Primes in Arithmetic Progressions to Moduli with a Large Power Factor

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 high- powers 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

 





 

 

 

 

1 1 2

, 1

, 1

max max ; , ,

B

A y x r qd

d x q L d q

Li y x

y qd r

qd q L



 

 

  

















1

3 3_{exp log log} qx  x

holds uniformly for moduli that are powers of . In this paper we are able to improve his

eywords: Primes; Arithmetic Progressions; Riemann Hypothesis

1. Introduction and Main Results

with

to count the number of primes in the arithmetic pro-

a

result.

K

Let p denote a prime number. For integer a q,

 

a q, 1, we introduce



x q; ,





mod  1

p x

p a q

a

 





gression a



modq



not exceeding x. For fixed q, we have





_{   }

1

; ,

x q a x q

 

 

as x tends to infinity. However the most important thin in this context is the range uniformity for the moduli q in terms of

g

x. The Siegel-Walfisz Theorem, see for e ample [1], shows that this estimate is true only if qLA, where and throughout this paper we denote

log

x

x by L. The Generalized Riemann Hypothesis for let L-functions could give a much better result: non-trivial estimate holds for

Dirich 1

2 2

qx L . Unfortunately the Generalized Riemann Hypo s 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.

direction the famous Bombieri-Vinogradov theorem [2, 3], states that

Theorem A. For any A0 there exists a constant

thesis ha

Therefore they try to find a satisfactory substitute. In this

 

0

BB A  such that

 



 

 



, 1

max max y; , A,

y x a q q Q

Li y

q a xL

q



  









where 

 

q is the Euler totient function, 1 2 B_, Qx L

and

 

2

d . log

y u

Li

u



Re

y 

cently 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 pre- cisely, 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

 





 

 

 

 

1 1 2

, 1

, 1

max ,r  Li y x



max ; ,

B

A y x r qd

d x q L d q

y qd

qd q L



 

 

  







3

3_{exp log log} qx  x holds uniformly for moduli





1

that are powers of

When result recovers the Bombieri- Vinogradov theorem. And obviously his result gives a

n of primes in arithmetic

st imp a. , his

1 q

deep insight into the distributio progressions.

The mo ortant 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 techniques estab- lis

l new

hed in the study of Waring-Goldbach problems. We shall prove the following resu t.

Theorem 1.1. Let a be an integer, a2. If A0, then there is a B A

 

0 such that



B

 











 

 

1 1 2

, 1

, 1

B

y x r qd d x q L

d q

qd q

 

 

  



max max  y qd r; ,  Li



_A,

L



holds uniformly for moduli

y x









2

3 5_{exp log log}

qx  x

that are powers of

When and an odd prime, our result gives that for t moduli with the form

a. 1 d

hes

a

e particular q





, 1, 2, 3,

n

q p n 



x q r; ,





1 O L

 

A



_{ }

 

, q





  Li x



holds uniformly for moduli









2

3 5_{exp log log .} qx  x Then the special case of our result shows that the least prime in these special progressions

satisfies

 

min ,

P q r



modq



nr

 

5 2

min , .

P q r q

This proves a former result given by Barb hudakov [5



result im an,

Linnik and Ts ],

 

8 3

min ,

P q r q 

where _q _pn,



_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.

rem 1.2. Let be a nteger, . If th

Theo a n i a2 A0,

en there is a BB A

 

0 such that



 

 







 

 

9 1 20

, 1

, 1

B

y x r qd d x q L

d q

qd q

 

 

  



max max  y; Li y x



qd r, _A,

L

 

holds uniformly for moduli









5

3 12_{exp log log}

qx  x

that are powers of

Then our resu ows that the least prime a.

lt sh Pmin

 

q r, in these special progressions nr



modq



satisfies

 

12 5 min ,

P q r 

It should be rem rk

.

q 

a ed that the Generalized Riemann

H ld allow



ypothesis for Dirichlet L-functions wou1 1

2 A

qdx L  with no further restriction pon the natu u re of . Therefore our Theorems 1.1 and 1.2 can be compared with the result under the G eralized Riemann H

2. Prelimin

q

en ypothesis.

ary Reduction

Let 

 

n denote von Mangoldt’s function, and for mutually prime integers w and r, let









; , .

w

y w r n

 







r

mod

n y n r

 

Fo _2wx3 4

and an integer q1, define

 

_{ }



_{   }



  , 1 , 1

1

max max ; , ; , .

r qd y x d w

d q

G w y qd r y q r d

 



 

 







Lemma 2.1. For any Then

0,1 4 1 2

K   , we have









 





3 1

6 1

1

exp log log log

2

log .

