Advances in Pure Mathematics, 2013, 3, 25-32 http://dx.doi.org/10.4236/apm.2013.37A003 Published Online October 2013 (http://www.scirp.org/journal/apm)
Primes in Arithmetic Progressions to Moduli with a Large Power Factor Ruting Guo Network Center, Shandong University, Jinan, China Email:
[email protected] Received July 8, 2013; revised September 9, 2013; accepted October 6, 2013 Copyright © 2013 Ruting Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT Recently Elliott studied the distribution of primes in arithmetic progressions whose moduli can be divisible by highpowers of a given integer and showed that for integer a 2 and real number A 0 . There is a B B A 0 such that max max y; qd , r
1 d x 2 q 1 L B
y x
r , qd 1
Li y
qd
x
q LA
,
d , q 1
1 3
holds uniformly for moduli q x exp log log x
3
that are powers of a . In this paper we are able to improve his
result. Keywords: Primes; Arithmetic Progressions; Riemann Hypothesis
1. Introduction and Main Results Let p denote a prime number. For integer a, q with a, q 1 , we introduce
x; q, a
1
p x p a mod q
to count the number of primes in the arithmetic progression a mod q not exceeding x . For fixed q , we have 1 x; q, a x q
Therefore they try to find a satisfactory substitute. In this direction the famous Bombieri-Vinogradov theorem [2, 3], states that Theorem A. For any A 0 there exists a constant B B A 0 such that Li y
max y; q, a max y x a , q 1 q
xL A ,
q Q
1
where q is the Euler totient function, Q x 2 L B , du . log u Recently in order to study the arithmetic functions on shifted primes, Elliott [4] studied the distribution of primes in arithmetic progressions whose moduli can be divisible by high-powers of a given integer. More precisely, he showed that Theorem B. Let a be an integer, a 2 . If A 0 , then there is a B B A 0 such that
and Li y
y
2
as x tends to infinity. However the most important thing in this context is the range uniformity for the moduli q in terms of x . The Siegel-Walfisz Theorem, see for example [1], shows that this estimate is true only if q LA , where and throughout this paper we denote log x by L . The Generalized Riemann Hypothesis for Dirichlet L-functions could give a 1much better result: non-trivial estimate holds for q x 2 L2 . Unfortunately the Generalized Riemann Hypothesis has withstood the attack of several generations of researchers and it is still out of reach. However number theorists still want to live a better life without the Generalized Riemann Hypothesis. Copyright © 2013 SciRes.
1
d x 2 q 1 L B d , q 1
max max y; qd , r y x
r , qd 1
Li y
qd
x , q LA
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R. T. GUO
1
holds uniformly for moduli q x 3 exp log log x that are powers of a . When q 1 , his result recovers the BombieriVinogradov theorem. And obviously his result gives a deep insight into the distribution of primes in arithmetic progressions. The most important thing Elliott concerned in [4] is that in Theorem B the parameter q may reach a fixed power of x . However we want to purse the widest uniformity in q by using some new techniques established in the study of Waring-Goldbach problems. We shall prove the following result. Theorem 1.1. Let a be an integer, a 2 . If A 0 , then there is a B B A 0 such that
1 d x 2 q 1 L B
max max y; qd , r y x
r , qd 1
Li y
qd
d , q 1
3
2
x
q LA
Li x
q
2 5
3
, 3
This result improves a former result given by Barban, Linnik and Tshudakov [5], Pmin q, r q8 3
where q p n , n 1, 2,3, . If we focus our attention on the least prime in arithmetic progressions with special moduli, we can prove the following result. Theorem 1.2. Let a be an integer, a 2 . If A 0 , then there is a B B A 0 such that
Li y
holds uniformly for moduli q x12 exp log log x that are powers of a . Then our result shows that the least prime Pmin q, r in these special progressions n r mod q satisfies Pmin q, r q12 5 . Copyright © 2013 SciRes.
For 2 w x 3 4 and an integer q 1 , define max max y; qd , r
d w
r , qd 1
y x
1 y; q, r . d
Then Lemma 2.1. For any K 0,1 4 1 2 , we have 3 1 G x q 1 L K G exp log log x log x 2
q q x log x 1
3
q x exp log log x , x 3 3
6 K
(1)
.
where
2 5 , if
9 20 1 2 and 5 12 , if 1 4 9 20 . Here
q n|q1 . For Dirichlet characters and real y 0 define
y, n n .
