Primes in Arithmetic Progressions to Moduli with a Large Power Factor

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Oct 6, 2013 - http://dx.doi.org/10.4236/apm.2013.37A003 Published Online October ...... we see from the orthogonality of Dirichlet characters that. (. (. ) yDr ψ. ).
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 L2 . 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

APM

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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 A1 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  m1m10  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 11 L  iy , ds  O L2 F s    s 2 i 11 L  iy 1 2  iy 1 2  iy 11 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 1n2 j  z



max

1 2  11 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 22 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  11 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

APM

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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  L1 . 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  nit N n2 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  

yx

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

30

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 yw   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

31

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



 

22 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  , A1  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

22 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

32

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

Copyright © 2013 SciRes.

APM