Publications Mathématiques de Besançon

David P. Roberts

A three-parameter clan of Hurwitz–Belyi maps

Alisa Sedunova

Publications mathématiques de Besançon
A l g è b r e  e t  t h é o r i e  d e s  n o m b r e s
Alisa Sedunova
A partial Bombieri–Vinogradov theorem with explicit constants
2018, p. 101-110.
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Publications mathématiques de Besançon – 2018, 101-110
A PARTIAL BOMBIERI–VINOGRADOV THEOREM WITH
EXPLICIT CONSTANTS
by
Alisa Sedunova
Abstract. — In this paper we improve the result of [1] with getting (log x) 72 instead of (log x) 92 .
In particular we obtain a better version of Vaughan’s inequality by applying the explicit variant
of an inequality connected to the Möbius function from [5].
Résumé. — (Aspects explicites d’un théorème de Bombieri–Vinogradov) Dans cet article, nous
améliorons un résultat de [1] en remplaçant le (log x) 72 par un (log x) 92 . En particulier, nous
obtenons une version améliorée de l’inégalité de Vaughan en appliquant une version explicite
d’une inégalité dans [5] liée à la fonction de Möbius.
1. Introduction
For integer number a and q ≥ 1, let
ψ(x; q, a) =
∑
n≤x
n≡a mod q
Λ(n),
where Λ(n) is the von Mangoldt function. The Bombieri–Vinogradov theorem is an estimate
for the error terms in the prime number theorem for arithmetic progressions averaged over
all q ≤ x1/2.
Theorem 1.1 (Bombieri–Vinogradov). — Let A be a given positive number and Q ≤
x1/2
(log x)B where B = B(A), then∑
q≤Q
max
2≤y≤x
max
a
(a,q)=1
∣∣∣∣ψ(y, q, a)− yϕ(q)
∣∣∣∣A x(log x)A .
The implied constant in this theorem is not effective, since we have to take care of characters,
associated with those q that have small prime factors. The main result of this paper is
2010 Mathematics Subject Classification. — 11N13, 11N05, 11N36.
Key words and phrases. — Primes in arithmetic progressions, Bombieri–Vinogradov theorem, large sieve.
102 A partial Bombieri–Vinogradov theorem with explicit constants
Theorem 1.2 (Bombieri–Vinogradov theorem with explicit constants). — Let x ≥
4, 1 ≤ Q1 ≤ Q ≤ x
1
2 . Let also l(q) denote the least prime divisor of q. Define F (x,Q,Q1) by
F (x,Q,Q1) =
14x
Q1
+ 4x
1
2Q+ 15x
2
3Q
1
2 + 4x
5
6 log Q
Q1
.
Then ∑
q≤Q
l(q)>Q1
max
2≤y≤x
max
a
(a,q)=1
∣∣∣∣ψ(y; q, a)− ψ(y)ϕ(q)
∣∣∣∣ < c1F (x,Q,Q1)(log x) 72 ,
where c1 = 42.2.
Previously the best result obtained by these methods in the literature is due to Akbary,
Hambrook (see [1, Theorem 1.3]), where they proved that under assumptions of Theorem 1.2
we have. ∑
q≤Q
l(q)>Q1
max
2≤y≤x
max
a
(a,q)=1
∣∣∣∣ψ(y; q, a)− ψ(y)ϕ(q)
∣∣∣∣ < c1F (x,Q,Q1)(log x) 92 ,
where F (x,Q,Q1) is defined by
F (x,Q,Q1) =
4x
Q1
+ 4x
1
2Q+ 18x
2
3Q
1
2 + 5x
5
6 log eQ
Q1
.