(2)
n y
Lemma 2.2. Let y, defined as in (2). Then
d Q mod Dd
max y, x x1 2 Q 2 D x 4 5QD1 2 Lc , y x
(3) holds uniformly for all integers D 1 and real numbers x 2, Q 1. Lemma 2.3. Let y, defined as in (2). Then
d Q mod Dd
x max max y; qd , r , y x r , qd 1 qd q LA 5
n y n r mod w
uniformly for positive integers
Pmin q, r q 5 2 .
9
Let n denote von Mangoldt’s function, and for mutually prime integers w and r , let y; w, r n .
d , q 1
holds uniformly for moduli q x exp log log x . Then the special case of our result shows that the least prime Pmin q, r in these special progressions n r mod q satisfies
d x 20 q 1 L B d , q 1
2. Preliminary Reduction
G w
,
holds uniformly for moduli q x 5 exp log log x that are powers of a . When d 1 and a an odd prime, our result gives that for these particular moduli q with the form q p n , n 1, 2,3,
x; q, r 1 O L A
It should be remarked that the Generalized Riemann Hypothesis for Dirichlet L-functions would allow 1 qd x 2 L A1 with no further restriction upon the nature of q . Therefore our Theorems 1.1 and 1.2 can be compared with the result under the Generalized Riemann Hypothesis.
max y, x x11 20 Q 2 D Lc . y x
(4)
holds uniformly for all integers D 1 and real numbers x 2, Q 1. Here the inner sum is taken over all primitive Dirchlet characters mod Dd .
3. Proof of Lemma 2.2 Let 2
X5 Y X APM
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R. T. GUO
S :
and M 1 , , M 10 be positive real numbers such that 1 5
Y M 1 M 10 X and 2M 6 , , 2 M 10 X .
(5)
For j 1, ,10 define log m, a j m 1, m ,
if j 1, if j 2, ,5,
(6)
if j 6, ,10,
where n is the Möbius function. Then we define the functions f j s,
a j m m
m M j
ms
a1 m1 m1 a10 m10 m10 ,
m1 M1 ,, m10 M10 y 2 m1m10 y
where denotes the vector M 1 , M 2 , , M 10 with M j as in (5). Obviously some of the intervals M j , 2M j may contain only integer 1. By using Perron’s summation formula with T y (see Proposition 5.5 in [1]), and then shifting the contour to the left, we have ys y 2 1 11 L iy , ds O L2 F s s 2 i 11 L iy 1 2 iy 1 2 iy 11 L iy 1 1 2 iy 1 2 iy O L2 . 1 1 L iy 2 i s
S
,
On using the trivial estimate
and F s, f1 s, f10 s, ,
where is a Dirchlet character, s a complex variable. Lemma 3.1. Let F s, be as in (7), and A 1 arbitrary. Then for any 1 R X 2 A and 0 T X A , T
r R mod r T d |r
1 F it , dt 2
3 1 1 R2 R T 1 2 T 2 X 10 X 2 log c X , d d
2 5
Y x , X x.
Define a j m , f j s, and F s, as above. To go further, we first recall Heath-Brown’s identity [8], which states that for any n 2 z k with z 1 and k 1, k
j 1
j 1
k log n1 n j 1 n2 j . j n1n2 n2 j n n j 1n2 j z
max
1 2 11 L
2Y 2 x y X x,
y, is a linear combination of O L of which is of the form
10
terms, each
y x 2 y 2 L x 10 L. y 1
x1 L
3
1
Then we have
S 1
1 2
1 y2 yF 2 it ,
it
y
1
y
y 2
y
1
y 22 1 it 2
it
3 dt O x10 L
3
1 dt F it , x10 L. 2 t 1
Noting that F s, does not depend on y , we have 3
x 1 dt x 10 L11 . max y , L10 x 2 x F it , 2Y < y x 2 t 1 (9) 1
On the other hand we have max y, Y .
(10)
y 2Y
From (9) and (10), we have
2 5
the integral on the two horizontal segments above can be estimated as y max F iy, 1 2 11 L y
d Q mod Dd
Then for
Copyright © 2013 SciRes.
