∫ ∞
1
( ∑
n≤x
f ∗(n)
nα
)
dx
xσ+1−α
= 1+o(1)
(1/2−α)
1
(2σ−1)1/2+o(1) . (5)
And similarly to (4), almost surely there exists a sequence σk → 1/2+ such that∫ ∞
1
∣∣∣∣ ∑
n≤x
f ∗(n)
nα
∣∣∣∣ dxxσ+1−α ≫ 1(2σk −1)0.99 . (6)
Just as before, (5) and (6) captures infinite sign changes, almost surely.
3.3. The case
∑
n≤x
f (n)
nα
This case can be treated similarly as the previous one, although we can proceed in a more
elementary way as in [4], that is, Theorem 3 is not needed here.
The proof of Theorems 1 and 2 are now complete.
Acknowledgements
This project started while I was a visiting Professor at Aix-Marseille Université, and I am warmly
thankful for their kind 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.
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Sign changes of the partial sums of a random multiplicative function II

Megane Bournissou

Sylvain Ervedoza

Marius Tucsnak

Exact controllability for systems describing plate vibrations. A perturbation approach

Matteo Ferrari

Comptes Rendus
Mathématique
Matteo Ferrari
Some properties of a modified Hilbert transform
Volume 362 (2024), p. 799-806
Online since: 17 September 2024
https://doi.org/10.5802/crmath.600
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. 799-806
https://doi.org/10.5802/crmath.600
Research article / Article de recherche
Numerical analysis / Analyse numérique
Some properties of a modified Hilbert
transform
Quelques propriétés d’une transformée de Hilbert
modifiée
Matteo Ferrari ,a
a University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090,
Vienna, Austria
E-mail: matteo.ferrari@univie.it (M. Ferrari)
Abstract. Recently, Steinbach et al. introduced a novel operator HT : L
2(0,T ) → L2(0,T ), known as the
modified Hilbert transform. This operator has shown its significance in space-time formulations related to
the heat and wave equations. In this paper, we establish a direct connection between the modified Hilbert
transform HT and the canonical Hilbert transform H . Specifically, we prove the relationship HTϕ=−H ϕ̃,
where ϕ ∈ L2(0,T ) and ϕ̃ is a suitable extension of ϕ over the entire R. By leveraging this crucial result, we
derive some properties of HT , including a new inversion formula, that emerge as immediate consequences
of well-established findings on H .
Résumé. Récemment, Steinbach et al. ont introduit un nouvel opérateur HT : L
2(0,T ) → L2(0,T ), connu
sous le nom de transformée de Hilbert modifiée. Cet opérateur a montré son importance dans les formu-
lations spatio-temporelles liées aux équations de la chaleur et des ondes. Dans cet article, nous établissons
un lien direct entre la transformée de Hilbert modifiée HT et la transformée de Hilbert canonique H . Plus
précisément, nous prouvons la relation HTϕ = −H ϕ̃, où ϕ ∈ L2(0,T ) et ϕ̃ est une extension appropriée de
ϕ sur l’ensemble de R. En tirant parti de ce résultat crucial, nous déduisons certaines propriétés de HT , y
compris une nouvelle formule d’inversion, qui émergent comme des conséquences immédiates de résultats
bien établis sur H .
2020 Mathematics Subject Classification. 44A15.
Funding. This work is partially supported by the INdAM-GNCS project “Metodi numerici per lo studio di
strutture geometriche parametriche complesse” (CUP E53C22001930001) and by the MIUR project “Diparti-
menti di Eccellenza 2018-2022” (CUP E11G18000350001). Moreover, this research was funded in part by the
Austrian Science Fund (FWF) project 10.55776/F65.
Manuscript received 25 September 2023, accepted 15 December 2023.
1. Introduction and main result
In [10], a modified Hilbert transform HT , associated with the bounded interval (0,T ), has been
defined. Given the Fourier series of ϕ ∈ L2(0,T )
ϕ(t ) =
∞∑
k=0
ϕk sin
((π
2
+kπ
) t
T
)
with ϕk =
2
T
∫ T
0
ϕ(s)sin
((π
2
+kπ
) s
T
)
ds,
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
800 Matteo Ferrari
the modified Hilbert transform is defined as
HTϕ(t ) =
∞∑
k=0
ϕk cos
((π
2
+kπ
) t
T
)
t ∈ (0,T ). (1)
This operator has been employed in the context of space-time discretizations of PDEs using
both finite element [2, 7, 10] and boundary element [9] methods. It is particularly well-suited
because, by defining the Sobolev space
H 1/20, (0,T ) =
{
ϕ ∈ H 1/2(0,T ) :ϕ(0) = 0} ,
it can be shown (see [10]) that −∂t HT : H 1/20, (0,T ) → [H 1/20, (0,T )]′ induces an equivalent norm in
H 1/20, (0,T ). Furthermore, it holds 〈ϕ,HTϕ〉L2(0,T ) ≥ 0 for all ϕ ∈ L2(0,T ).
An alternative integral representation of HT suited for numerical schemes has been presented
in [11, Lemma 2.1].
Lemma 1. For ϕ ∈ L2(0,T ), the operator HT allows the integral representation
HTϕ(t ) = 1
2T
Pϕ
∫ T
0
ϕ(s)
[
csc
(
π(s + t )
2T
)
+csc
(
π(s − t )
2T
)]
ds, t ∈ (0,T ) (2)
as Cauchy principal value integral. Moreover, if ϕ ∈ H 1(0,T ) it holds
HTϕ(t ) =− 2
π
ϕ(0) log
(
tan
(
πt
4T
))
− 1
π
∫ T
0
∂tϕ(s) log
(
tan
(
π(s + t )
4T
)
tan
(
π|s − t |
4T
))
ds, (3)
for t ∈ (0,T ) as a weakly singular integral.
The Hilbert transform H of a function ϕ is defined as Cauchy principal value integral
Hϕ(t ) = 1
π
p.v.
∫
R
ϕ(s)
t − s d s,
whenever it exists (see [1, Chapter 9], [5] and references therein).
The following relationship between the Hilbert transform H and its modified version HT has
been established in [8, Theorem 4.3].
Theorem 2. For ϕ ∈ L2(0,T ), it holds
HTϕ=−H ϕ̄+Bϕ in L2(0,T ),
where B : L2(0,T ) → L2(0,T ) is a compact operator, and ϕ̄ is the reflection (see Figure 1)
ϕ̄(s) =