Here we reduce this power to (log x)
7
2 by applying an explicit version for an upper bound for
bk =
∑
d≤V
d|k
µ(d),
where µ(d) is Mobius function, V is a given number. This version can be found in [5, Chap-
ter 6], namely we have
Lemma 1.3 (Helfgott [5]). — There exists a constant L ≤ 0.441, such that for V large
enough we have ∑
k≤Y
|bk|2 = Y (L+O∗(C)) +O∗(V 2),
where C = 0.000023 and O∗(x) means that it is less in absolute value than x.
This Lemma is a variant of the sum considered in [3], where it is shown that∑
d1,d2≤Y
µ(d1)µ(d2)
gcd(d1, d2)
tends to a positive constant as Y → ∞. It is also suggested without proving that L can be
about 0.440729.
Notice, that by sharpening the inequality in Lemma 1.3 we will not be able to reduce the
power of log x, since the upper bound is optimal there, so by these methods the power 72 is the
best possible. Going further seems to be a hard problem which involves among simpler things
a very careful analysis of the logarithmic mean of Möbius function twisted by a Dirichlet
character.
Publications mathématiques de Besançon – 2018
A. Sedunova 103
Corollary 1.4. — Let Q = x
1
2
(log x)B , where B >
7
2 . Then Theorem 1.2 gives us the following
bound ∑
q≤Q
l(q)>Q1
max
2≤y≤x
max
a
(a,q)=1
∣∣∣∣ψ(y; q, a)− ψ(y)ϕ(q)
∣∣∣∣ < c1 (14xQ1 (log x) 72 + 19x(log x) 72−B
)
.
Remark. — It would be very good for applications to get (log x)2 in Theorem 1.2, however
it seems impossible to get by present methods.
Corollary 1.5. — Define
π(x) =
∑
p≤x
1 and π(x; q, a) =
∑
p≤x
p≡amod q
1.
Then Theorem 1.2 under the same assumptions can be also formulated for π(x) and π(x; q, a):∑
q≤Q
l(q)>Q1
max
2≤y≤x
max
a
(a,q)=1
∣∣∣∣π(y; q, a)− π(y)ϕ(q)
∣∣∣∣ < c2F (x,Q,Q1)(log x) 72 ,
where c2 = 1 + 2c1log 2 .
Proof of Corollary 1.5 is exactly the same as in [1], we just have to change the power of log.
The key tool for the proof of Theorem 1.2 is Vaughan’s identity, which we have to get in an
explicit version for our goal. Define
ψ(y, χ) =
∑
n≤y
Λ(n)χ(n),
the twisted summatory function for the von Mangoldt function Λ and a Dirichlet character
χ modulo q. One of two main results of this paper is
Proposition 1.6 (Vaughan’s inequality in an explicit form). — For x ≥ 4
∑
q≤Q
q
ϕ(q)
∗∑
χ(q)
max
y≤x
|ψ(y, χ)| < c0
(
7x+ 2Q2x
1
2 + 5Q
3
2x
2
3 + 4Qx
5
6
)
(log x)
5
2 ,
where Q is any positive real number and
∑∗
χ(q) means a sum over all primitive characters
χ(mod q).
The goal is to get an explicit version of f(x,Q) by applying an improved version of Pólya–
Vinogradov inequality (see [6]), that will reduce the coefficients of f(x,Q) and then we can
apply Lemma 1.3.
2. Proof of Proposition 1.6
Fix arbitrary real numbersQ > 0 and x ≥ 4. In this section, we shall establish Proposition 1.6,
which is the main ingredient in the proof of Theorem 1.2. Here we follow the ideas of [1] and
Publications mathématiques de Besançon – 2018
104 A partial Bombieri–Vinogradov theorem with explicit constants
applying the results from [5]. The main tool in the proof is the large sieve inequality (see, for
example [4, Lemma 2])
∑
q≤Q
q
ϕ(q)
∗∑
χ(q)
∣∣∣∣∣∣
m0+M∑
m=m0+1
amχ(m)
∣∣∣∣∣∣
2
≤ (M +Q2)
m0+M∑
m=m0+1
|am|2,(2.1)
from which it follows (see [2, Theorem 8.8.3] or [1, Lemma 6.1]) that
(2.2)
∑
q≤Q
q
ϕ(q)
∗∑
χ(q)
max
y
∣∣∣∣∣∣∣
M∑
m=m0
N∑
n=n0
mn≤y
ambnχ(mn)
∣∣∣∣∣∣∣
≤ c3(M ′ +Q2)
1
2 (N ′ +Q2)
1
2
(
M∑
m=m0
|am|2
) 1
2
(
N∑
n=n0
|bn|2
) 1
2
L(M,N),
where c3 = 2.65, L(M,N) = log(2MN) and M ′ = M −m0 + 1, N ′ = N − n0 + 1 are the
number of terms in the sums overm and n respectively. Here the am, bn are arbitrary complex
numbers.