1 M 11 L M 21 M 10 x1 L,
(8)
where c 0 is an absolute constant independent of A , but the constant implied in depends on A. Proof of Lemma 3.1. This lemma with d 1 was established in [6], and in this general form [7]. We mention that in general the exponent 3/10 to X in the second term on the right-hand side is the best possible on considering the lack of sixth power mean value of Dirchlet L-functions. Now we complete the proof of Lemma 2.2. Proof of Lemma 2.2. In (5), we take
n 1
F iy, f1 iy, f10 iy,
(7)
max y, y x
d Q mod Dd
L10 x
1 2
max y ,
2Y < y x
d Q mod Dd
d Q mod Dd
x
x
max y , y 2Y
1 dt Q 2 Dx 5 . F it , t 2 1 2
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R. T. GUO
Further let q Dd and then we obtain
d Q mod Dd
y s y 2 1 b iy F s , ds O xy 1 L2 y, 2 i b iy s s
max y, y x
0 T x 1 R QD
2T 1 1 F it , dt T T 1 q R mod q 2 D| q
2 5
1 y s y 2 1 b iy F s, ds O x 2 L2 , 2 i b iy s
where 0 b L1 . If we let b 0, we have
Q 2 Dx .
y, yF it , y
From Lemma 3.1, we have
d Q mod Dd
max y,
max y, x F it , x
1 2
Y y x
2
3 1 1 QD 2 QD 1 c Lx 1 2 T 1 2 x 10 x 2 T 1 D D
1 dt O x 2 L2 . t 1
(12)
1
max y, Y x 2 .
1 2
2 5
From (12) and (13), we have
x x Q 2 D x QD Lc .
This completes the proof of Lemma 2.2.
4. Proof of Lemma 2.3
T
11 dt N HN 20 Lc ,
where c is an absolute constant, H mr 1Q 2T and L log HN . Now we complete the proof of Lemma 2.3. 1 Proof of Lemma 2.3. Let Y x 2 and X x . We define F s,
n n n s .
Y n X
If y satisfies Y y X,
(11)
we apply Perron’s summation formula with T y (see Proposition 5.5 in [1]), and then obtain Copyright © 2013 SciRes.
max y, y x
d Q mod Dd
max y,
Y y x
d Q mod Dd
dt
max y, y 2Y
1
xF it , t 1 Q 2 Dx 2 L2 . d Q mod Dd
x
Further let q Dd and then we obtain
Firstly we recall one result of Choi and Kumchev [9] about mean value of Dirichlet polynomials. Let m 1, r 1, and Q r , Let m, r , Q denote the set of character modulo mq , where is a character modulo m and is a primitive character modulo q with r q Q , r q and q, m 1. Then the result of Choi and Kumchev states as follows. Lemma 4.1. Let m 1, r 1, T 2, N 2, and m, r , Q be a set of characters as described as above, Then
T n n nit N n2 N m , r ,Q
d Q mod Dd
1 2
(13)
y 2Y
4 5
1 2
On the other hand we have
Q 2 Dx 5
Q 2 Dx
1 dt O xy 1 L2 . t 1
Noting that F s, does not depend on y , we have
y x
1 3 1 1 R 2 R L x max max T 1 2 T 2 x 10 x 2 0 T x 1 R QD T 1 D D c
s
1 2
L12 x max max
d Q mod Dd
max y, y x
1
2T 1 T F it , dt Q 2 Dx 2 L2 . 0 T x 1 R QD T 1 q R mod Dd
max max
D|q
Lemma 4.1 with m = 1 gives that 11 R 2T 20 c F it , dt x x L . D
T
2T
q R mod Dd D| q
(14)
From (14), we have
d Q mod Dd
max y, y x
11 1 1 R 2T 20 c x x L Q 2 Dx 2 L2 0 T x 1 R Q T 1 D 1 QD 2 11 1 Lc x 20 x T 1 Q 2 Dx 2 L2 D 11 x x 20 Q 2 D Lc .
max max
This completes the proof of Lemma 2.3. APM
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R. T. GUO
5. Proof of Lemma 2.1
y; qd , r
We partition the moduli qd as qd1d 2 , where the prime factor of d 2 not exceed LK and those of d1 do not. If n denotes the number of distinct prime divisors of the integer n , and t 2 K log log x log log log x , with estimate x; q, r x q , for 1 4 1 2, we
1 y; q, r d
1 1 y; q , r y; qd1 , r d 2 d1
y; Dd 2 , r
have
d q 1 x
x; qd , r
x 1 1 q d1 x d1 d2 x d 2 d1 t
d1 t
x log x 1 1 m q k t k ! p LK m 1 p
k
d1 d L 2
x log x exp t 1 o 1 log log log x , q
Moreover the corresponding sum, taken over those moduli d for which d1 is divisible by the v th power of some prime, v 8, is x 1 1 q d2 x d 2 p LK p v
m x
O q xL
K
,
We denote
d 2 L K qd1
too. by
.