−ϕ(s +2T ) s ∈ (−2T,−T ),
−ϕ(−s) s ∈ (−T,0),
ϕ(s) s ∈ (0,T ),
ϕ(2T − s) s ∈ (T,2T ),
0 else.
(4)
In this paper, we prove that HT is, in fact, precisely the Hilbert transform H applied to a
specific odd periodic extension with alternating signs.
Theorem 3. For ϕ ∈ L2(0,T ), it holds
HTϕ=−H ϕ̃ in L2(0,T ),
where ϕ̃ is the periodic reflection (see Figure 2)
ϕ̃(s) =

−ϕ(s +2T −4kT ) s ∈ ((4k −2)T, (4k −1)T ), k ∈Z,
−ϕ(4kT − s) s ∈ ((4k −1)T,4kT ), k ∈Z,
ϕ(s −4kT ) s ∈ (4kT, (4k +1)T ), k ∈Z,
ϕ(4kT +2T − s) s ∈ ((4k +1)T, (4k +2)T ), k ∈Z.
(5)
Matteo Ferrari 801
T 2T
−T−2T
Figure 1. For ϕ(x) = x2, ϕ : [0,T ] →R, it is plotted ϕ̄ :R→R defined in (4).
T 2T
−T−2T 3T 4T
−3T−4T
Figure 2. For ϕ(x) = x2, ϕ : [0,T ] →R, it is plotted ϕ̃ :R→R defined in (5).
Example 4. From (3), we obtain that for ϕ(s) = 1 with s ∈ (0,T ), we have HTϕ(t ) =
− 2π log | tan
(
πt
4T
)|. It can be readily seen that, for s ∈ R, ϕ̃(s) = sgn(sin(as)) with a = π2T , and for
this function, it is well-known that H ϕ̃(t ) = 2π log | tan
( at
2
)| (see [5, Equation 6.9]).
Example 5. A function that satisfies a periodicity of the form (5) is ϕ(s) = sin(πs) with T = 12 .
In this case, on R, we have the trivial extension ϕ̃(s) = sin(πs). It is well-known that H ϕ̃(t ) =
−cos(πt ) (see [5, Equation 1.8]). From (2), we can calculate
HTϕ(t ) = p.v.
∫ 1
2
0
sin(πs)
[
csc(π(s + t ))+csc(π(s − t ))]ds
= p.v.
∫ t+ 12
t
sin(π(s − t ))
sin(πs)
ds +p.v.
∫ 1
2 −t
−t
sin(π(s − t ))
sin(πs)
ds = cos(πt ).
2. Proofs of the main result
In this section, we present three distinct proofs of Theorem 3. The first proof relies on the integral
representation 2, while the second proof utilizes an alternative definition of the Hilbert transform
H specifically designed for periodic functions. Finally, the third proof employs Fourier series.
802 Matteo Ferrari
2.1. Proof based on the integral representation
The first proof is established by leveraging the Laurent series expansion of the cosecant function,
as presented in [4, Formula (9.3.30)]
csc(z) = 1
z
+2z
∞∑
k=1
(−1)k
z2 −k2π2 . (6)
This series converges absolutely and uniformly for |z| <π, in fact for each N ∈N it holds∣∣∣∣csc(z)− 1z
∣∣∣∣≤ 2|z|π2 N∑k=1 1∣∣∣( zkπ )2 −1∣∣∣
1
k2
≤ |z|
3
.
