A mean value on the sum of two primes
in arithmetic progressions
Yuta Suzuki
1
Introduction
In this note, we consider the sum of two primes in arithmetic progressions,
which is a generalization of the Goldbach problem. Although it is somewhat an
indirect way, we shall consider this problem in some average sense as Rüppel
did in [12, 13]. For this additive problem, we use the representation function
given by
∑
R(n, q, a) = R(n, q1 , a1 , q2 , a2 ) :=
Λ(m1 )Λ(m2 ),
m1 +m2 =n
mi ≡ai (mod qi )
where Λ(n) is the von Mangoldt function, a1 , a2 , q1 , q2 , n are positive integers
satisfying (a1 , q1 ) = (a2 , q2 ) = 1 and q = (q1 , q2 ), a = (a1 , a2 ). Let us also
introduce
R(n, q, a, b) := R(n, q, a, q, b)
for positive integers a, b, q satisfying (ab, q) = 1. In 2009, Rüppel [12] studied
the mean value of this representation function, i.e.
∑
R(n, q, a, b).
(1)
n≤X
In particular, she obtained, under a weakened variant of the Generalized Riemann Hypothesis (GRH), an asymptotic formula for the mean value (1). More
precisely, she assumed GRH except the existence of real zeros. In this note, we
assume a hypothesis similar to, but even weaker than, Rüppel’s one.
Hypothesis (GRH with real zeros). Every complex non-trivial zero of Dirichlet
L functions in the strip 0 < σ < 1 lies on the critical line σ = 1/2.
We call this hypothesis GRHR as an abbreviation.
Assuming this hypothesis GRHR, Rüppel [12] proved
∑
n≤X
R(n, q, a, b) =
(
)
X2
+ O X 1+δ (log q)2 ,
2φ(q)2
(2)
where δ = 1/2 unless real zeros exist for the modulus q, in which case we let δ
be the largest one among these real zeros. She considered the mean value (1),
1
but we can also obtain the corresponding result for the mean value
∑
R(n, q1 , a1 , q2 , a2 )
(3)
n≤X
by the same method. Moreover, her method can be used to prove
∑
X2
−
2φ(q)2
R(n, q, a, b) =
n≤X
+
∑∑
∑
χ (mod q)
χ(a) + χ(b) ∑ X βχ +1
φ(q)2
βχ (βχ + 1)
χ(a)ψ(b) ∑ ∑ Γ(βχ )Γ(βψ ) X βχ +βψ
χ, ψ (mod q)
φ(q)2
βχ , βψ
Γ(βχ + βψ ) βχ + βψ
βχ
(
+O X
3/2
2
(log q)
(4)
)
,
where βχ runs through all real zeros of L(s, χ) with βχ > 1/2.
These results correspond to the result of Fujii [4] for the ordinary Goldbach
Problem. Fujii [4] proved, for the representation function
∑
Λ(m1 )Λ(m2 )
R(n) =
m1 +m2 =n
of the ordinary Goldbach problem, an asymptotic formula
∑
n≤X
R(n) =
(
)
X2
+ O X 3/2
2
under the Riemann Hypothesis (RH). This was improved by Fujii [5] himself to
∑
R(n) =
n≤X
(
)
∑ X ρ+1
X2
−2
+ O (X log X)4/3
2
ρ(ρ + 1)
ρ
(5)
under RH, where ρ runs through all non-trivial zeros of the Riemann zeta function. After this pioneering work of Fujii, the error term on the right-hand side
of (5) was improved to
≪ X(log X)5
(by Bhowmik and Schlage-Puchta [1]),
≪ X(log X)
(by Languasco and Zaccagnini [9]).
3
Of course we assume RH in all of these results. We note that Bhowmik and
Schlage-Puchta proved also the omega result
= Ω(X log log X)
(6)
for this error term. This omega result is independent from RH or GRH.