K

K

G x q L G x x

q q x x





 

 

  

  

 

 





(1)

uniformly for positive integers







3



exp log log , 3

qx  x x where  2 5 , if

9 20  1 2 and 5 12, if 1 4  9 20. Here

 

q 1

 



_{n q|} .

For Dirichlet characters  and real y0 define





  

n y

y n

  



 



,



n .

Lemma 2. t

(2)

2. Le 



y,



defined as in (2). Then

 









1 2 2 5 1 2

max y, x x Q D QD Lc



 



 4

mod

, y x

d Q Dd

x

 





ho ers

Lemma 2.3. Let

(3) lds uniformly for all integers D1 and real numb

2, 1.

x Q



y,



  defined as in (2). Then

 









11 20 2

mod

max , c.

y x

d Q Dd

y x x Q D



 

  



 

 L

ds uniformly for all integers

the inn all

racters

3. Proof of Lemma 2.2

(4)

hol D1 and real numbers

2,Q 1.

  Here er sum is taken over hlet cha



modDd



.

x

primitive Dirc

Let

2 5

and M1,,M10 be positive real numbers such that 1 5

1 10 and 2 6, , 2 10 .

Y M M X M  M X (5) For define

where is the Möbius function. Then we define the fu

1, ,10

j 

 

 

1, if 2, , 5, , if 6, ,10,

j

a m j

m j

 

_ 

 _ _

log ,m if j 1,

 



(6)

 

n



nctions



,



   

j

j

j s

m M

a m m f s

m



 

_



,

,

and



,



1



,



10



,



F s   f s  f s  (7) where  is a Dirchlet character, s a complex variable.

mma 3.1. Let

Le F s



,



be as in (7), and A1

0 ,

arbitrary. Then for any 1 R X2A and TXA

mod  |

3 2

log d r

R R

T

 



 



)

1 1

10

2 2

1 2

2

,

r R r T

c

T X X X

d d

   



 

 





(8

where is an absolute constant independent of

1

, d

T

F it  t

  



 

 

0

c 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 We mention that in general the exponent 3/10 to

[7].

X in the o

ix n value of

D

second term on the right-hand side is the best possible n considering the lack of s th power mea

irchlet L-functions.

Now we complete the proof of Lemma 2.2.

Proof of Lemma 2.2. In (5), we take 2

5_{, .}

Y x X x

Define a_j

  

m ,f_j s,



and F s



,



as above. To

go s identity [8],

whic and

further, we first recall Heath-Brow

k

n’

2

n z with h states that for any z1 1,

k

 



  

1 2 2 1 2

1 1 2

1

lo

j j j

k

 

1





1 j g _{j j}

j n n n n

n n z

n n n

j





 



 





 k

n  

  

   _{ }



 



The for

.

n

2 5

2Y 2x  y X x,



y,



 

of which is of t

 

   

   

1 1 10 10

1 10

1 1 1 10 10 10

, , 2

: ,

m M m M

y m m y

a m  m a m  m

 

 

 

 

 

S

where



 denotes the vector



M M1, 2,,M10



with

j

M as in (5). Obviously some of the intervals



Mj, 2Mj may contain on using

mmation form

Propo-sition 5.5 in [1]), and then shifting the contour to the left, we have

ly integer 1. By ula with T y (see Perron’s su

 









 





 

1 1 ₂

1 1

1 2 1 2 1 1 ₂

1 1 1 2 1 2 2 1

, d

2 1

. 2

s s

L iy

L iy

iy iy L

L iy iy iy

y y

iy

F s s O L

i s

O L i

 



   

  

   



 

   











On usin

 S

g the trivial estimate

















1 10

1 1 1 1

1 2 10

, ,

,

F iy f iy f iy,

M L M  M  x L

     

   

    

  

the integral on the two horizontal segments above can be estimated as





1 2 1 1

3 1 1

1 _{2 2 10}

1 2max1 1L .

max ,

L

y F iy

y

y

x L x y L x L y

   

  

Then we ha

 

 

 

  

 





ve

 





1 1

3

2 ₂

10

3 1

10 2

2

1 1

, d

1

2 2

2

1 d

, .

2 1

it _it

y

y

y

y

y y

F it t O x L

it

t

y F it x L t

 



 _





 



 

     _ _

  _{  }

 _  _

 _{ }

 







 S



,



F s  does not depend on y, we have Noting that





10 12 103 11

2 <

1 d

max , , .