Starting from the definitions of Hilbert transform H and ϕ̃ in (5), we can write
H ϕ̃(t ) = 1
π
[
. . .+p.v.
∫ −T
−3T
ϕ̃(s)
t − s ds +p.v.
∫ T
−T
ϕ̃(s)
t − s ds +p.v.
∫ 3T
T
ϕ̃(s)
t − s ds + . . .
]
= 1
π
[
. . .−p.v.
∫ T
−T
ϕ̃(s)
t − s −2T ds +p.v.
∫ T
−T
ϕ̃(s)
t − s ds −p.v.
∫ T
−T
ϕ̃(s)
t − s +2T ds + . . .
]
where we have used the alternating signs periodicity of ϕ̃. Therefore, we can compactly write
H ϕ̃(t ) = 1
π
+∞∑
k=−∞
p.v.
∫ T
−T
ϕ̃(s)
(−1)k
t − s +2kT ds.
By interchanging the integral and the summation (which is possible due to the uniform conver-
gence of the series), and utilizing (6), we obtain
H ϕ̃(t ) = 1
π
p.v.
∫ T
−T
ϕ̃(s)
[ +∞∑
k=−∞
(−1)k
t − s +2kT
]
ds
= 1
π
p.v.
∫ T
−T
ϕ̃(s)
[
1
t − s +2(t − s)
+∞∑
k=1
(−1)k
(t − s)2 −4k2T 2
]
ds
= 1
2T
p.v.
∫ T
−T
ϕ̃(s)csc
(
π(t − s)
2T
)
ds.
From the latter, we obtain the modified Hilbert transform HT in integral form (2) by recalling the
definition (5), and writing
H ϕ̃(t ) =− 1
2T
p.v.
∫ 0
−T
ϕ(−s)csc
(
π(t − s)
2T
)
ds + 1
2T
p.v.
∫ T
0
ϕ(s)csc
(
π(t − s)
2T
)
ds
=− 1
2T
p.v.
∫ T
0
ϕ(s)csc
(
π(t + s)
2T
)
ds − 1
2T
p.v.
∫ T
0
ϕ(s)csc
(
π(t − s)
2T
)
ds
=−HTϕ(t ).
2.2. Proof based on the Hilbert transform for periodic functions
Given ϕ ∈ L2(0,T ), we observe that ϕ̃ as in (5) is actually periodic with a period of 4T , and it is
known that the Hilbert transform H for a periodic function can be calculated using (see [6]):
H ϕ̃(t ) = 1
4T
p.v.
∫ 2T
−2T
ϕ̃(s)cot
(
π(t − s)
4T
)
ds. (7)
Matteo Ferrari 803
Let us expand the four contributions of the integral above to recover HTϕ
H ϕ̃(t ) =− 1
4T
[
p.v.
∫ −T
−2T
ϕ(s +2T )cot
(
π(t − s)
4T
)
ds −p.v.
∫ 0
−T
ϕ(−s)cot
(
π(t − s)
4T
)
ds
+p.v.
∫ T
0
ϕ(s)cot
(
π(t − s)
4T
)
ds +p.v.
∫ 2T
T
ϕ(2T − s)cot
(
π(t − s)
4T
)
ds
]
=− 1
4T
p.v.
∫ T
0
ϕ(s)cot
(
π(t − s +2T )
4T
)
ds − 1
4T
p.v.
∫ T
0
ϕ(s)cot
(
π(t + s)
4T
)
ds
+ 1
4T
p.v.
∫ T
0
ϕ(s)cot
(
π(t − s)
4T
)
ds + 1
4T
p.v.
∫ T
0
ϕ(s)cot
(
π(t + s −2T )
4T
)
ds.
Finally, using the trigonometric formula
2csc(2x) =−cot
(
x ± π
2
)
+cot(x), |x| < π
2
, (8)
we again conclude that HTϕ=−H ϕ̃ in L2(0,T ).
2.3. Proof based on Fourier series
For f with period 2T in L2(−T,T ), the circular Hilbert transform is defined as (see [5, Section 6.4])
H f (t ) = i
∞∑