2.1. Sieving and Vaughan’s identity. —We reduce to the case 2 ≤ Q ≤ x1/2. If Q < 1,
then the sum on the left-hand side of (1.6) is empty and we are done. Next, 1 ≤ Q < 2 then
only the q = 1 term exists and we have
(2.3)
∑
q≤Q
q
ϕ(q)
∗∑
χ(q)
max
y≤x
|ψ(y, χ)| = max
y≤x
∣∣∣∣∣∣
∑
n≤y
Λ(n)
∣∣∣∣∣∣ = ψ(x) ≤ A0x,
which is better than the theorem. As in [1] we use Vaughan’s identity
Λ(n) = λ1(n) + λ2(n) + λ3(n) + λ4(n),
where
λ1(n) =
{
Λ(n), if n ≤ U,
0, if n > U,
λ2(n) =
∑
hd=n
d≤V
µ(d) log h,
λ3(n) = −
∑
mdr=n
m≤U,d≤V
Λ(m)µ(d), λ4(n) = −
∑
mk=n
m>U,k>V
Λ(m)
∑
d|k
d≤V
µ(d).
Assume y ≤ x, q ≤ Q, and χ is a character mod q. We use the above decomposition to write
ψ(y, χ) = s1 + s2 + s3 + s4,
where
si =
∑
n≤y
λi(n)χ(n).
Let U , V be non-negative functions of x and Q to be set later and denote the contributions
to our main sum by
Si =
∑
q≤Q
q
ϕ(q)
∗∑
χ(q)
max
y≤x
|si|.
Publications mathématiques de Besançon – 2018
A. Sedunova 105
Easily we obtain ∑
q≤Q
q
ϕ(q)
∗∑
χ(q)
max
y≤x
|ψ(y, χ)| ≤ S1 + S2 + S3 + S4.
The heart of the proof of Theorem 1.3 in [1] are the following estimates:
Lemma 2.1 (Akbary, Hambrook [1], Section 7). — We have
S1 ≤ A0UQ2, S2 <
(
x+Q
5
2V
)
(log xV )2, S3 < S
′
3 + S
′′
3 ,
S
′
3 < (x+Q
5
2U)(log xU)2,
S
′′
3 <
c3
log 2
(
x+Qx
1
2U
1
2V
1
2 + 2
1
2QxU−
1
2 +Q2x
1
2
)
(log 2UV )2(log 4x),
S4 <
2
3
2A
1
2
0 c3
log 2 (x+QxV
− 12 + 2
1
2QxU−
1
2 +Q2x
1
2 )
(
log 2x
V
) 3
2
(log e3V )(log 4x).
where c3 = 2.65 and A0 = maxx>0 ψ(x)x =
ψ(113)
113 < 1.04.
Here we improve the estimate for S4 in Lemma 2.1 above.
Lemma 2.2. — Under previous notations we have
S4 ≤
c3
log 2(2A0)
1
2
(
x+ 2
1
2Q
1
2xU−
1
2 +QxV −
1
2 +Q2x
1
2
)
(log 4x)
(
log 2x
V
) 3
2
.
