Journal de Théorie des Nombres de Bordeaux

Barry Mazur

Karl Rubin

Controlling Selmer groups in the  higher core rank case

Bryden Cais

Christopher Davis

Jonathan Lubin

A characterization of strictly APF extensions

Florin Nicolae

Florin NICOLAE
On the determination of number fields by L-functions
Tome 28, no 2 (2016), p. 431-433.
<http://jtnb.cedram.org/item?id=JTNB_2016__28_2_431_0>
© Société Arithmétique de Bordeaux, 2016, tous droits réservés.
L’accès aux articles de la revue « Journal de Théorie des Nom-
bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord
avec les conditions générales d’utilisation (http://jtnb.cedram.
org/legal/). Toute reproduction en tout ou partie de cet article sous
quelque forme que ce soit pour tout usage autre que l’utilisation à
fin strictement personnelle du copiste est constitutive d’une infrac-
tion pénale. Toute copie ou impression de ce fichier doit contenir la
présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Journal de Théorie des Nombres
de Bordeaux 28 (2016), 431–433
On the determination of number fields by
L-functions
par Florin NICOLAE
Résumé. Nous montrons qu’une extension galoisienne du corps
des nombres rationnels est déterminée par toute fonction L d’Artin
corespondant à un caractère du groupe de Galois qui contient tous
les caractères irréductibles.
Abstract. We prove that a Galois number field is determined
by any Artin L-function corresponding to a character of the Galois
group which contains all irreducible characters.
Let K/Q be a finite Galois extension with the Galois group G, χ1, . . . , χr
the irreducible characters of G with the dimensions d1 := χ1(1), . . . , dr :=
χr(1), and L(s, χ1,K/Q), . . . , L(s, χr,K/Q) the corresponding Artin L-
functions. The Dedekind zeta function ζK(s) is the Artin L-function corre-
sponding to the regular character RegG =
∑r
j=1 djχj :
ζK(s) = L(s,RegG,K/Q)
= L(s, χ1,K/Q)d1 · . . . · L(s, χr,K/Q)dr .
It is known that ζK(s) determines the field K: if L/Q is a finite normal
extension of Q such that
ζL(s) = ζK(s)
then
L = K.
Since dj > 0, the regular character RegG contains all irreducible characters
χj , j = 1, . . . , r. Starting from this observation, we prove that the field K
is determined by any Artin L-function corresponding to a character of the
