Divisors of Fourier coefficients of two newforms

For a pair of distinct non-CM newforms of weights at least 2, having rational integral Fourier coefficients $a_{1}(n)$ and $a_{2}(n)$, under GRH, we obtain an estimate for the set of primes $p$ such that $$ \omega(a_1(p)-a_2(p)) \le [ 7k+{1}/{2}+k^{1/5}],$$ where $\omega(n)$ denotes the number of distinct prime divisors of an integer $n$ and $k$ is the maximum of their weights. As an application, under GRH, we show that the number of primes giving congruences between two such newforms is bounded by $[7k+{1}/{2}+k^{1/5} ]$. We also obtain a multiplicity one result for newforms via congruences.


Introduction and statement of the results
For an elliptic curve E/‫ޑ‬ and a prime p of good reduction, let N p (E) := p+1−a( p) be the number of points of the reduction of E modulo p. Assume that E is not ‫-ޑ‬isogenous to an elliptic curve with torsion.Then Koblitz's conjecture [7] says that the number of primes p ≤ X for which N p (E) is prime is asymptotically equal to C E (X/(log X ) 2 ), where C E is a positive constant depending on E. In particular, N p (E) is prime infinitely often when p runs over the set of primes.This conjecture is still open but there are many results towards this in the literature (see [17]).Indeed, Koblitz's conjecture can be seen as a variant of the twin prime conjecture (for more details, see [7]).
Inspired by Koblitz's conjecture, Kirti Joshi [6] studied the prime divisors of N p ( f ) := p k−1 + 1 − a( p), where a( p) is the (integer) p-th Fourier coefficient of a newform f ∈ S k (N ), the space of cusp forms of weight k and level N .Note that for the Ramanujan delta function ∈ S 12 (1), ω(N p ( )) ≥ 3 for any p ≥ 5, where ω(n) is the number of distinct prime divisors of an integer n.This shows that, in general, the obvious variant of Koblitz's conjecture is not true for modular forms of higher weights.In fact, Joshi shows that there exist infinitely many cusp forms f k i (which need not be eigenforms) of increasing weight k i and of level 1 such that ω(N p ( f k i )) ≥ 2 for all primes p.
If f is a non-CM newform of weight k ≥ 4 then in the same paper Joshi gives an estimate for the primes p for which N p ( f ) is an almost prime, i.e., has few prime divisors.More precisely, under GRH and Artin's holomorphy conjecture, he uses a suitably weighted sieve due to Richert to prove that  p ≤ X : ω(N p ( f )) where [ • ] is the greatest integer function.He also proves a similar result for the function (N p ( f )), where (n) counts the number of prime divisors of n with multiplicity.
One can interpret N p ( f ) as the difference of p-th Fourier coefficients of the normalized Eisenstein series E k and the newform f .This leads us to study the number of prime divisors of the difference between the p-th Fourier coefficients of any two distinct cuspidal newforms which allows us to deduce many interesting consequences about congruences between newforms, multiplicity-one results, etc.More precisely, we prove the following.
Theorem 1.1.Let f 1 ∈ S k 1 (N 1 ) and f 2 ∈ S k 2 (N 2 ) be non-CM newforms with integer Fourier coefficients a 1 (n) and a 2 (n), respectively, of weights at least 2. We also assume that f 1 and f 2 are not character twists of each other if k 1 = k 2 .Put k = max{k 1 , k 2 }.Then under GRH, we have If k ≥ 6 then the term k 1/5 appearing in (1-2) can be replaced with the smaller term √ log k.
We remark that because of Deligne's estimate of Fourier coefficients, for any p, N p ( f ) in (1-1) never vanishes, whereas a 1 ( p) − a 2 ( p) may be zero.Therefore we remove such primes from (1-2).
Remark 1.2.In Theorem 1.1 and all the subsequent results in this section, by GRH, we mean the generalized Riemann hypothesis holds for all the number fields L h , h ≥ 1 (see Section 2B for the definition of L h ), i.e., the Dedekind zeta functions associated with L h have no zeros in the complex region Re(s) > 1  2 for all h.We now state a few applications of our main result.Unless stated otherwise, throughout the paper we shall work with forms f 1 and f 2 as in Theorem 1.1.We also assume that a newform is always normalized so that its first Fourier coefficient is 1.An immediate consequence of Theorem 1.1 is the following.
Corollary 1.3.Let f 1 and f 2 be newforms as in Theorem 1.1.Then under GRH there exist infinitely many primes p such that a 1 ( p) ̸ = a 2 ( p) and . We now recall a multiplicity-one result which says that if a 1 ( p) = a 2 ( p) for all but finitely many primes p, then f 1 = f 2 .Rajan [14] has extensively generalized this result by proving that if a 1 ( p) = a 2 ( p) for a set of primes p of positive upper density, then f 1 is a character twist of f 2 .This is known as a strong multiplicity-one result.Recently, in [12], a variant of this result for normalized Fourier coefficients has been obtained.In this direction, we prove in Proposition 4.2 that, under GRH, if for any ϵ > 0, then f 1 is a character twist of f 2 .As a consequence of Theorem 1.1, we obtain the following interesting result that can be seen as a variant of a multiplicityone result in terms of congruences.