In [17], we improved Rüppel’s result (2) up to the accuracy of the result of
Languasco and Zaccagnini [9]. For a given Dirichlet character χ (mod q), we
introduce the set of “non-trivial zeros” of L(s, χ)
{
{ ρ = β + iγ | 0 < β < 1, ζ(ρ) = 0 } ∪ {1} (when χ is principal),
Z(χ) :=
{ ρ = β + iγ | 0 < β < 1, L(ρ, χ) = 0 }
(otherwise)
2
and Z0 (χ) := Z(χ) ∩ (1/2, 1]. For ρ ∈ Z(χ), we let
{
1
(when ρ = 1),
A(ρ) :=
−m(ρ) (when ρ ̸= 1),
where m(ρ) is the multiplicity of ρ as a zero of L(s, χ). For ℜκ, ℜµ > 0 and
X ≥ 2, we let
W0 (κ, µ) = W0 (X, κ, µ) :=
Γ(κ)Γ(µ) X κ+µ
,
Γ(κ + µ) κ + µ
and for ρ1 ∈ Z(χ1 ), ρ2 ∈ Z(χ2 ) and X ≥ 2, we let
W (ρ1 , ρ2 ) = W (X, ρ1 , ρ2 ) := A(ρ1 )A(ρ2 )W0 (X, ρ1 , ρ2 ).
Throughout this note, E(X) denotes error terms which can be estimated as
E(X) ≪ X(log X)(log q1 X)(log q2 X).
Then the main result in [17] is:
Theorem 1. Assume GRHR. For X ≥ 2, we have
∑ ∑ χ1 (a1 )χ2 (a2 ) ∑ ∑
∑
R(n, q, a) =
W (ρ1 , ρ2 ) + E(X),
φ(q1 )φ(q2 )
n≤X
ρ1 ∈Z(χ1 )
ρ2 ∈Z(χ2 )
β1 +β2 >1
χ1 (mod q1 )
χ2 (mod q2 )
where the implicit constant in the error term is absolute.
The main aim of this note is to give an alternative proof of Theorem 1.
We remark that the main term of Rüppel’s result (2) appears in Theorem
1 as the term corresponding to the pair (1, 1) ∈ Z(χ0 ) × Z(χ0 ). Note that
W (1, 1) = X 2 /2. Moreover, the main and oscillating terms on the right-hand
side of (4) correspond to the pairs of “real non-trivial zeros” in Z0 (χ1 )×Z0 (χ2 ).
The other terms in Theorem 1 was included in the error terms of (2) and (4).
Hence Theorem 1 is an improvement of these results (2) and (4) of Rüppel.
Our previous proof [17] of Theorem 1 follows the argument of Languasco and
Zaccagnini [9], which uses the circle method with power series. Recently, Goldston and Yang succeeded in obtaining the result of Languasco and Zaccagnini
even via finite trigonometric polynomials. The key point of their argument is a
quite neat preliminary averaging which will be carried out in (8). This preliminary averaging enables us to save some logarithms from the estimate of K(U, h)
defined by (19). See the proof of Lemma 3. In this note, we sketch an alternative
proof of Theorem 1 by using this technique of Goldston and Yang [7].
2
The circle method and the Goldston-Yang trick
Instead of the original representation function R(n, q, a), we use the twisted
representation function
∑
χ1 (m1 )Λ(m1 )χ2 (m2 )Λ(m2 )
R(n, χ1 , χ2 ) :=
m1 +m2 =n
3
for Dirichlet characters χ1 (mod q1 ), χ2 (mod q2 ). We use two trigonometric
polynomials
∑
S(α, U, χ) :=
χ(n)Λ(n)e(nα),
n≤U
∑
Tβ (α, U ) :=
n
β−1
e(nα),
T (α) := T1 (α, X),
n≤U
where e(α) := exp(2πiα), U ≥ X and χ (mod q) is a Dirichlet character. Clearly,
∑
∫
R(n, χ1 , χ2 ) =
1/2
−1/2
n≤X
S(α, U, χ1 )S(α, U, χ2 )T (−α)dα.