Proof. — We estimate S4 contribution with the use of Lemma 1.3. Writing s4 as a dyadic
sum we have
s4 = −
∑
M=2α
1
2U<M≤x/V
∑
U<m≤x/V
M<m≤2M
∑
V <k≤x/M
mk≤y
Λ(m)
∑
d|k
d≤V
µ(d)
χ(mk).
Using the triangle inequality
S4 ≤
∑
M=2α
1
2U<M≤x/V
∑
q≤Q
q
ϕ(q)
∗∑
χ(q)
max
y≤x
∣∣∣∣∣∣∣∣∣
∑
U<m≤x/V
M<m≤2M
∑
V <k≤x/M
mk≤y
ambkχ(mk)
∣∣∣∣∣∣∣∣∣ ,
where am = Λ(m), and, as it was defined in the introduction bk =
∑
d|k, d≤V µ(d). Now apply
the large sieve inequality (2.2) to get
S4 ≤ c3
∑
M=2α
1
2U<M≤x/V
(M ′ +Q2)
1
2 (K ′ +Q2)
1
2σ1(M)
1
2σ2(M)
1
2L(M)
where
σ1(M) =
∑
V <k≤x/M
|bk|2, σ2(M) =
∑
U<m≤x/V
M<m≤2M
|am|2,
and
L(M) = log
(2x
M
min
(
x
V
, 2M
))
≤ log 4x,
Publications mathématiques de Besançon – 2018
106 A partial Bombieri–Vinogradov theorem with explicit constants
where M ′ and K ′ denote the number of terms in the sums over m and k, respectively. From
the definition of M ′ and N ′ we conclude
M ′ = min
(
2M, x
V
)
−max (M + 1, U + 1) ≤M,
K ′ = x
M
− (V + 1) + 1 ≤ x
M
.
By Chebyshev estimate we have an upper bound
σ2(M) ≤
∑
m≤2M
Λ(m)2 ≤ ψ(2M) log 2M ≤ 2A0M log 2M.
Thus by Cauchy inequality
(2.4) S4 ≤ c3(log 4x)
∑
M=2α
1
2U<M≤x/V
(M +Q2)
1
2
(
x
M
+Q2
) 1
2
(2A0M log 2M)
1
2σ1(M)
1
2 .
Further
M(M +Q2)
(
x
M
+Q2
)
= Mx+Q2x+M2Q2 +MQ4
and
(log 2M)
1
2 ≤
(
log 2x
V
) 1
2
.
Using Lemma 1.3 we get
(σ1(M))
1
2 ≤ x
M
(L+ C)− V (L+ C) + 2V 2,
that implies
S4 ≤ c3(2A0)
1
2
(
x+ 2
1
2Q
1
2xU−
1
2 +QxV −
1
2 +Q2x
1
2
)
(log 4x)
(
log 2x
V
) 1
2 ∑
M=2α
1
2U<M≤x/V
1.
Since ∑
M=2α
1
2U<M≤x/V
1 ≤
log 2xV
log 2 ,
then
S4 ≤
c3
log 2(2A0)
1
2
(
x+ 2
1
2Q
1
2xU−
1
2 +QxV −
1
2 +Q2x
1
2
)
(log 4x)
(
log 2x
V
) 3
2
. 