2 1

x

x Y y x

t y L x F it x

t

  

 

 _  _

 _{ }

 



 L

(9) hand we

On the other have





2

max , .

yY  y  Y

 (10)

From (9) and (10), we have

 





 





 





mod

max ,

y x

Q Dd

y



 

is a linear combination of terms, each

he form

 

10 O L

 

2 <

mod mod

2 2 5 2

mod

max ,

d

, .

2 1

d

Y y x

d Q Dd d Q Dd

x

d Q Dd

y y

t

L x F it Q Dx

t

 



2

max ,

y Y

1

10 x 1

  





 

 

 

 





 

 _{ }

 



 

 

  

 



 



Further let qDd and then we obtain

 





  mod

1

2 12 ₂

0 1

mod | 2

2 5_.

Q Dx 

max ,

1 1

max max , d

1 2

y x

d Q Dd

T T T x R QD

q R q

D q y

L x F it t

T





 



  

    

 _ 

 

  

 

  





From Lemma 3.1, we have

 





 

_

_

_

_

mod

3

1 1 1

2 2

10

2 2

1 2 0 1

2 2

3 1

1 10

2 ₂ 2

1 2

2 2 5

4

1 1

2 5

2 2

max ,

1 max max

1

1 1

. y x

d Q Dd

c

T x R QD

c

c y

R R

L x T T x x

T D D

QD QD

L x T x x T

D D

Q Dx

x x Q D x QD L



 

  

   

 

 

 _{ } 

 

  

 

 _{   } 

 

 

 



 

 

 

 

 

 







This completes the proof of Lemma 2.2.

4. Proof of Lemma 2.3

Firstly we recall one result of Choi and Kumchev [9] about mean value of Dirichlet polynomials. Let

and Let denote the er

5

2

1 1

Q Dx 

1, 1, m r set of charact

,

Qr 



m r Q, ,



  modulo mq, where  is a character modulo m and  is a primitiv

,

e c acter with

har modulo q r q Q r q and



q m,



1.

as fo s. Then the result of Choi and Kumchev states llow

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

 

   

11 20

, , 2

d ,

T _{it c}

T

m r Q N n N

n n n t N HN L



 



  

 

 _  _













where c is an absolute constant, 1 2



H mr Q T and

plete the

logHN.

 Now we com 2.3.

Lemma 2.3. Let

L

Proof of

proof of Lemma1 2

Y x and X x. We define



,



   

s.

Y n X

F s  n  

 





 n n

If satisfies

w rmula

y

,

Y  y X (11) e apply Perron’s summation fo with T y (see Proposi

























1 2

1 2 2

ds O x L ,

s _ _

2 1

, , d

2

2 ,

2

s s

b iy b iy

s s

iy b iy

y y

y F s s O xy L

i s

y y

F s i

  



 

 _







 

 



 





1 b

where _{0 b L}1_.

If we let b0, we have









1



1 2



, , d

1 y

y

y F it t O xy

t

   

 _ 



 L .

Noting that F s



,



does not depend on y, we have









d 1₂ 2

max , , .

1 x

x

Y y x 



 it   O x L_ _

t

y F

t

   



 (12)

On the other hand we have





12

2

max , .

yY  y  Y x (13)

From (12) and (13), we have

 





 





 





tion 5.5 in [1]), and then obtain

 





mod

2

mod mod

1 2 2

max ,

max ,

d

, .

y x

d Q Dd

Y y x y Y

d Q Dd d Q Dd

x y

y

t

F it Q Dx L



 

 

2

mod x 1

d Q Dd t

max y,





 

 



  



  

 

 _

 

 

  

 

Further let



 

qDd and then we obtain

 





 





mod

1

2 _{2 2}

2

0 1 _mod

|

max ,

1

max max , d .

1 y x

d Q Dd

T T

T x R QD _{q R Dd}

D q y

F it t Q Dx L

T





 



  



     

 

  





Lemma 4.1 with m = 1 gives that

 





11 2 2

20 mod

|

, d

T _c

T

q R Dd

D q

R T

.

F it t x x L

D





  _ 

 

 

 

  





(14)

From (14), we have

 





 

_

_

mod y x

d Q Dd

11 1

2

2 2

20 2

0 1

2 _{11 1}

1 2 ₂ 2

11 2 20

max ,

1 max max

1

1

.

c T x R Q

c y

R T

20

c

x x L Q Dx

T D

QD

x T Q Dx L

x x Q D L

 



   



 

 _  _

 

 _ _

 

  _

 

 



 

 

 

 





This completes the proof of Lemma 2.3.