(7)
Here we note that this equation (7) holds for any U ≥ X. Hence we can take
an average over X ≤ U ≤ 2X. Then we have
∑
n≤X
∫
1
R(n, χ1 , χ2 ) =
X
2X
∫
1/2
−1/2
X
We introduce the approximation
∑
S(α, U, χ) =
S(α, U, χ1 )S(α, U, χ2 )T (−α)dα dU.
(8)
A(β)Tβ (α, U ) + R(α, U, χ).
β∈Z0 (χ)
Then the integral in (8) can be expanded as
∑
∑
A(β1 )Iβ1 S2 +
=
∑∑
−
A(β2 )Iβ2 S1
β2 ∈Z0 (χ2 )
β1 ∈Z0 (χ1 )
A(β1 )A(β2 )Iβ1 β2 + IR ,
β1 ∈Z0 (χ1 )
β2 ∈Z0 (χ2 )
where
Iβi Sj :=
1
X
∫
2X
X
∫
1/2
−1/2
Tβi (α, U )S(α, U, χj )T (−α)dα dU,
∫
∫
1 2X 1/2
Iβ1 β2 :=
Tβ1 (α, U )Tβ2 (α, U )T (−α)dα dU,
X X
−1/2
∫
∫
1 2X 1/2
R(α, U, χ1 )R(α, U, χ2 )T (−α)dα dU.
IR :=
X X
−1/2
3
(9)
(10)
(11)
The estimate of Bhowmik and Schlage-Puchta
We first estimate the integral (11). This kind of estimate without characters was
obtained by Bhowmik and Schlage-Puchta [1] and improved by Languasco and
4
Zaccagnini [9]. We have to generalize their result to the case of the integral with
Dirichlet characters. Originally, the improvement of Languasco and Zaccagnini
[9] was based on the circle method with power series, which is also applicable
to our integral with characters. However, as we mentioned in Section 1, our
exposition follows an alternative method of Goldston and Yang [7], which uses
only finite trigonometric polynomials instead of power series.
By the Cauchy-Schwarz inequality, we have
IR ≪ J(χ1 )1/2 J(χ2 )1/2 ,
where
∫
1/2
J(χ) =
−1/2
and
E(α, χ) :=
(12)
1
X
∫
E(α, χ) |T (α)| dα
2X
(13)
2
|R(α, U, χ)| dU.
(14)
X
Note that T (α) ≪ min (X, 1/|α|) for |α| ≤ 1/2. Hence we obtain
∫
∫
dα
J(χ) ≪ X
E(α, χ)dα +
E(α, χ) .
|α|
|α|≤1/X
1/X<|α|≤1/2
The latter integral is
∫
=
(
1/2
dξ
ξ2
E(α, χ) 2 +
1/X<|α|≤1/2
∫
≪
∫
(∫
∫
|α|
E(α, χ)dα +
|α|≤1/2
)
dα
)
E(α, χ)dα
|α|≤ξ
1/X<ξ≤1/2
Thus we have
1
J(χ) ≪ (log X)
sup
1/X≤ξ≤1/2 ξ
dξ
.
ξ2
∫
E(α, χ)dα.
As a result our estimate is reduced to the estimate of the integral
∫
∫
∫
1 2X
|R(α, U, χ)|2 dα dU.
E(α, χ)dα =
X
|α|≤ξ
X
|α|≤ξ
By Gallagher’s lemma [6, Lemma 1], we have
2
∫ ∞
∫
∑♯
2
2
|R(α, U, χ)| dα ≪ ξ
χ(n)Λ(n) dx,
−∞ |α|≤ξ
x<n≤x+(2ξ)−1
n≤U
where
∑♯
n
χ(n)Λ(n) :=
∑
χ(n)Λ(n) −
n
∑
β∈Z0 (χ)
5
(15)
|α|≤ξ
A(β)
∑
n
nβ−1 .