Arguing
similarly for d x; q, r , we have 1
1
d q x d1
max max y; qd , r
r , qd 1
y x
1
d
x mod Dd 2
1
D1 D mod D1
1
qd1d 2
,
max y, * y x
1 Dd 2
max | y, | y x
1 . d 2 0 mod Dd 2
Here L K D 1 x , and the innermost bounding sum 1 is D log x. We cover the range of with adjoining intervals U 2U , subject to LK U L K D 1 x . When 1 2 , by Lemma 2.2 a typical interval contributes
y; q , r
x . q LK
4 1 1 x log 5x log log x x 2UD1 x 5 D12 . D U
Since 3 1 x1 5 exp log log x , the 4 whole sum over is 12 1
D
We collect together those moduli qd with a fixed value of d1 not exceeding and set D qd1 . Noting that
y; D , r
yx
Interchanging summations,
2
3 1 exp log log x 2
x mod Dd 2
max y,
y, y, * O log Dd 2 y .
1 m
With v 4 K 3 log log x , this is 1
qd1
1
is q q 1 x log x . For a fixed value of D qd1 , we collect together those terms involving the characters induced by a particular primitive character * mod D1 , where D1 D and d 2 . Since and * differ on at most the integers n for which n, D1 1 but n, Dd 2 1,
q 2 v 2 x log x . 1
K
6 K
which is O q xL K . 1
1 r y, O log d 2 y . d Dd 2 mod Dd2 2
where ' denotes that if we factorise as 1 2 defined mod D , 2 defined mod d 2 , then the character 2 is not principal. In order to establish Lemma 2.1 it will therefore suffice to prove that the sum S given by
k
x log x e log K log L O 1 k q k t
1 y; D , r d2
n y , n , d 2 1 n r mod D
n O log yd 2 ,
we see from the orthogonality of Dirichlet characters that Copyright © 2013 SciRes.
D1 2 qd1
12
D 1 x log x
5 K
log log x.
Arguing similarly for 9 20 , by Lemma 2.3 the 5 K whole sum over is also D 1 x log x log log x. Noting that APM
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R. T. GUO
d1
D D
q q 1
d1
d1
for any fixed positive A. We employ the representation
d1
p p2 q q 1 1 p p2 p 1 q q 1 1 p p
2
q q 1 log
n n E y
n y
2
6
summation over d1 delivers the desired bound on S. This completes the proof of Lemma 2.1.
6. Zeros of Dirichlet L-Functions Lemma 4.1. Let L s, , s it , denote an L-function formed with a Dirichlet character mod q , q 3, h p q p. With l log q t 3 , define
34
.
1
(15)
Proof of Lemma 4.2. This is result of Landau, which can be found at Satz 6.4, p. 127, of Prachar [11]. Lemma 4.3. For any modulus D, 0 1, T 0, let N , T , D denote the number of zeros, counted with multiplicity, of all functions L s, formed with a character mod D , that lie in the rectangle Re s 1, Im s T . Then we have 12
N , T , D DT 5
1
,
uniformly for 0 1, T 2. Proof of Lemma 4.3. This is Theorem of HeathBrown [12], on p. 249.
7. Proof of Theorems 1.1 and 1.2 We shall first provide a version of the theorem with y; qd , r in place of y; qd , r . After Lemma 2.1 it will suffice to establish the bound
G x q log x
Copyright © 2013 SciRes.
A 1
,
valid for all characters mod D , where y T 2 ; E is 1 if is principal, zero otherwise; i runs through all the zeros of L s, in the rectangle 0 Re s 1, Im s T with a half disc 1 s c4 log D 0, Re s 0 removed. This representation is a slightly modified version of that given in Satz 4.6, pp. 232-234 of Prachar [11]. Since L s, has log DT zeros in the strip 0 Re s 1, T Im s T 1, cf. Prachar [11], Satz 3.3, p. 220,
Then there can be at most one non-principal character mod q for which the corresponding L-function has a zero in the region 1 . Moreover such a character would be real and the zero would be real and simple. Proof of Lemma 4.1. This is Theorem 2 of Iwaniec, [10]. Lemma 4.2. Let j mod D j , j 1, 2 be distinct primitive real characters. There is a positive real c1 so that at most one of the functions L s, formed with these characters can vanish on the line segment 1 c1 log D1 D2 1, t 0.
y log Dy 2 O y1 4 log Dy , T
q q 1 log log x .