L

L x

D

 



 



 

5. Proof of Lemma 2.1

We partition the moduli as , where the prime factor of not exceed

qd qd d_{1 2}

2

d LK

ber of

an of do not. If de tes the num ct prim divisors of

and

d those distin

1

d

e

 

n



the integer no

n, t2Klog logx log l

 

og logx, with

estimate 



x q r; ,



x q , for 1 4  1 2 , we have





 

 

 

 

 

 

 





 

 





 





1

1 2

1

1 2

1 1

; ,

log 1 1

lo g

g

exp 1 1 l log ,

d x d x

d q x

d t d t

m

x x qd r

q d d

x x

x

t o x

q



 



  



  



 



 

 

 











which is

Moreover the corresponding sum, taken over those moduli for which is divisible by the power of some me,

1

k

1 ! K

k t p L m

q k p





 

 

 



log g 1

lo k

k t

e K L O x x

q k

 



 

 

 



lo

og log x



 



1



. K

O  q  xL

d pri

1

d

is

th v 8,

v

 

 

_{ }

 

 





2 2

1 2 2

1 1

2 log .

K v

d x p L m x

v

x

q d p

q x x



   



  

 











1 m

With v_4



K3 log log



x_, this is



1



,

K

xL

 

too.

 

O  q

We denote exp 1



log log



3

2 x

 

 

  by . Arguing

similarly for 

 

d 1



x q r; ,



, we have

 



   



 

1 1

, 1

1

max max ; , ; ,

.

r qd y x d q x

d

K

y qd r y q r d

x q L

   



  

 







We collect together those moduli with a fixed value of n eeding set

qd

and 1

d ot exc  Dqd1.

Noting that

we see from the orthogonality of Dirichlet characters that



y D r







_

_{ }

   2 

2 , , 1

mod

; , log ,

n y n d

n r D

n O yd

  





 



_{   }



  

   





_{   }







 2

  



 

1

2 1

2

2

; , ; ,

g 1

y Dd r y D r

d

d y

 



 



 2

mod

2 2

1

; , ; ,

1 1

; , ; ,

1

lo

, .

Dd

y qd r y q r

d

y qd r y q r

d d

r y O

Dd  d

 



 

 

  

 



 

 

 _  _

 

 



 _{  }

 

 



where ' denotes that if we factorise  as  1 2 defined



modD



, ₂ defined



₂



, the character 2

modd n the

 is not principal.

In ma

fice to prove that the sum S given by

order to establish Lem 2.1 it will therefore suf-

 1  



 



1 2 1 mod 2 1 2

1

max , ,

K y x

d _{d L qd x} Dd

y

qd d



 





 

 _









 

1





6

log K.

q q x x

  



is For a fixed value of



1



D qd , we collect togethe characters

r those terms involving the

 induce character

d by a particular primitive





*

1

modD ,

  where D D1 and  d2 . Since  and * differ on at most the integers n for which



n D, ₁



1 but



n Dd, 2



1,









*



_O





2

, , log

y y Dd

   y .

Interchanging summations,

   









 





 





1

2

2 1

1 1

mod 2

mod

1

max ,

1

max | , | .

K Dd y x

d L qd x

y x

D D D

y

Dd

y



  

 



 



 



 





 





Here

2 0 mod 2

d    Dd

1

K L D x

*





_  

, and the innermost bounding sum is 

 

D 1log .x We cover the range of  with adjoining intervals U  2U subjec o

.

K K

L U L D x

, t t

1 

 

  When  1 2, by Lemma .2 a typical

2 interval contributes

4

1 1 5

5

2 2

1 1

log log log

.

x x

x

x UD x D

U

 

 D

 

 

 





Since

 

1 2





3

1 2 1 2 1 5

1 1

1

exp log log ,

4

D D  qd x _ x _

  the

whole sum over

 is



logx



log log .x

Arguing similarly for 5

1 K

D x 



9 20

 

_{ }

 

 

 

 

 

  



  



1 1

1 1

1 2 1

2

2

2 1

6 1

1

1 1

log log .

d d

p

p

1

log d

D d

p p q q

p p

q q q q

p

q q x



  

 



 

 



 

 

 

 _{  } 

 

 

 

 _  _ 







 

 

summation over delivers the desired b und on S. f of Lem

ic

unctions

Lemma 4.1. Let , denote an

L-function form et character

D

q q



 









1

d o

This completes the proo ma 2.1.