(16)
(17)
Now we separate some irregular parts of the integral on the right-hand of (17):
2
∫ ∞
∑♯
dx
χ(n)Λ(n)
(18)
−∞ −1
x<n≤x+(2ξ)
n≤U
≪ I((2ξ)
where
∫
h
I(h) :=
0
−1
) + J(U, (2ξ)−1 ) + K(U, (2ξ)−1 ),
2
∑
♯
J(U, h) :=
χ(n)Λ(n) dx,
0 x<n≤x+h
2
∫ U ∑
♯
dx,
χ(n)Λ(n)
(19)
K(U, h) :=
U −h U <n≤x+h
2
∑
♯
χ(n)Λ(n) dx,
n≤x
∫
U
and we assume 1 ≤ h ≤ X/2 in what follows. We estimate these integrals. First
we recall the follwoing explicit formula which can be derived from Theorem 12.5
and Theorem 12.10 of [10]:
ψ(x, χ) =
∑
ρ∈Z(χ)
|γ|≤T
A(ρ)
xρ
L′
+ (1, χ) + O (log qX) ,
ρ
L
(20)
where 2 ≤ x ≤ X, T = X(log qX) and χ is primitive. Moreover, we need the
following lemma:
Lemma 1. For any non-principal character χ (mod q), we have
L′
1
(1, χ) =
+ O(log q),
L
1 − β̃
∑ 1
∑ 1
=
≪ log q,
β
1−β
β̸=1−β̃
(21)
(22)
β̸=β̃
1
≪ q 1/2 (log q)2 ,
1 − β̃
(23)
where β runs thorough all real zeros of L(s, χ) counted with multiplicity and β̃
is the possible exceptional zero for χ (mod q).
Proof. For the assertion (21), see Theorem 11.4 [10]. Note that
∑ 1
∑ ( 1 )
≤
ℜ
.
1−β
1−ρ
β̸=β̃
ρ̸=β̃
Then the estimate (22) can be obtained via the proof of (11.13) of [10]. The last
estimate (23) is the famous estimate of Page, see Corollary 11.12 of [10].
6
The estimate for the integral I(h) is essentially the famous result of Cramér.
For the original result of Cramér, see [2] or [10, Theorem 13.5].
Lemma 2. Assume GRHR. Then we have I(h) ≪ h2 (log 2q)2 .
Proof. (Sketch) Combining (20) with Lemma 1, we have
(
)
∑ xρ
∑♯
χ(n)Λ(n) =
+ O h1/2 log 2q ,
ρ
n≤x
0<|γ|≤T
where 2 ≤ x ≤ h and T = h log qh. Hence it is sufficient to estimate the integral
2
∫ h ∑
ρ
x
dx.
2 0<|γ|≤T ρ By expanding square and using GRHR, we have this integral is
≪
∑
∑
0<|γ1 |≤T 0<|γ2 |≤T
≪ h2 (log 2q)
h2
1
·
(1 + |γ1 |)(1 + |γ2 |) 1 + |γ1 − γ2 |
∑
0<|γ1 |≤T
(log(1 + |γ1 |))2
≪ h2 (log 2q)2 .
1 + |γ1 |2
This proves the lemma.
The integral K(U, h), although it seems to be not so important at first sight,
is quite annoying part to estimate, which was a main obstacle to reduce the
logarithms from the final error term as Languasco and Zaccagnini [9] did. This
is exactly the place where we launch the weapon of Goldston and Yang [7].
Lemma 3. Assume GRHR. Then we have
∫
1 2X
K(U, h)dU ≪ hX(log q)2 .
X X
Proof. The left-hand side of the assertion is
2
∫
∫
1 2X U +h ∑♯
=
χ(n)Λ(n) dx dU
X X
U
U <n≤x
2
2
∫
∫ 3X ∑
♯
h
h 2X ∑♯
dU +
dx.
χ(n)Λ(n)
χ(n)Λ(n)
≪
X X X X n≤U
n≤x
Therefore the lemma immediately follows from Lemma 2
The estimate of the “main-part” integral J(U, h) is classical. Actually, the
special case without Dirichlet characters was already obtained by Selberg [15].
The estimate necessary for Theorem 1 is obtained via the techniques of Saffari
and Vaughan [14]. See also Prachar [11] or Yuan and Zun [16].