Galois group G which contains all irreducible characters.
Manuscrit reçu le 26 mars 2014, révisé le 16 juillet 2014, accepté le 18 septembre 2014.
Mathematics Subject Classification. 11R42.
Mots-clefs. Galois number field, Artin L-function.
432 Florin Nicolae
Theorem 1. Let K/Q, L/Q be finite Galois extensions, G the Galois group
of K/Q, χ1, . . . , χr the irreducible characters of G, k1 > 0, . . . , kr > 0 inte-
gers, χ =
∑r
j=1 kjχj, H the Galois group of L/Q, ϕ1, . . . , ϕt the irreducible
characters of H, l1 > 0, . . . , lt > 0 integers, ϕ =
∑t
j=1 ljϕj. If
L(s, χ,K/Q) = L(s, ϕ, L/Q)
then
K = L.
Proof. Let M/Q be a finite Galois extension which contains both K and
L. For j = 1, . . . , r let
χ̃j : Gal(M/Q)→ C, χ̃j(σ) := χj(σGal(M/K)),
where we identify G with the factor group Gal(M/Q)/Gal(M/K). Then χ̃j
is an irreducible character of Gal(M/Q) and
(1) Gal(M/K) = ∩rj=1Ker (χ̃j),
where Ker (χ̃j) is the kernel of the character χ̃j . For j = 1, . . . , t let
ϕ̃j : Gal(M/Q)→ C, ϕ̃j(σ) := ϕj(σGal(M/L)),
where we identify H with the factor group Gal(M/Q)/Gal(M/L). Then ϕ̃j
is an irreducible character of Gal(M/Q) and
(2) Gal(M/L) = ∩tj=1Ker (ϕ̃j).
It is a fundamental property of Artin L-functions that
L(s, χj ,K/Q) = L(s, χ̃j ,M/Q), j = 1, . . . , r
and that
L(s, ϕj , L/Q) = L(s, ϕ̃j ,M/Q), j = 1, . . . , t
hence
L(s, χ,K/Q) =
r∏
j=1
L(s, χj ,K/Q)kj =
r∏
j=1
L(s, χ̃j ,M/Q)kj ,
and
L(s, ϕ, L/Q) =
t∏
j=1
L(s, ϕj , L/Q)lj =
t∏
j=1
L(s, ϕ̃j ,M/Q)lj ,
therefore
(3)
r∏
j=1
L(s, χ̃j ,M/Q)kj =
t∏
j=1
L(s, ϕ̃j , L/Q)lj .
Artin proved that the L-functions corresponding to irreducible characters
are multiplicatively independent ([1], Satz 5, P. 106). Thus, from the mul-
tiplicative relation (3) we deduce that
{χ̃1, . . . , χ̃r} = {ϕ̃1, . . . , ϕ̃t}.
Determination of number fields 433
Together with (1) and (2), this implies that
Gal(M/K) = Gal(M/L),
so
K = L. 
References
[1] E. Artin, “Über eine neue Art vonL-Reihen”, Abh. Math. Sem. Univ. Hamburg 3 (1924),
no. 1, p. 89-108.
Florin Nicolae
Technische Universität Berlin
Institut für Mathematik
Strasse des 17. Juni 136
D-10623 Berlin, Germany
and
Institute of Mathematics of the Romanian Academy
P.O.BOX 1-764
RO-014700 Bucharest, Romania
E-mail: Florin.Nicolae@imar.ro
URL: http://www.imar.ro/