Combining Lemma 2.2 with results of Lemma 2.1 we get
(2.5) S =
∑
q≤Q
q
ϕ(q)
∗∑
χ(q)
max
y≤x
|ψ(y, χ)| ≤ c4 Rx,Q,U,V Gx,V,U ,
Publications mathématiques de Besançon – 2018
A. Sedunova 107
where
c4 = max
{
A0,
c3
log 2 ,
c3
log 2(2A0)
1
2
}
= c3log 2(2A0)
1
2 ,
Rx,Q,U,V = 4x+Q2
(
2x
1
2 + U
)
+Q
5
2 (U + V ) + 2
1
2Q
1
2x
U
1
2
+ 2
1
2Qx
U
1
2
+Qx
1
2U
1
2V
1
2 + Qx
V
1
2
,
Gx,V,U = max
{
(log xV )2, (log xU)2, (log 2UV )2 log 4x,
(
log 2x
V
) 3
2
log 4x
}
,
Now let’s specify U and V . If x
1
3 ≤ Q ≤ x
1
2 , then U = V = x
2
3Q−1. Then putting that into
previous expression we get for the factor
Rx,Q,U,V = R1(x,Q) = 4x+ 2Q2x
1
2 +Qx
2
3
(
1 + 2
1
2
)
+Q
3
2x
2
3
(
2 + 2
1
2 + 1
)
+ x
7
6
≤ 4x+ 2Q2x
1
2 + 2Qx
5
6 +Q
3
2x
2
3
(
2 + 2
1
2 + 1
)
.
where we used the fact that x
7
6 ≤ Qx
5
6 and Qx
2
3 ≤ Qx
5
6 . Working in the same manner with
G and keeping in mind the condition x ≥ 4 we find that
Gx,V,U = G1(x, V, U) ≤
(4
3 log x
) 3
2
2 log x = 2
4
3
3
2
(log x)
5
2 .
If Q ≤ x
1
3 , we let U = V = x
1
3 and get
R2(x,Q) = 4x+ 2Q2x
1
2 +Q2x
1
3 + 2Q
5
2x
1
3 + 2
1
2Q
1
2x
5
6 +Qx
5
6 (2
1
2 + 2)
≤ x
(
5 + 2
1
2
)
+ 2Q2x
1
2 + 2Q
3
2x
2
3 +Qx
5
6
(
2
1
2 + 2
)
,
where we used Q2x
1
3 ≤ x, Q
5
2x
1
3 ≤ Q
3
2x
2
3 and Q
1
2x
5
6 ≤ x. Similarly we get for
G2(x, V, U) ≤ 2
(7
6
) 3
2
(log x)
5
2 .
Finally, we have in (2.5)
S ≤ c4
24
3
3
2
(
7x+ 2Q2x
1
2 + 5Q
3
2x
2
3 + 4Qx
5
6
)
(log x)
5
2 ,
as demanded.
3. Proof of Theorem 1.2
Let y ≥ 2,(a, q) = 1. By orthogonality of characters modulo q, we have
ψ(y; q, a) = 1
ϕ(q)
∑
χ
χ(a)ψ(y, χ).
Define ψ′(y, χ) = ψ(y, χ) if χ 6= χ0 and ψ′(y, χ) = ψ(y, χ)−ψ(y) otherwise, χ0 is the principal
character mod q. Then
ψ(y, q, a)− ψ(y)
ϕ(q) =
1
ϕ(q)
∑
χ
χ(a)ψ′(y, χ).
Publications mathématiques de Besançon – 2018
108 A partial Bombieri–Vinogradov theorem with explicit constants
For a character χ (mod q), we let χ∗ be the primitive character modulo q∗ inducing χ. Follow
the way of [1] we obtain
ψ′(y, χ∗)− ψ′(y, χ) = ψ(y, χ∗)− ψ(y, χ) =
∑
pk≤y
(log p)(χ∗(pk)− χ(pk)).
If p|q then (pk, q∗) = 1, and hence χ∗(pk) = χ(pk). If p|q then χ(pk) = 0. Therefore
|ψ′(y, χ∗)− ψ′(y, χ)| ≤
∑
pk≤y
p|q
(log p) ≤ (log y)
∑
p|q
1 ≤ (log qy)2.
Denote the quantity we want to estimate as
M =
∑
q≤Q
l(q)>Q1
max
2≤y≤x
max
a
(a,q)=1
∣∣∣∣ψ(y; q, a)− ψ(y)ϕ(q)
∣∣∣∣ .