7
Lemma 4. Assume GRHR. Then we have J(U, h) ≪ hU (log qU )2 .
Proof. (Sketch) We first decompose the integral as
∫ h ∫ U
J(U, h) =
+
= J1 + J2 , say.
0
h
The first integral J1 is estimated by Lemma 2 as
J1 ≪ h2 (log 2q)2 ≪ hU (log qU )2 ,
which is admissible. The integral J2 can be estimated by the techniques of
Saffari and Vaughan [14] as
J2 ≪ hU (log qU )2 .
For the details, see Lemma 5 and Lemma 6 of [14]. This completes the proof.
Applying (18), Lemma 2 and Lemma 4 to (17), we have
∫
|R(α, U, χ)|2 dα ≪ ξX(log qX)2 + ξ 2 K(U, (2ξ)−1 ).
|α|≤ξ
Hence by (15), (16) and Lemma 3, we have
J(χ) ≪ X(log X)(log qX)2 .
Substituting this estimate into (12), we arrive at
IR ≪ E(X).
(24)
This completes the estimate of IR .
4
Detection of main and oscillating terms
We next calculate the integral Iβi Sj defined by (9). Clearly,
∫ 1/2
∑
Iβi Sj =
Tβi (α, X)S(α, X, χj )T (−α)dα =
χj (m)Λ(m)nβi −1
−1/2
m+n≤X
We now calculate this last sum explicitly. The result is
Lemma 5. Assume GRHR. For any real numbers X ≥ 2 and 1/2 < µ ≤ 1, we
have
∑
∑
χ(m)Λ(m)nµ−1 =
A(ρ)W0 (ρ, µ) + O(XQ(χ)(log 2q)(log X)),
m+n≤X
where
Q(χ) :=
ρ∈Z(χ)
β+µ>1
{
q 1/2 (log 2q) (when χ (mod q) is an exceptional character),
1
(otherwise).
8
Proof. Recalling 1/2 < µ ≤ 1, we have
∑
∑
χ(m)Λ(m)nµ−1 =
χ(m)Λ(m)
m+n≤X
m≤X
=
∑
nµ−1
n≤X−m
∫
X−m
uµ−1 du + O(X)
χ(m)Λ(m)
0
m≤X
∫
∑
X
ψ(u, χ)(X − u)µ−1 du + O(X).
=
0
Now we use the explicit formula (20). We can remove the restriction that
χ (mod q) is primitive with the error of the size (log 2q)(log X). By (20), we
have
∑
∑ A(ρ) ∫ X
X µ L′
ψµ (m, χ) =
uρ (X − u)µ−1 du +
· (1, χ) + E1 (X)
ρ
µ
L
0
m≤X
ρ∈Z(χ)
|γ|≤T
∑
=
A(ρ)W0 (ρ, µ) +
ρ∈Z(χ)
|γ|≤T
X µ L′
· (1, χ) + E1 (X),
µ
L
where E1 (X) ≪ X(log 2q)(log X). Now we remove the restriction |γ| ≤ T from
the above sum over Z(χ). Note that Stirling’s formula implies
(
)
Γ(ρ)Γ(µ) X ρ+µ
X β+µ
1
W0 (ρ, µ) =
≪
1+
,
(25)
Γ(ρ + µ) ρ + µ
(1 + |γ|)1+µ
|ρ|
where we have to care about the pole ρ = 0 of the factor Γ(ρ). Then we can
estimate the extended part of the sum by
∑
≪ X µ+1
ρ
|γ|>T
1
X µ+1
≪
(log qT ) ≪ X(log 2q)(log X).
|γ|1+µ
Tµ
Hence we obtain
∑
∑
X µ L′
ψµ (m, χ) =
A(ρ)W0 (ρ, µ) +
· (1, χ) + E1 (X).
µ
L
m≤X
(26)
ρ∈Z(χ)
By (21) and (23), we have
X µ L′
· (1, χ) ≪ X µ Q(χ)(log 2q) ≪ XQ(χ)(log 2q).