1 4.104 log h l log 2l
T
y
1 2 | |T
1 y1 2 log 2 D m T m m 1
y
y1 2 log Dy m 1 y1 2 log Dy , 2
m T
and at the expense of raising y1 4 log Dy to 2 y1 2 log Dy we may confine the zeros to the halfplane Re s 1 2. From the orthogonality of Dirichlet characters
D y; D, r y
mod D
r n n E y
1 2 T
n y y y 2 y1 2 D log Dy , T
where it is understood that the i are the zeros of the L-function formed with the character of the outer summation. We replace y by z and average over the interval A 2 y z y w with w y log y to obtain 1 yw D z; D, r z dz w y y y y 2 y1 2 log Dy . 2 w 1 2 T T
Replacing z in the integrand by y introduces an error of w D
y n y w n r mod D
w log y w D 1 log y, D
and we may remove the integral averaging: APM
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R. T. GUO
D y; D, r y log y
A 2
Rqd x log x
y
1 2 T
with the same uniformity in d . If there is an exceptional zero
y 2 y . y1 2 D log Dy A 1 T log y
y log y 2 y 1, y; D , r D D
which is valid for all positive y. With these bounds y x
x log x
i
mod D 1 2 T
2
x 2 x x1 2 D log x , A 1 T log x
4
22 k
mod D 1 2 2k 1
1 2k T
4
1 2k T
c log d log 2q T 3 log log 2q T 3
3 4 1
12 1
12
1 u
x exp log x
78
32
x exp log x
19
12 1
D c log x
x
q log x
A
,
, this again allows the argument
to proceed. We may therefore assume that A 2 D log x and remove the restriction '' from the above summation over d at an expense of
x log x x log x x . A qd qD q log x
y x
2
y x , A1 q q log x
and in this case there is no exceptional zero. By substraction we see that 32
,
uniformly for 2T and d . Moreover, N 1 2, , D D 2 log D 2 , Prachar [11], Satz 3.3, p. 220, as earlier. Altogether Copyright © 2013 SciRes.
d
max y; q, r
xu log xdu.
with restriction q x 5 12 2 we have 78 D12 5 x exp log x , then the integral is
qd
A modified version of this argument delivers the bound
N u , , D x u log xdu
x1 2 N 1 2, , D c2 1 2 D 5
A 1
where '' indicates that the moduli are not divisible by the (possibly non-existent) modulus D . A theorem of Siegel shows that for any 0 there is a positive constant c so that an L-function fromed with a real character mod D has no zero on the line-segment 1 c D Re 1, Im( s ) 0; cf. Prachar [11], Satz 8.2, p.144. Unless
d 4 ad 0 mod D
1
1 2
log x
1 2 xu dN u , , D 1
1
x
.
In view of Lemma 4.3, typically
d
where 1 is the largest value of taken over all the zeros i in the rectangle 0 Re s 1, Im s 2T . Supposing for the moment that D qd and that there is no zero that is exceptional in the sense of Lemma 4.1, then we may take
1
y; qd , r max y x qd
, D x3 4 . We
22 k 1 2 xu dN u, 2k 1 , D
x u N u , , D
of some 4ad with d , and an application of Lemma 4.2 shows that there is no further L-function formed with a real character mod D , D 4a, that has a real zero on the line-segment
y
x
1
1 c1 2 log 4a , and the corresponding function L s, is attached to a real character induced by a primitive character mod D , then D is a divisor
1
y D
A 2
for which
1 c1 2 log 4a Re s 1, Im s 0 unless that character is also induced by mod D . In particular, D will be divisible by D . For those moduli qd for which 4ad is not a multiple of D we may choose the same as before and recover the above estimate for Rqd . Hence
A 2
holds uniformly for 2 T x log x set T x1 2 . The double-sum does not exceed
mod qd ,
1
This bound will be satisfactory for y > x(log x) A 2 . Otherwise, we shall employ the crude bound
RD max D y; D, r
A 1
2
G x q log x
A 1
indeed holds for every fixed
A 0. Since q log q, an application of Lemma 2.1 shows that with B A 6, APM
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R. T. GUO
qd x log x
B
max max y; qd , r
r , qd 1
y x
y qd
REFERENCES
x
q log x
, A 1
uniformly for moduli q x exp log log x that are powers of a where 2 5 , if 1 2 and 5 12, if 9 20. Replacing y; qd , r in this bound by
y; qd , r
3
qd x1 2 log x
qd x1 2 log x
B 2 m log x
B
x1 2 xq 1 log x
A 6
max y; D, r
y0 ; D, r
A 1
.