On the determination of number fields by L-functions

David P. Roberts

Lightly ramified number fields with Galois group $S.M_{12}.A$

Xiaodong Lü

Quanwu Mu

On Waring–Goldbach problem of mixed powers

Fabrizio Andreatta

Carlo Gasbarri

Deformation of torsors under monogenic group schemes

Mariusz Urbański

Anna Zdunik

Continuity of the Hausdorff Measure of Continued Fractions and Countable Alphabet Iterated Function Systems

Alain Togbé

Alain TOGBÉ
Corrigendum to "Complete Solutions of a Family of Cubic Thue Equations"
[Journal de Théorie des Nombres de Bordeaux 18 (2006), 285-298]
Tome 28, no 1 (2016), p. 287-288.
<http://jtnb.cedram.org/item?id=JTNB_2016__28_1_287_0>
© Société Arithmétique de Bordeaux, 2016, tous droits réservés.
L’accès aux articles de la revue « Journal de Théorie des Nom-
bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord
avec les conditions générales d’utilisation (http://jtnb.cedram.
org/legal/). Toute reproduction en tout ou partie de cet article sous
quelque forme que ce soit pour tout usage autre que l’utilisation à
fin strictement personnelle du copiste est constitutive d’une infrac-
tion pénale. Toute copie ou impression de ce fichier doit contenir la
présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Journal de Théorie des Nombres
de Bordeaux 28 (2016), 287–288
Corrigendum to "Complete Solutions of a Family
of Cubic Thue Equations" [Journal de Théorie des
Nombres de Bordeaux 18 (2006), 285-298]
par Alain TOGBÉ
Résumé. Dans notre article original [4], le lemme 2.2 n’était pas
correctement démontré. L’erreur a depuis été réparée par Lee–
Louboutin [3]. Cette erreur n’affecte cependant pas le résultat final
de l’article.
Abstract. In our original paper [4], Lemma 2.2 was not prop-
erly proved. This was done by Lee–Louboutin [3]. The gap doesn’t
affect the final result of the paper.
In [4], using Baker’s method we studied the family of Thue equations
(1)
Φn(x, y) = x3 + (n8 + 2n6 − 3n5 + 3n4 − 4n3 + 5n2 − 3n+ 3)x2y
− (n3 − 2)n2xy2 − y3
= ±1,
for n ≥ 0. To do so, we considered the number field Kn related with φn(x)
defined by
(2)
φn(x) = x3 + (n8 + 2n6 − 3n5 + 3n4 − 4n3 + 5n2 − 3n+ 3)x2
− (n3 − 2)n2x− 1.
See also [2], pages 100-103. One can see that φn has three real roots
θ(1), θ(2), θ(3). For any solution (x, y) of (1), we have
(3) Φn(x, y) =
3∏
j=1
(
x− θ(j)y
)
= NQ(θ(1))/Q
(
x− θ(j)y
)
= ±1.
This means theat x − θ(j)y is a unit in the order O := Z[θ(1), θ(2)]. We
proved the following result (see Lemma 2.2 of [4]).
Manuscrit reçu le 6 novembre 2014, accepté le 21 novembre 2014.
Mathematics Subject Classification. 11D59, 11Y50.
Mots-clefs. Parametric Thue equations, Baker’s method.
288 Alain Togbé
Lemma 2.2. Let us consider O := Z[θ(1), θ(2)] and < −1, θ(1), θ(2) > a
subgroup of the unit group. We have
(4) I := [O× :< −1, θ(1), θ(2) >] < 3,
for n ≥ 29.
There was a gap in the proof of Lemma 2.2 due to the misinterpretation
of Cusick’s result, see Proposition 2.1 of [1]. Lee and Louboutin filled the
gap and confirmed Lemma 2.2, for any integer n by proving Lemma 6.4,
page 293 of [3]. As Lemma 2.2 is correct, this doesn’t affect the main result
obtained in our paper.
Remark. The example given on page 288 of [3] has no link with polyno-
mial (2).
Acknowledgments. We thank Professor Stéphane Louboutin for sending
us the manuscript of their paper.
References
[1] T. W. Cusick, “Lower bounds for regulators”, in Number theory, Noordwijkerhout 1983
(Noordwijkerhout, 1983), Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, p. 63-73.
[2] Y. Kishi, “A family of cyclic cubic polynomials whose roots are systems of fundamental
units”, J. Number Theory 102 (2003), no. 1, p. 90-106.
[3] J. H. Lee & S. R. Louboutin, “On the fundamental units of some cubic orders generated
by units”, Acta Arith. 165 (2014), no. 3, p. 283-299.
[4] A. Togbé, “Complete solutions of a family of cubic Thue equations”, J. Théor. Nombres
Bordeaux 18 (2006), no. 1, p. 285-298.
Alain Togbé
Mathematics Department
Purdue University North Central
1401 S, U.S. 421
Westville IN 46391
USA
E-mail: atogbe@pnc.edu


Corrigendum to “Complete Solutions of a Family of Cubic Thue Equations” [Journal de Théorie des Nombres de Bordeaux 18 (2006), 285-298]