µ
L
By using (22), (23), (25) and GRHR, we have


∑
∑1 ∑
1

A(ρ)W0 (ρ, µ) ≪ X 
+
3/2
β
(1
+
|γ|)
ρ
β
ρ∈Z(χ)
β+µ≤1
≪ XQ(χ)(log 2q).
9
(27)
(28)
Substituting (27) and (28) into (26), we obtain the lemma.
Finally, the integral Iβ1 β2 (10) is obviously
∫
Iβ1 β2 =
=
1/2
−1/2
Tβ1 (α, X)Tβ2 (α, X)T (−α)dα
∑
mβ1 −1 nβ2 −1 = W0 (β1 , β2 ) + O(X).
(29)
m+n≤X
5
Completion of the proof
By Lemma 5 and (29), we have
∑
∑
A(β1 )Iβ1 S2 +
β1 ∈Z0 (χ1 )
β2 ∈Z0 (χ2 )
=
∑∑
∑∑
A(β1 )A(β2 )Iβ1 β2
β1 ∈Z0 (χ1 )
β2 ∈Z0 (χ2 )
W (β1 , ρ2 ) +
β1 ∈Z0 (χ1 )
ρ2 ∈Z(χ2 )
β1 +β2 >1
−
∑∑
A(β2 )Iβ2 S1 −
∑∑
W (ρ1 , β2 )
ρ1 ∈Z(χ1 )
β2 ∈Z0 (χ2 )
β1 +β2 >1
W (β1 , β2 ) + (Q(χ1 ) + Q(χ2 ))E(X).
β1 ∈Z0 (χ1 )
β2 ∈Z0 (χ2 )
Since the condition β1 + β2 > 1 implies that β1 > 1/2 or β2 > 1/2, we find that
{ (ρ1 , ρ2 ) ∈ Z(χ1 ) × Z(χ2 ) | β1 + β2 > 1 }
⊂ Z0 (χ1 ) × Z(χ2 ) ∪ Z(χ1 ) × Z0 (χ2 )
and that
Z0 (χ1 ) × Z(χ2 ) ∩ Z(χ1 ) × Z0 (χ2 ) = Z0 (χ1 ) × Z0 (χ2 ).
∑∑
Therefore
+
β1 ∈Z0 (χ1 )
ρ2 ∈Z(χ2 )
β1 +β2 >1
=
ρ1 ∈Z(χ1 )
β2 ∈Z0 (χ2 )
β1 +β2 >1
Thus we have
∑
A(β1 )Iβ1 S2 +
β1 ∈Z0 (χ1 )
∑∑
∑
ρ1 ∈Z(χ1 )
ρ2 ∈Z(χ2 )
β1 +β2 >1
A(β2 )Iβ2 S1 −
β2 ∈Z0 (χ2 )
=
∑∑
∑∑
+
∑∑
.
β1 ∈Z0 (χ1 )
β2 ∈Z0 (χ2 )
∑∑
A(β1 )A(β2 )Iβ1 β2
β1 ∈Z0 (χ1 )
β2 ∈Z0 (χ2 )
W (ρ1 , ρ2 ) + (Q(χ1 ) + Q(χ2 ))E(X).
ρ1 ∈Z(χ1 )
ρ2 ∈Z(χ2 )
β1 +β2 >1
10
Substituting this equation and (24) into the result of Section 2, we obtain
∑
∑∑
R(n, χ1 , χ2 ) =
W (ρ1 , ρ2 ) + (Q(χ1 ) + Q(χ2 ))E(X).
ρ1 ∈Z(χ1 )
ρ2 ∈Z(χ2 )
β1 +β2 >1
n≤X
By the orthogonality of Dirichlet characters, we have
∑
R(n, q, a) =
n≤X
∑ ∑ χ1 (a1 )χ2 (a2 ) ∑ ∑
W (ρ1 , ρ2 ) + Q · E(X), (30)
φ(q1 )φ(q2 )
ρ1 ∈Z(χ1 )
ρ2 ∈Z(χ2 )
β1 +β2 >1
χ1 (mod q1 )
χ2 (mod q2 )
where
Q :=
∑ ∑ Q(χ1 ) + Q(χ2 )
.