Li y
D
Li y
D
[3]
A. I. Vinogradov, “The Density Hypothesis for Dirichlet L-Series,” Izvestiya Rossiiskoi Akademii Nauk SSSR. Seriya Matematicheskaya, Vol. 29, 1965, pp. 903-934. http://dx.doi.org/10.1007/s11139-006-0250-4
[4]
P. D. T. A. Elliott, “Primes in Progressions to Moduli with a Large Power Factor,” The Ramanujan Journal, Vol. 13, No. 1-3, 2007, pp. 241-251.
[5]
M. B. Barban, Y. V. Linink and N. G. Chudakov, “On Prime Numbers in an Arithmetic Progression with a Prime-Power Difference,” Acta Arithmetica, Vol. 9, No. 4, 1964, pp. 375-390.
[6]
J. Y. Liu and M. C. Liu, “The Exceptional Set in Four Prime Squares Problem,” Illinois Journal of Mathematics, Vol. 44, No. 2, 2000, pp. 272-293.
[7]
J. Y. Liu, “On Lagrange’s Theorem with Prime Variables,” Quarterly Journal of Mathematics (Oxford), Vol. 54, No. 4, 2003, pp. 453-462. http://dx.doi.org/10.1093/qmath/hag028
[8]
D. R. Heath-Brown, “Prime Numbers in Short Intervals and a Generalized Vaughan’s Identity,” Canadian Journal of Mathematics, Vol. 34, 1982, pp. 1365-1377. http://dx.doi.org/10.4153/CJM-1982-095-9
[9]
S. K. K. Choi and A. Kumchev, “Mean Values of Dirichlet Polynomials and Applications to Linear Equations with Prime Variables,” Acta Arithmetica, Vol. 123, No. 2, 2006, pp. 125-142. http://dx.doi.org/10.4064/aa123-2-2
1
We may therefore confine our attention to maxima over the range y0 y x. Integration by parts shows that y0 y x
E. Bombieri, “On the Large Sieve,” Mathematika, Vol. 12, No. 2, 1965, pp. 210-225. http://dx.doi.org/10.1112/S0025579300005313
,
y0 qd log y0 y0 q
x q log x
[2]
log p
p x1 m p m r mod qd
1
d x1 2 q 1
H. Iwaniec and E. Kowalski, “Analytic Number Theory,” American Mathematical Society, Providence, 2004.
log p
the congruence condition p m r mod qd having been ignored. Employing the Brun-Titchmarsh bound 1 y; D, r y D log y , valid unifromly for 1 D y 3 4 , r , D 1, We see that the contribution to the sum in the theorem that arises from maxima that A occur in the range 0 y y0 x log x is
[1]
p y p r mod qd
introduces an error
Jianya Liu and Guangshi Lü for their encouragements.
y 1 max y; D, r . D log x y0 y x
The theorems hold with B A 6.
8. Acknowledgements
[10] H. Ivaniec, “On Zeros of Dirichlet’s L-Series,” Inventiones Mathematicae, Vol. 23, No. 2, 1974, pp. 97-104. http://dx.doi.org/10.1007/BF01405163 [11] K. Prachar, “Primzahlverteilung,” Grund. der math. Wiss., Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957. [12] D. R. Heath-Brown, “Zero-Free Regions for Dirichlet L-Functions and the Least Prime in an Arithmetic Progressions,” Proceedings of the London Mathematical Society, Vol. 64, No. 2, 1992, pp. 265-338. http://dx.doi.org/10.1112/plms/s3-64.2.265
This work is supported by IIFSDU (Grant No. 2012JC020). The author would like to thank to Professor
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