Gautami Bhowmik

Olivier Ramaré

Jan-Christoph Schlage-Puchta

Gautami BHOWMIK, Olivier RAMARÉ et Jan-Christoph SCHLAGE-PUCHTA
Tauberian Oscillation Theorems and the Distribution of Goldbach numbers
Tome 28, no 2 (2016), p. 291-299.
<http://jtnb.cedram.org/item?id=JTNB_2016__28_2_291_0>
© Société Arithmétique de Bordeaux, 2016, tous droits réservés.
L’accès aux articles de la revue « Journal de Théorie des Nom-
bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord
avec les conditions générales d’utilisation (http://jtnb.cedram.
org/legal/). Toute reproduction en tout ou partie de cet article sous
quelque forme que ce soit pour tout usage autre que l’utilisation à
fin strictement personnelle du copiste est constitutive d’une infrac-
tion pénale. Toute copie ou impression de ce fichier doit contenir la
présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Journal de Théorie des Nombres
de Bordeaux 28 (2016), 291–299
Tauberian Oscillation Theorems and the
Distribution of Goldbach numbers
par Gautami BHOWMIK, Olivier RAMARÉ et Jan-Christoph
SCHLAGE-PUCHTA
Résumé. Soit D(s) =
∑
ann
−s une série de Dirichlet qui, en
approchant par la droite un point non-réel sur la frontière de
son domaine de convergence, tend vers ∞. Nous présentons un
théorème d’oscillation localisé pour
∑
n≤x an, et une application
à
∑
n1+···+nk≤x Λ(n1) · · ·Λ(nk).
Abstract. Let D(s) =
∑
ann
−s be a Dirichlet-series which
tends to ∞ when approaching some non-real point on the bound-
ary of its domain of convergence from the right. We give a lo-
calized oscillation theorem for
∑
n≤x an, and apply this result to∑
n1+···+nk≤x Λ(n1) · · ·Λ(nk).
1. Introduction and Results
Let D(s) =
∑
ann
−s be a function represented by a Dirichlet series,
holomorphic in a half-plane < s > σ0. A large class of Tauberian theorems
deal with the question of recovering the behaviour of
∑
n≤N an from the
analytic behaviour ofD. Under suitable assumptions one finds that the limit
s ↘ σ0 gives a main term, while the behaviour of D on the line < s = σ0
determines the error term. If one is looking for more precise information,
then the behaviour ofD on the above line can be turned into oscillation type
results. The first theorem of this type is Landau’s theorem stating that if an
is always positive, then D has a singularity at its abscissa of convergence.
This result translates into oscillation type results for arithmetical functions,
e.g. one immediately obtains Ψ(x) = x+ Ω±(x1/2−ε) by exploiting the fact
that ζ
′
ζ (s) is regular at 1/2. Landau’s original theorem may be used to prove
the existence of infinitely many sign changes of a function. However it is
impossible to deduce from it further information on the number of sign
changes, the least sign change, localisation of sign changes and so forth. To
do so would require more information on D and the use of less elementary
methods. For a more detailed analysis of this problem we refer the reader
Manuscrit reçu le 8 février 2014, accepté le 4 septembre 2015.
Mathematics Subject Classification. 11M45, 11N37, 11P32.
Mots-clefs. Dirichlet series, Omega-results, Goldbach numbers.