k=1
[
f−k e−ikπ
t
T − fk e ikπ
t
T
]
with fk =
1
2T
∫ T
−T
f (s)e−ikπ
s
T ds.
Let us verify that this definition is consistent with the original definition of the modified Hilbert
transform HT by Steinbach and Zank (1). That is, we will demonstrate that for a given function
ϕ ∈ L2(0,T ), we have H ϕ̃ = −HTϕ in L2(0,T ), with ϕ̃ as in (5). Recalling that ϕ̃ is a periodic
function with period 4T , let us begin by writing
H ϕ̃(t ) = i
∞∑
k=1
[
ϕ̃−k e−ikπ
t
2T − ϕ̃k e ikπ
t
2T
]
,
and readily one can verify
ϕ̃k =
1
4T
∫ 2T
−2T
ϕ̃(s)e−ikπ
s
2T ds = 1
4T
∫ 2T
0
ϕ̃(s)
[
e−ikπ
s
2T −e ikπ s2T
]
ds
=− i
2T
∫ 2T
0
ϕ̃(s)sin
(
kπ
s
2T
)
ds =−ϕ̃−k .
Hence, we continue
H ϕ̃(t ) =−
∞∑
k=1
[
e−ikπ
t
2T +e ikπ t2T
] 1
2T
∫ 2T
0
ϕ̃(s)sin
(
kπ
s
2T
)
ds
=−
∞∑
k=1
2cos
(
kπ
t
2T
)
1
2T
∫ 2T
0
ϕ̃(s)sin
(
kπ
s
2T
)
ds.
Moreover, we can now write
1
2T
∫ 2T
0
ϕ̃(s)sin
(
kπ
s
2T
)
ds = 1
2T
∫ T
0
ϕ(s)sin
(
kπ
s
2T
)
ds + 1
2T
∫ 2T
T
ϕ(2T − s)sin
(
kπ
s
2T
)
ds
= 1
2T
∫ T
0
ϕ(s)
[
sin
(
kπ
s
2T
)
− sin(kπ)sin
(
kπ
s
2T
)]
ds
=
0 k even,1
T
∫ T
0
ϕ(s)sin
(
kπ
s
2T
)
ds k odd.
804 Matteo Ferrari
Finally, we conclude
H ϕ̃(t ) =−
∞∑
k=0
cos
(
(2k +1)π t
2T
)
2
T
∫ T
0
ϕ̃(s)sin
(
(2k +1)π s
2T
)
ds =−HTϕ(t ).
3. Consequences of the main result
In this section we show simple consequences of Theorem 3.
3.1. Inversion formula
For f ∈ L2(−T,T ) and periodic with period 2T , the inversion formula holds (see [5, Formula
(6.35)])
H 2 f (t ) =− f (t )+ 1
2T
∫ T
−T
f (s)ds, in L2(−T,T ). (9)
For ϕ ∈ L2(0,T ), we can calculate
H
(
HTϕ
)
(t ) =−H 2ϕ̃(t ) = ϕ̃(t )− 1
4T
∫ 2T
−2T
ϕ̃(s)ds =ϕ(t ), in L2(0,T ),
since ϕ̃ is an odd function.
3.2. Alternative formula
For an odd function f ∈ L2(−T,T ) with period 2T , it can be shown that
H f (t ) = 1
T
p.v.
∫ T
0
f (s)
sin
(
π sT
)
cos
(
π sT
)−cos(π tT )ds, in L2(−T,T ).
Therefore, we obtain the alternative formula in L2(0,T )
HTϕ(t ) =−H ϕ̃(t ) =− 1
2T
p.v.
∫ 2T
0
ϕ̃(s)
sin
(
π s2T
)
cos
(
π s2T
)−cos(π t2T )ds
=−cos
(
π t2T
)
T
p.v.
∫ T
0
ϕ(s)
sin
(
π s2T
)
cos2
(
π s2T
)−cos2 (π t2T )ds.
This formula can also be deduced from equation (2) using trigonometric identities.
3.3. Integral representation
Let suppose that ϕ ∈ H 1(0,T ), then the derivative of ϕ̃ is
∂t ϕ̃(s)|(−2T,2T ) =