Since ∣∣∣∣ψ(y, q, a)− ψ(y)ϕ(q)
∣∣∣∣ ≤ 1ϕ(q) ∑χ |ψ′(y, χ)| ≤ (log qy)2 +
1
ϕ(q)
∑
χ
|ψ′(y, χ∗)|,
then
M≤ Q(logQx)2 +
∑
q≤Q
l(q)>Q1
1
ϕ(q)
∑
χ
max
2≤y≤x
|ψ′(y, χ∗)|.
We have to take care just of the second term in the inequality above, since the first one is
smaller than the desired bound. It remains to prove
N =
∑
q≤Q
l(q)>Q1
1
ϕ(q)
∑
χ
max
2≤y≤x
|ψ′(y, χ∗)| ≤ (c1 − 1)F (x,Q,Q1)(log x)4,
where F (x,Q,Q1) is the function from Theorem 1.2. A primitive character χ∗mod q∗ induces
characters of moduli dq∗ and ψ′(y, χ∗) = 0 for χ principal, we observe
N =
∑
q≤Q
l(q)>Q1
1
ϕ(q)
∑
q∗|q
q∗6=1
∗∑
χ(q∗)
max
2≤y≤x
|ψ′(y, χ)| ≤
∑
q∗≤Q
l(q∗)>Q1
∗∑
χ(q∗)
max
2≤y≤x
|ψ′(y, χ)|
∑
k≤ Q
q∗
1
ϕ(kq∗) .
As it was noted in [1] for x > 0 ∑
k≤x
1
ϕ(k) ≤ E0 log(ex)
and as q∗ ≤ Q ≤ x1/2, ϕ(k)ϕ(q∗) ≤ ϕ(kq∗) and x ≥ 4, we have∑
k≤ Q
q∗
1
ϕ(kq∗) <
5E0
4ϕ(q∗) log x.
For q > 1 and χ primitive character (mod q), we know that χ is non-principal and ψ(y, χ) =
ψ′(y, χ). Since we assumed Q1 ≥ 1 then we can replace ψ′(y, χ) by ψ(y, χ) inside the internal
Publications mathématiques de Besançon – 2018
A. Sedunova 109
sum for N . Combining it with an expression for N we get
N ≤ 5E04 (log x)
∑
q≤Q
l(q)>Q1
1
ϕ(q)
∗∑
χ(q)
max
2≤y≤x
|ψ(y, χ)| = R.
Thus it remains to show that
R ≤ 4(c1 − 1)5E0
F (x,Q,Q1)(log x)
5
2 .
Let
R(q) = q
ϕ(q)
∗∑
χ(q)
max
2≤y≤x
|ψ(y, χ)|.
Partial summation gives us
∑
Q1<q≤Q
1
ϕ(q)
∗∑
χ(q)
max
2≤y≤x
|ψ(y, χ)| = 1
Q
∑
q≤Q
R(q)− 1
Q1
∑
q≤Q1
R(q) +
∫ Q
Q1
∑
q≤t
R(q)
 dt
t
.
Now we apply Theorem 1.6 ∑
q≤Q
R(q) < c0f(x,Q)(log x)
5
2 ,
where f(x,Q) = 7x+ 2Q2x
1
2 + 5Q
3
2x
2
3 + 4Qx
5
6 . Then∑
Q1<q≤Q
1
ϕ(q)
∗∑
χ(q)
max
2≤y≤x
|ψ(y, χ)| < c0
(
∆f (Q,Q1) +
∫ Q
Q1
f(x, t)dt
t
)
(log x)
5
2 ,
where
∆f (Q,Q1) =
f(x,Q)
Q
− f(x,Q1)
Q1
≤ 7x
Q1
+ 2x
1
2Q+ 5x
2
3Q
1
2 .
Calculating the integrals gives us∫ Q
Q1
f(x, t)dt
t
<
7x
Q1
+ 2x
1
2Q+ 10x
2
3Q
1
2 + 4x
5
6 log Q
Q1
.