292 Gautami Bhowmik, Olivier Ramaré, Jan-Christoph Schlage-Puchta
to the introduction of [7], which could be seen as the first example of the
use of non-elementary methods in this area.
To prove an effective version of Landau’s theorem one requires more
explicit data for D. For the rest of this article we fix the following notation.
Put F (s) =
∫∞
1 B(x) dx/xs+1, where B is a real function satisfying
(1.1)
2U∫
U
B(x)2 dx U2σ0+1η(U),
for some positive, increasing, continuous function η with η(U)  U ε.
Clearly, F is holomorphic for < s > σ0. Assume that there exists a real
number t0 6= 0 such that
(1.2) c = lim sup
σ↘σ0
(σ − σ0)|F (σ + it0)| > 0,
and
(1.3) a = lim sup
σ↘σ0
(σ − σ0)F (σ) <∞, b = lim inf
σ↘σ0
(σ − σ0)F (σ) > −∞.
We denote the Lebesgue measure of a set of real numbers by µ. Then
Kaczorowski & Szydło [6] proved the following :
Theorem 1.1. Let B be a real function satisfying (1.1). Define F as above,
a and b by (1.3), and assume that (1.2) holds true. Then for every ε > 0
we have
µ ({x ∈ [1, T ] : B(x) > (b+ c− ε)xσ0}) = Ω( T
η(T ))
and
µ ({x ∈ [1, T ] : B(x) < (a− c+ ε)xσ0}) = Ω( T
η(T )).
This theorem has the disadvantage that the points at which B is large
may come in large blocks, so we get no explicit information on the location
or even the number of sign changes. Here we prove the following.
Theorem 1.2. Let B be a real function satisfying (1.1). Define F as above,
a and b by (1.3), and assume that (1.2) holds true. Then we have, for every
fixed ε > 0 and as δ goes to 0,
(1.4)
∫ ∞
1
µ
(
{x ∈ [u, 2u] : B(x) > (b+ c− ε)xσ0}η(4u)du
u2+δ
= Ω(δ−1)
and
(1.5)
∫ ∞
1
µ
(
{x ∈ [u, 2u] : B(x) < (a− c+ ε)xσ0}η(4u)du
u2+δ
= Ω(δ−1).
Oscillation Theorems and Goldbach Numbers 293
As an application we consider the error term for the summatory function
of the number of representations of an integer as the sum of k primes. Put
Gk(n) =
∑
n1+···+nk=n Λ(n1) · · · · · Λ(nk), Hk(x) = −k
∑
ρ
xk−1+ρ
ρ(1+ρ)...(k−1+ρ) ,
where the summation runs over all non-trivial zeros of the Riemann ζ func-
tion and define the error term ∆k(x) by means of the formula∑
n≤x
Gk(n) =
xk
k! +Hk(x) + ∆k(x).
The function ∆k was first considered by Fujii [5] who showed that un-
der RH we have ∆2(x) = O((x log x)4/3), which was later improved [3] to
O(x log5 x) and finally by Languasco & Zaccagnini [8] to O(x log3 x) and for
k ≥ 3 to O(xk−1 logk x) . On the other hand we have ∆2(x) = Ω(x log log x)
(see [2]). The proof of the Ω-result uses the fact that G2 gets surprisingly
large, though ∆2(x) is expected to be much more regular for most x. In
fact, we have the following.
Proposition 1.3. Assume RH. Then we have
lim
σ↘k
(σ − k)
∫ ∞
1
∆k(x)
xσ
dx = ck
for some constant ck. When k = 2, we have
c2 =
∑
ρ
π
sin(πρ)
where the sum runs over all non-trivial zeros of ζ.
On the other hand we can apply Theorem 1.2 to show that the relation
∆k(x)  ckxr−1 holds only in a very weak sense.
Proposition 1.4. There exists a constant c′k > 0, such that, as δ goes to
0, we have∫ ∞
1
µ
(
{x ∈ [u, 2u] : ∆k(x) > (ck + c′k)xk−1}
)(log u)6du
u2+δ
= Ω(δ−1)
and ∫ ∞
1
µ
(
{x ∈ [u, 2u] : ∆k(x) < (ck − c′k)xk−1}