φ(q1 )φ(q2 )
χ1 (mod q1 )
χ2 (mod q2 )
This Q can be estimated as
Q≪1
(31)
by recalling Landau’s theorem on the exceptional characters. See Corollary 11.8
of [10]. Substituting (31) into (30), we arrive at Theorem 1.
Acknowledgments
First, the author would like to express his gratitude to Prof. Yuichi Kamiya and
Prof. Hideaki Ishikawa for giving the opportunity to give a talk. Second, the
author also would like to express his gratitude to Prof. Kohji Matsumoto for
his suggestion of this problem, advice and encouragement. Finally, the author
would like to express his gratitude to Prof. Alessandro Languasco for pointing
out some inaccuracies in my original preprint.
References
[1] G. Bhowmik and J-C. Schlage-Puchta, Mean representation number of integers
as the sum of primes, Nagoya Math. Journal 200 (2010) 27–33.
[2] H. Cramér, Some theorems concerning prime numbers, Arkiv för Mat. Astr. Fys.
15 (5), 33 pp; Collected works, Vol. 1, (Springer-Verlag, 1994), pp. 138–170.
[3] S. Egami and K. Matsumoto, Convolutions of the von Mangoldt function and
related Dirichlet series, Proceedings of the 4th China-Japan Seminar held at
Shangdong, ed. S. Kanemitsu and J.-Y. Liu, (World Sci. Publ., 2007), pp. 1–23.
[4] A. Fujii, An additive problem of prime numbers, Acta Arith. 58 (1991) 173–179.
[5] A. Fujii, An additive problem of prime numbers. II, Proc. Japan Acad. Ser. A
Math. Sci. 67 (1991) 248–252.
[6] P. X. Gallagher, A large sieve density estimate near σ = 1, Invent. Math. 11
(1970) 329–339.
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[7] D. A. Goldston and Liyang Yang, The average number of Goldbach representations, preprint (2016), arXiv:1601.06902.
[8] A. Languasco and A. Perelli, On Linnik’s theorem on Goldbach numbers in short
intervals and related problems, Ann. Inst. Fourier 44 (2) (1994) 307–322.
[9] A. Languasco and A. Zaccagnini, The number of Goldbach representations of
an integer, Proc. Amer. Math. Soc. 140 (2012) 795–804.
[10] H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I. Classical
Theory, (Cambridge University Press, 2007).
[11] K. Prachar, Generalization of a theorem of A. Selberg on primes in short intervals, Topics in Number Theory, Colloq. Math. Soc. János Bolyai 13 (1974), pp.
267–280.
[12] F. Rüppel, Convolution of the von Mangoldt function over residue classes, Diplomarbeit, Univ. Würzburg, (2009).
[13] F. Rüppel, Convolution of the von Mangoldt function over residue classes,
Šiauliai Math. Semin. 7 (15) (2012) 135–156.
[14] B. Saffari and R. C. Vaughan, On the fractional parts of x/n and related sequences II, Ann. Inst. Fourier 27 (2) (1997) 1–30.
[15] A. Selberg, On the normal density of primes in small intervals, and the difference
between consecutive primes, Arch. Math. Naturvid. 47 (1943) 87–105; Collected
Papers, Vol. 1, (Springer Verlag, 1989), pp. 160–178.
[16] Wang Yuan and Shan Zun, A conditional result on Goldbach problem, Acta
Math. Sinica, New Series 1 (1) (1985) 72–78.
[17] Y. Suzuki, A mean value of the representation function for the sum of two primes
in arithmetic progressions, preprint (2015), arXiv:1504.01967.
Yuta Suzuki
Graduate School of Mathematics, Nagoya University
Chikusa-ku, Nagoya 464-8602, Japan
e-mail: [email protected]
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