6. Zeros of Dir hlet L-F



,



,

L s  s  it

ed with a Dirichl



mod



, 3, .

p q

q q h p

  



With llogq t



3 ,



define







3 4



1 _4.104 _{logh llog 2l .}

 _{ }

Then there can be at most one non-principal character for which the corresponding L-function has a

region



modq



zero in the   1 . Moreover such a character would be real and the zero would be real and simple.

, [1

Lemma 4.2. Let be distinct e real so

Proof of Lemma 4.1. This is Theorem 2 of Iwaniec 0].



modD



,j 1, 2

 

v functions L s



,



j j

primitive real characters. There is a positi c1 that at most one of the  formed with these characters can vanish on the line segment

1, t



 _{ }

)





1

1 1 2

1c logD D  0. (15

Proof of Lemma 4.2. This is result of La which can be found at Satz 6.4, p. 127, of Prachar [1 ].

ndau, 1

Lemma 4.3. For any modulus D, 0  1,T 0, let



, ,T D



denote the number of zeros, counted with multiplicity, of all functions L s



,



N

 formed with a character 



modD



, that lie in the rectangle

Res 1, Ims T.

   Then we have



  

12 

1 5

, , ,

N  T D  DT 

uniformly for 0  1,T2.

Proof of Lemma 4.3. This is m of Heath- Brown [12], on p. 249.

Theore

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

ed positive A. We employ the repr

it will suffice to establish the boun

 



 



1

lo A ,

G x  q



 

for any fix

esentation

d



gx

   





2

1 4 log

log , y Dy

O y Dy

 

 

 

n y T

y n n E y

T

 







  

 

 





valid for all characters



modD



, 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 ha disc lf





1

 

4 log 0, Re 0

s c D   s  remove versi

d. Th o in Satz 4.6, pp. 232-234 of Prachar [11].

is given representation is a slightly modified n of that

Since L s



,



has logDT zeros in the strip

 

 

0Re s 1,T  Im s  T 1, 220,

cf. Prachar [11], Satz 3.3, p.









1 2

1 2 1

| |

2

1 2 1 1 2

1 log 2

log log ,

m T m m T

m T y

y D

y Dy m y Dy



 

      

 

 



 

 

 



 





 

and at the expense of raising 1 4

log

y Dy to





2

log

y1 Dy 2 we may confine the zeros 

to the half- plane Re

 

s 1 2.

of Dirichlet characters From the orthogonality

  



 

 













mod

2 1 2

1 2

log ,

D n y

T

y y

y D Dy

T



  









 

 

 

 

_  _

 





 

where it is understood that the ; ,

D y D r y

  



  r  n n E y_ 





i



   

with the character are the zeros of the L-function formed  of the outer summation.

We replace y by z and average over the interval

y  z y w with wy



logy



 A2 to obtain

  











2

1 2 2

1 2 1

; , d

log .

y w

y z D r z z

w D

T

y y y

y Dy

w T

  



 





 



 

_  _

 

 



Replacing in the integrand by introduces an error of



z y

 

 

mod 

n r D

and we may remove the integral averaging:

log 1 log ,

y n y w

w

w D y w D y

D

 

  

 

  _  _

 



  















2

2 1 2

, log A

T

D y D r y y

2 1 2

1

;

log .

log A

y

y y

y D Dy

T y

  



 



  

 

 



 

_  _ 

 

This bound will be satisfactory for Otherwise, we shall employ the crude b

. ) log (

> x x A2 y

ound



y; ,r

_{  }

log



2



1,

D D





 y  y y 

D 

which is valid for all positive y. With these bounds

  

_{  }

 













2 log A x x 

2 mod 1 2

2 1 2

1

max ; ,

log ,

log

D y x

D T

A

y R D y D r

D

i

x x

x D x

T _x

 



 



 



 



 



 

_  _ 

 

 



holds uniformly for 2 T x



logx



 A 2,Dx3 4. We set T x1 2.