)(log u)6du
u2+δ
= Ω(δ−1).
Corollary 1.5. We have ∆2(x) = Ω±(x) and ∆k(x)− ckxk−1 = Ω±(xk−1)
for k ≥ 3.
We mention that the same result was obtained in [3] with Ω in place
of Ω±.
294 Gautami Bhowmik, Olivier Ramaré, Jan-Christoph Schlage-Puchta
2. Proof of Theorem 1.2
We first note that (1.5) corresponds to (1.4) applied to −B instead of B.
We set a′ = a− ε and consider the auxiliary functions
g(x) = B(x)− a′xθ, G(s) =
∫ ∞
1
g(x)dx
xs+1
= F (s)− a
′
s− θ
.
We introduce the positive and negative parts of g:
g+(x) = max(g(x), 0), g−(x) = max(−g(x), 0), (x ≥ 1),
and the corresponding Mellin transforms
G+(s) =
∫ ∞
1
g+(x)dx
xs+1
, G−(s) =
∫ ∞
1
g−(x)dx
xs+1
.
The reader will readily check that the three transforms G, G+ and G− and
absolutely convergent when <s > θ. We clearly have G(s) = G+(s)−G−(s).
The set we want to investigate is
A = {x ≥ 1 : g(x) > 0}.
We define the following function:
(2.1) m#(T ) = η(T )µ{T ≤ x ≤ 2T : g(x) > 0}/T.
Turning to the core of the proof, we assume that
(2.2)
∑
`≥0
m#(2`)
2δ` = o(δ
−1)
when δ > 0. Our aim is to obtain a contradiction. We will then convert the
negation of (2.2) into a “continuous” statement.
Let t be a real number and select σ = σ0 + δ for some δ > 0. We want
to show that (σ − θ)|G+(σ + it)| is small. In order to do so, we write
(σ − σ0)|G+(σ + it)| ≤ δ G+(σ)
= δ
∫
A
g(x)dx
xθ+δ+1
≤ δ
∑
k≥0
∫
A∩[2k,2k+1]
g(x)dx
xθ+δ+1
≤ δ
∑
k≥0
(
η(2k)
∫
A∩[2k,2k+1]
dx
x2δ+1
)1/2
.
Oscillation Theorems and Goldbach Numbers 295
As a consequence, on simply using
∫
A∩[2k,2k+1]
dx
x2δ+1
≤
∫
A∩[2k,2k+1]
dx
2(2δ+1)k ,
we have obtained
(2.3)
(σ − σ0)|G+(σ + it)| ≤ δ
∑
k≥0
√
m#(2k)
2kδ
≤ δ
(
δ−1sumk≥0
m#(2k)
2kδ
)1/2
= o(1).
We have thus established that
lim
σ→σ+
(σ − σ0)|G+(σ + it)| = 0.
To treat G−(σ + it) we simply use
∀σ > σ0, |G+(σ + it)| ≤ G−(σ).
It remains for us to use our hypothesis concerning the existence of a singu-
larity. There exists a sequence of σ tending to σ0 from above for which
c ≤ (σ − σ0)|F (σ + it0)|+ o(1)
≤ (σ − σ0)
∣∣∣G(σ + it0) + a′
σ + it0 − θ
∣∣∣+ o(1)
≤ (σ − σ0)|G(σ + it0)|+ o(1)
since t0 6= 0. We bound above |G(σ + it0)| by |G(σ)|, which is also −G(σ),
up to o(1/(σ − δ)). We then employ −G(σ) = −F (σ) + a′/(σ − σ0). We
finally reach
c ≤ a′ − lim inf
σ→θ+
(σ − σ0)F (σ) + o(1)
which contradicts our hypotheses.
To sum up, we have established that
(2.4)
∑
`≥0
m#(2`)
2δ` = Ω(δ
−1)
and now we convert this to a ’continuous’ form.
(2.5)
∫ ∞
0
m#(et)e−δtdt = Ω(δ−1).
296 Gautami Bhowmik, Olivier Ramaré, Jan-Christoph Schlage-Puchta
The proof of (2.5) follows from (2.4). We notice that∫ `
`−1
m#(2u)η(2
u+2)du
η(2u)
=
∫ `
`−1
∫
2u≤t≤2u+1,
g(t)>0
η(2u+2)dt
2u du
= η(2`)
∫
2`−1≤t≤2`,
g(t)>0
∫
t/2≤2u≤t
du
2u dt+ η(2
`+1)
∫
2`≤t≤2`+1,
g(t)>0
∫
t/2≤2u≤t
du
2u dt
≥ m#(2`) +m#(2`−1).
The conclusion is now easy.
3. Application to Goldbach numbers
Let us put Φk(s) =
∑
n≥1
Gk(n)
ns .
Lemma 3.1. Suppose the RH holds. Then for any k ≥ 3 there exist rational
functions f1,k(s), . . . , f4,k(s), such that
Φk(s) = f1,k(s)ζ(s− k + 1) + f2,k(s)ζ(s− k + 2)
+ f3,k(s)
ζ ′
ζ
(s− k + 1) + f4,k(s)Φ2(s− k + 2) +R(s),