Finally
N ≤ 4(c1 − 1)5E0
(14x
Q1
+ 4x
1
2Q+ 15x
2
3Q
1
2 + 4x
5
6 log Q
Q1
)
(log x)
5
2 .
3.1. Proof of Corollary 1.5. —Define two functions
π1(y) =
∑
2≤n≤y
Λ(n)
logn and π1(y; q, a) =
∑
2≤n≤y
n≡a(mod q)
Λ(n)
logn.
Since
π1(y; q, a)− π(y; q, a) =
∑
2≤k≤ log ylog 2
∑
pk≤y
pk≡a(mod q)
1
k
≤
∑
2≤k≤ log ylog 2
π(y
1
2 )
2 < 2y
1
2 ,
where we used the fact that for x > 1(see for example [1, Lemma 3.1])
π(x) < 1.25506 xlog x.
Publications mathématiques de Besançon – 2018
110 A partial Bombieri–Vinogradov theorem with explicit constants
Similarly, π1(y)− π(y) < 2y
1
2 . Thus by partial summation we obtain the bound∣∣∣∣π1(y; q, a)− π1(y)ϕ(q)
∣∣∣∣ = ∣∣∣∣ψ(y; q, a)− ψ(y)/ϕ(q)log y −
∫ y
2
ψ(t; q, a)− ψ(t)/ϕ(q)
t log2 t
dt
∣∣∣∣
≤ 1log 2
∣∣∣∣ψ(y; q, a)− ψ(y)ϕ(q)
∣∣∣∣+ max2≤t≤y
∣∣∣∣ψ(t; q, a)− ψ(t)ϕ(q)
∣∣∣∣ ( 1log 2 − 1log y
)
.
We have∑
q≤Q
l(q)>Q1
max
2≤y≤x
max
a
(a,q)=1
∣∣∣∣π(y; q, a)− π(y)ϕ(q)
∣∣∣∣
≤ 2log 2
∑
q≤Q
l(q)>Q1
max
2≤y≤x
max
a,(a,q)=1
∣∣∣∣ψ(y, q, a)− ψ(y)ϕ(q)
∣∣∣∣+ 2x 12 ∑
q≤Q
l(q)>Q1
(
1 + 1
ϕ(q)
)
<
2c1
log 2F (x,Q,Q1)(log x)
7
2 + 2x
1
2
∑
q≤Q
l(q)>Q1
(
1 + 1
ϕ(q)
)
,
where we used Theorem 1.2 to estimate the first summand. For x ≥ 4
2x
1
2
∑
q≤Q
l(q)>Q1
(
1 + 1
ϕ(q)
)
<
2c1
log 2F (x,Q,Q1)(log x)
7
2 .
and we are done.
References
[1] A. Akbary & K. Hambrook, “A variant of the Bombieri-Vinogradov theorem with explicit
constants and applications”, Math. Comput. 84 (2015), no. 294, p. 1901-1932.
[2] A. C. Cojocaru & M. R. Murty, An introduction to sieve methods and their applications, Lon-
don Mathematical Society Student Texts, vol. 66, Cambridge University Press, 2006, xii+224 pages.
[3] F. Dress, H. Iwaniec & G. Tenenbaum, “Sur une somme liée à la fonction de Möbius”, J.
Reine Angew. Math. 340 (1983), p. 53-58.
[4] P. X. Gallagher, “A large sieve density estimate near σ = 1”, Invent. Math. 11 (1970), p. 329-
339.
[5] H. A. Helfgott, “The ternary Goldbach problem”, https://arxiv.org/abs/1501.05438, 2015.
[6] C. Pomerance, “Remarks on the Pólya-Vinogradov inequality”, Integers 11 (2011), no. 4, p. 531-
542.
Alisa Sedunova, Mathematisches Institut, Bunsenstraße 3-5, Göttingen, Germany
E-mail : alisa.sedunova@phystech.edu
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