The double-sum does not exceed

 





1

2

mod 1 2 1 2

2

1

2k u k 1

1 2 1 2

4 2

4 2 d , 2 ,

k

k

k

k

D T

T

x

x N u D



 





  

  



 





where

 

  

 

 

1 is the largest value of  taken over all the zeros  i in the rectangle

 

 

0 Re s 1, Im s 2 .T

Supp for the moment that and that there is no zero that is exceptional in th a 4.1, then we may take

 

osing Dqd

e sense of Lemm















_{3 4}



1

log log 2 3 log log 2 3 .

c d q T q T

     

In view of Lemma 4.3, typically

















 

 

1 _u

1 2

1 1

1 2 1 2

12

1 1

1 2

5 2 1 2

d , ,

, , , , lo d

1 2 , , log d .

u u

u u

x N u D

g

x N u D N u D x x u

x N D c D x x u

 





 

 



  



 



  









   

with restriction 5 12 2

qx  we have







7 8



integral is 12 5 _{exp log ,}

D x  x then the







7 8



3 2







1 9



3 2

exp log exp ,

x  x  x  logx 

 

uniformly for 2T and d   . Moreover,











 2 ,



her

1 2 , ,

N  D D 2 l 

s earlier.





1

log A

qd

R x x  

ogD

 Prachar [11],

Satz 3.3, p. 220, a Altoget

with

for which

, and the corresponding function the same uniformity in d.

If there is an exceptional zero



modqd



,





1

1

1 c 2 log 4a

 

  



,



L s  is attached to a real character induced by a itive character

prim 



modD



, , d

then is a di sor with

D vi

of some 4ad   a ows that there is no

character e line-se

nd an ication of fu her L-function that n th

0 appl rt

,

D D

Lemma 4.2 sh

formed with a real



mod



has a real zero o gment

4a ,

 





1

 

 

1

1c 2 log 4a  Re s 1, Im s  unless that character is also induced by . In particul ,

le by



modD



  ar

D will be divisib D. For those m le of D we m

oduli qd for which 4ad is not ay choose the same

a multip

 as before and recover the above estima for .

qd

R

te

Hence



_{  }





1

 

 



,

logx A d



 qd  q logx A

 

where '' icates that the moduli are not divisible by the (possibly non-existent) modulus D.

A theorem of Siegel shows that fo ere is a positive constant

max y qd r; , 



1 y x

d   qd

 y

x x

ind

r any 0 th

 

c  so that an L-f fromed with a real ch

unction aracter



modD



has no zero on the line-segment 1c

 

D Re

 

1, Im( )s 0; cf. Prachar [11], Satz 8.2, p.144. Unless





 

 





1 1 2

log ,

D  c x  this again allows the argument to p ceed. We may therefore assume that







ro 2

A

D logx 

the above summ

and remove the restriction from ation over at an expense o

 ''

f

d

 









2

4 0 mod

log log

. log A

d

ad D

x x

x x x

qd qD q x

  





  

A modified version of this argument delivers the bound



_{     }

A1

o excepti tion we see that

max ; , ,

log

y x  y q r  _q  _{q x}

and in this case there is n onal zero. By substrac

 

 



y x





1 lo A G x  q gx



  indeed holds for every fixed 0.

A

Since 

 

q log ,q an application of Lemma 2.1 6,

   



  

 



, 1 log

1

max max ; ,

, log

B r qd y x

qd x x

A

y y qd r

qd

x q x









  











of wh

uniformly for moduli qxexp







log logx



3



that are powers a ere  2 5 , if  1 2 and

5 12 ,

  if  9 20.

Replacing 



y qd r; ,



in this bound by



; ,



log

p y

y qd r p







mo 

p r dqd



introduces an error

 

 

 





1 2

1 2

log

6 1 2 1

log

log

log ,

B m

B

qd x x

A

qd x x

p

x xq x

 



  













the congruence condition



havi ignored.

Employing the Brun-Titchmarsh bound

valid unifromly for

1

2 log mod

m

m x p x

p r qd

 





 



mod

m

p r qd ng been







 



1

; , log ,

y D r y D y

   





3 4

1Dy , r D, 1, bution to the sum in the theorem s from that

e range y y A

occur in th

We see that the contri that arise maxima

g is





0

0  x lo x

 





 

 







1 2 1

1

log

d y  y 1

0 0 0

1

log .

d x q

A

y q q

x q x







 





e confine our attention to maxima over the range

Integration by s that



 

We may therefor

0 .

y  y x

parts show





_{ }

 





_{ }

 



_{  }

0

max ; ,

1

max ;

y y x

Li y y D r

D

y D

D









  

The theorems hold with

0 log y y x

D x

  

6.

0; , , .

Li y y

y D r r

   



B A

8. Acknowledgements

This work is supported by IIFSDU (Grant No 2012JC020). The author would like to thank to Prof

Jianya Liu and Guangshi Lü for their encouragements.

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