(3.1)
where R(s) is holomorphic in the half-plane <s > k− 1− 1/10, and all the
poles and zeros of the rational functions are real. Moreover, if we define
A2k(x) =
∑
n≤xGk(n)(x−n), then there exist polynomials g1,k, . . . g4,k, such
that
(3.2)
∑
n≥1
A2k(n)n−s = g1,k(s)ζ(s− k − 1) + g2,k(s)ζ(s− k)
+ g3,k(s)
ζ ′
ζ
(s− k − 1) + g4,k(s)Φ2(s− k) + R̃(s),
where R̃ is holomorphic in the half-plane <s > k + 1− 1/10.
Proof. This is essentially [3, Theorem 3] together with the observation that
the polynomials in [3, Lemma 2] have only real roots. 
The following is due to Egami-Matsumoto [4, Equation 4.3].
Lemma 3.2. Assume the RH. Then we have
Φ2(s) = −
1
Γ(s)
∑
ρ,ρ′
Γ(s+ 1− ρ)Γ(ρ)
(s− ρ− ρ′)ρ′ +R(s),
where R is a function holomorphic in the entire complex plane.
Oscillation Theorems and Goldbach Numbers 297
We now prove the first half of Proposition 1.3 and Corollary 1.5.
Let γ0 = 14.134 . . . be the imaginary part of the first non-trivial zero of
ζ. We claim that taking t0 = 2γ0 in Theorem 1.2 yields a non-trivial result.
To do so we have to check that the conditions (1.2) and (1.3) are satisfied.
It follows from Stirling’s formula that |Γ(12 + it)| < e
−π|t|/4. Hence if ρ, ρ′
are non-trivial zeros of ζ, and s is a complex number with real part > 1
such that |s− ρ− ρ′| < |ρ′|/2 we have
Γ(s+ 1− ρ)Γ(ρ)
(s− ρ− ρ′)ρ′ ≤
1
< s− 1e
−π|ρ−1/2|/4,
whereas for |s− ρ− ρ′| ≥ |ρ′|/2 we have
Γ(s+ 1− ρ)Γ(ρ)
(s− ρ− ρ′)ρ′ ≤
2e−π|ρ−1/2|/4
|ρ′|2
.
Clearly the sum over all ρ, ρ′ of the latter form converges absolutely. For
each fixed ρ there are O(|ρ| log |ρ|) pairs of the former type, hence after
multiplying with <s− 1 the sum over these pairs converges absolutely and
uniformly as well, and we obtain
lim
σ↘1
(σ − 1)
∑
ρ,ρ′
Γ(σ + it+ 1− ρ)Γ(ρ)
(σ + it− ρ− ρ′)ρ′
=
∑
ρ,ρ′
lim
σ↘1
(σ − 1)Γ(σ + it+ 1− ρ)Γ(ρ)
(σ + it− ρ− ρ′)ρ′
If ρ+ ρ′ 6= 1 + it, then the denominator is bounded away from 0, while the
numerator tends to 0, and we see that the limit is 0. If ρ+ ρ′ = 1 + it, then
σ+ it− ρ− ρ′ = σ− 1, and using the functional equation of the Γ-function
we obtain
lim
σ↘1
(σ − 1)
∑
ρ,ρ′
Γ(σ + it+ 1− ρ)Γ(ρ)
(σ + it− ρ− ρ′)ρ′ =
∑
ρ+ρ′=1+it
Γ(ρ′)Γ(ρ)
Putting t = 0 we obtain
(σ − 1)Φ2(σ + it) ∼
1
Γ(1)
∑
ρ
|Γ(ρ)|2 = 3.259 . . . · 10−19,
which implies the "k = 2"-part of Proposition 1.3.
For t = 2γ0 the sum contains no pair ρ, ρ′ with |=ρ′| < 1000 different
from ρ = ρ′ = 12 + iγ0. To estimate the sum over zeros we use the estimates
N(T + 1) − N(T ) ≤ log T , where N(T ) denotes the number of zeros of ζ
with imaginary part in (0, T ], and |Γ(1/2 + it)| ≤ e−
π
4 t. This bound follows
from the work of Backlund [1], details have been given in [9, Lemma 3].
298 Gautami Bhowmik, Olivier Ramaré, Jan-Christoph Schlage-Puchta
Hence∑
ρ+ρ′=1+2iγ0
ρ 6=ρ′
Γ(ρ)Γ(1 + 2γ1 − ρ) ≤
∑
=ρ>100
Γ(ρ)Γ(1 + 2γ1 − ρ)
≤
∑
γ>100
e−
π
2 (γ−γ1)
≤
∑
n≥100
e−
π(n−15)
2 logn
≤ e34
∑
103≤n≤104
e−
πn
2 + e24
∑
n>104
e−n
≤ e
34−50π
1− e−π/2
+ e
24−104
1− e−1
≤ e−122.
Hence
(3.3)
lim
σ↘1
(σ − 1)Φ2(σ + 2iγ0) =
Γ2(12 + iγ0) + θe
−122
Γ(1 + 2iγ0)
= 0.023049 . . .− 0.47088 . . . i+ θe−122.
where θ is a complex number satisfying |θ| ≤ 1. Hence (1.2) holds true with
