Primes in Arithmetic Progressions to Moduli with a Large Power Factor

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  nd real number A  here is a B   ch that 2 a T su 0 .   0 B A             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

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

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   , 1 a q  , we introduce  ; , q not exceeding x .For fixed q , we have 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 A q L  , 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 0 A  there exists a constant thesis ha Therefore they try to find a satisfactory substitute.In this is the Euler totient function, 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 precisely, he showed that Theorem B. Let a be an integer, 2 a  .If 0 A  , then there is a 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 establis 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, that are powers of When and an odd prime, our result gives that for t moduli with the form Then the special case of our result shows that the least prime in these special progressions satisfies that are powers of Then our resu ows that the least prime a .lt sh   min , P q r in these special progressions   mod n r q  satisfies   , 1 1 max max ; , ; , .

 
, y   defined as in (2).Then lds uniformly for all integers 1 D  and real numb 2, 1.
ds uniformly for all integers the inn all racters

Proof of
where  is a Dirchlet character, s a complex variable.mma 3.1.Let Le   , F s  be as in (7), and 1 A  0 , arbitrary.Then for an and where is an absolute constant independent of mplied in depends on e constant i  .A Proof of Lemma 3.1.This lemma with 1 d  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 To go s identity [8], whic and further, we first recall Heath-Brow

S
where   denotes the vector (5).Obviously some of tervals the in  , 2 j j M M   may contain on using mmation form Proposition 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 the integral on the two horizontal segments above can be estimated as hand we On the other have From ( 9) and ( 10), we have Further let and then we obtain max , This completes the proof of Lemma 2.2.

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

 
, , m r Q  be a set of characters as described as above, Then e apply Perron's summation fo with T y  (see Proposi On the other hand we have From ( 12) and (13), we have ), and then obtain .
 and then we obtain From ( 14), we have This completes the proof of Lemma 2.3.

Proof of Lemma 2.1
We partition the moduli as , where the prime factor of not exceed  x q r x q Moreover the corresponding sum, taken over those moduli for which is divisible by the power of some me, ; , log , .

Dd y qd r y q r d y qd r y q r
where denotes that if we factorise ' In ma fice to prove that the sum S given by order to establish Lem 2.1 it will therefore suf- For a fixed value of    differ on at most the integers for n , and the innermost bounding sum is Arguing similarly for , by Lemma 2.3 the whole sum over  is also

Zeros of Dir hlet
Then there can be at most one non-principal character for which the corresponding L-function has a region   mod q zero in the 1 .

 
  Moreover such a character would be real and the zero would be real and simple. , primitive real characters.There is a positi 1 c that at most one of the  formed with these characters can vanish on the line segment Proof of Lemma 4.2.This is result of La which can be found at Satz 6.4, p. 127, of Prachar [1 ].
and at the expense of raising This bound will be satisfactory for Otherwise, we shall employ the crude b which is valid for all positive y.With these bounds

T x 
The double-sum does not exceed

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