−∂tϕ(s +2T ) s ∈ (−2T,−T ),
∂tϕ(−s) s ∈ (−T,0),
∂tϕ(s) s ∈ (0,T ),
−∂tϕ(2T − s) s ∈ (T,2T ).
(10)
We note that if ϕ ∈ H 1(0,T ), the only possible discontinuity of ϕ̃|(−2T,2T ) is in the point s = 0.
Matteo Ferrari 805
Starting from (7), integrating by parts in the intervals of continuity (−2T,0) and (0,2T ) of ϕ̃, we
obtain
HTϕ(t ) =−H ϕ̃(t ) =− 1
π
∫ 2T
−2T
∂t ϕ̃(s) log
(
sin
(
π|t − s|
4T
))
ds
+ 1
π
[(
ϕ̃(0−)− ϕ̃(0+)) log(sin( πt
4T
))
+ (ϕ̃(2T −)− ϕ̃(−2T +)) log(cos( πt
4T
))]
=− 1
π
∫ 2T
−2T
∂t ϕ̃(s) log
(
sin
(
π|t − s|
4T
))
ds − 2
π
ϕ(0) log
(
tan
(
πt
4T
))
.
This formula, in the caseϕ(0) = 0 is the original form in which Hilbert wrote the Hilbert transform
in [3]. Let split the integral in the four usual intervals, and use (10),
HTϕ(t ) = 1
π
∫ T
0
∂tϕ(s) log
(
cos
(
π(t + s)
4T
))
ds − 1
π
∫ T
0
∂tϕ(s) log
(
sin
(
π(t + s)
4T
))
ds
− 1
π
∫ T
0
∂tϕ(s) log
(
sin
(
π|t − s|
4T
))
ds + 1
π
∫ T
0
∂tϕ(s) log
(
cos
(
π(t − s)
4T
))
ds
− 2
π
ϕ(0) log
(
tan
(
πt
4T
))
=− 1
π
∫ T
0
∂tϕ(s) log
(
tan
(
π(t + s)
4T
)
tan
(
π|t − s|
4T
))
ds − 2
π
ϕ(0) log
(
tan
(
πt
4T
))
.
We have obtained exactly the result of Lemma 3 in an alternative way.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organisation
that could benefit from this article, and have declared no affiliation other than their research
organisations.
References
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no. e23070.
[3] D. Hilbert, “Grundzüge einer allgemeinen theorie der linearen integralgleichungen”, Nachr.
Ges. Wiss. Göttingen, Math.-Phys. Kl. 3 (1904), pp. 213–259.
[4] E. Hille, Analytic function theory. Vol. 1, Ginn and Company, 1959.
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[7] I. Perugia, C. Schwab and M. Zank, “Exponential convergence of hp-time-stepping in
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[10] O. Steinbach and M. Zank, “Coercive space-time finite element methods for initial bound-
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806 Matteo Ferrari
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