some c ≥ 0.4713
We conclude that ∆2(x) − 3.259 · 10−19 is infinitely often larger than
0.4713 and smaller then −0.4713. Since 0.4713 > 3.259 · 10−19, the first
part of Corollary 1.5 follows.
For the general case of Proposition 1.3 we use (3.1) to see that there
exists some integer A, such that
(σ − k)
∫ ∞
1
∆k(x)
xσ logA x
dx = ck,
where A+1 equals the total order of the pole of the right hand side of (3.1).
Strictly speaking, Φ2 does not have a pole at 1, however, we have just
seen that as σ ↘ 1, Φ2(σ) behaves like 3.259...·10
−19
(σ−1) , which is sufficiently
pole-like. However, it then follows from the fact that the functions gi,k are
polynomials, that A = 0, thus the general case of Proposition 1.3 follows.
To obtain Corollary 1.5 for k ≥ 3, we first note that Proposition 1.3
implies that Condition (3) is satisfied, while (3.3) together with the fact
that the zeros of the functions fi,k are all real implies that Condition (2) is
satisfied for σ0 = k− 1 and t0 = 2γ. Hence we can apply Theorem 1.2, and
our claim follows.
Oscillation Theorems and Goldbach Numbers 299
References
[1] R. J. Backlund, “Über die Nullstellen der Riemannschen Zetafunktion”, Acta Math. 41
(1916), no. 1, p. 345-375.
[2] G. Bhowmik & J.-C. Schlage-Puchta, “Mean representation number of integers as the sum
of primes”, Nagoya Math. J. 200 (2010), p. 27-33.
[3] ———, “Meromorphic continuation of the Goldbach generating function”, Funct. Approx.
Comment. Math. 45 (2011), no. part 1, p. 43-53.
[4] S. Egami & K. Matsumoto, “Convolutions of the von Mangoldt function and related Dirich-
let series”, in Number theory, Ser. Number Theory Appl., vol. 2, World Sci. Publ., Hackensack,
NJ, 2007, p. 1-23.
[5] A. Fujii, “An additive problem of prime numbers. II, III”, Proc. Japan Acad. Ser. A Math.
Sci. 67 (1991), no. 7, p. 248-252, 278-283.
[6] J. Kaczorowski & B. Szydło, “Some Ω-results related to the fourth power moment of the
Riemann zeta-function and to the additive divisor problem”, J. Théor. Nombres Bordeaux 9
(1997), no. 1, p. 41-50.
[7] S. Knapowski & P. Turán, “Comparative prime-number theory. I. Introduction”, Acta
Math. Acad. Sci. Hungar. 13 (1962), p. 299-314.
[8] A. Languasco & A. Zaccagnini, “The number of Goldbach representations of an integer”,
Proc. Amer. Math. Soc. 140 (2012), no. 3, p. 795-804.
[9] J.-C. Schlage-Puchta, “Sign changes of π(x, q, 1) − π(x, q, a)”, Acta Math. Hungar. 102
(2004), no. 4, p. 305-320.
Gautami Bhowmik
Laboratoire Painlevé
LABEX-CEMPI
Université Lille 1
59655 Villeneuve d’Ascq Cedex
FRANCE
E-mail: bhowmik@math.univ-lille1.fr
Olivier Ramaré
Laboratoire Painlevé
LABEX-CEMPI
Université Lille 1
59655 Villeneuve d’Ascq Cedex
FRANCE
E-mail: bhowmik@math.univ-lille1.fr
Jan-Christoph Schlage-Puchta
Mathematisches Institut
Universität Rostock
18051 Rostock
GERMANY
E-mail: jan-christoph.schlage-puchta@uni-rostock.de


Tauberian Oscillation Theorems and the Distribution of Goldbach numbers

Takayuki Morisawa

Ryotaro Okazaki

Height and Weber’s Class Number Problem

Not a member yet

Controlling Selmer groups in the higher core rank case