Corollary 1.4.Let f 1 and f 2 be non-CM normalized newforms of weight k 1 and k 2 with integer Fourier coefficients a 1 (n) and a 2 (n), respectively.Put k = max{k 1 , k 2 } and assume GRH.If there exist primes ℓ 1 , ℓ 2 , . . ., ℓ n such that n > 7k + 1 2 + k 1/5 and, for each for all p except for a set of primes of order o(X/(log X ) 2 ), then k 1 = k 2 and f 1 is a character twist of f 2 .
Proof.On the contrary, assume that f 1 is not a character twist of f 2 .For 1 In particular, But from our assumptions in (1-4) we have |B(X )| = o(X/(log X ) 2 ) and this contradicts Theorem 1.1.□ We now mention the last application of Theorem 1.1 which is related to the number of congruence primes of a newform.Recall that for a newform f 1 ∈ S k (N 1 ) with integer Fourier coefficients a 1 (n), a positive integer D is called a congruence divisor if there exists another newform f 2 ∈ S k (N 2 ) with integer Fourier coefficients a 2 (n) which is not a character twist of f 1 such that f 1 and If D is a prime, then D is called a congruence prime and we refer to [3] for a nice overview of the subject.A congruence divisor of a newform is an important object to study as it is connected to many well-known problems.To name a few, a bound of the largest congruence divisor is related to the ABC conjecture, and if k = 2, then the congruence primes for f 1 are related to the prime divisors of the minimal degree of the modular parametrization to the elliptic curve attached to f 1 via the Eichler-Shimura mapping (see [11, p. 179-180]).It would be also of great interest to bound the number of congruence primes of a newform (see remark on page 180 of [11]).However, if we fix two newforms, then the following result gives a bound on the number of congruence primes which is immediate by Corollary 1.3.
Corollary 1.5.Let f 1 and f 2 be newforms as in Theorem 1.1.Suppose there exists a positive integer D such that f 1 and f 2 are congruent modulo D. Then under GRH Each prime divisor of D gives a congruence between f 1 and f 2 ; therefore, Corollary 1.5 ensures that the number of primes giving congruences between two newforms is bounded uniformly in terms of their weights and not on the levels.This is the novelty of this result.
We now discuss some results about the function (a 1 ( p) − a 2 ( p)), where p varies over the set of primes.Using a similar idea as the proof of Theorem 1.1, we obtain the following.
Theorem 1.6.Let f 1 and f 2 be as in Theorem 1.1.Then under GRH, we have p ≤ X : a 1 ( p) ̸ = a 2 ( p) and (a It is clear that Theorem 1.6 also has applications of similar nature to that of Theorem 1.1 mentioned above and we would not repeat it here.
Remark 1.7.It is possible to obtain an upper bound of the right order of magnitude for the estimate in Theorem 1.6.In fact, we can do so by using Selberg's sieve and the ideas used in the proof of [6,Theorem 2.3.1].More precisely, under GRH, one can obtain that if f 1 and f 2 are as in Theorem 1.1, then From the above estimate, it follows that under GRH In particular, the natural density of the set { p : a 1 ( p) − a 2 ( p) is prime} is zero.It would be interesting to obtain a suitable lower bound of this set or at least to know whether there are infinitely many primes p for which a 1 ( p) − a 2 ( p) is a prime.
In fact, all the above results are valid even if we replace a 1 ( p) − a 2 ( p) with a 1 ( p) + a 2 ( p).Also, similar results but with better bounds hold in Theorems 1.1 and 1.6 if we assume Artin's holomorphicity conjecture in addition to GRH.It is also worth mentioning that the full strength of GRH is not essential to prove our theorems.Rather, a quasi-GRH, which assumes a zero-free region for the associated Dedekind zeta functions in the region Re(s) = 1−ϵ for some ϵ ∈ 0, 1  2 , is sufficient for our purpose (see the discussion and results proved in [13]).In this case, we obtain similar results to Propositions 2.1 and 4.1, with the only difference being that the exponent of x becomes 1 − ϵ instead of 1  2 .However, it then requires a more careful analysis of handling the error terms in the subsequent part of the proof of our results, which will not be carried out here.
Contents and structure of the paper.The theorem of Deligne connecting the theory of ℓ-adic Galois representations to Fourier coefficients of newforms opens the door for obtaining many new results regarding the arithmetical nature of these coefficients.This connection and the Chebotarev density theorem play a prominent role in this paper.These are recalled in Section 2. To prove our results, we first establish Proposition 4.3 which gives an asymptotic formula for the number of primes p up to X for which a 1 ( p) ̸ = a 2 ( p) and a 1 ( p) − a 2 ( p) is divisible by a fixed positive integer.Proof of Proposition 4.3 requires computations of the image of the product Galois representations attached to f 1 and f 2 and this is obtained in Section 3. Finally, we apply a suitably weighted sieve due to Richert, recalled in Section 5, to prove our results.To establish the sieve conditions with the required uniformity of parameters, Proposition 4.3 plays a crucial role.We use the ideas employed in [6; 17] to prove our main results in Sections 6 and 7.
Notation.For any real number X ≥ 2, π(X ) denotes the number of primes less than or equal to X .Along with the standard analytic notation ≪, ≫, O, o, ∼ (the implied constants will often depend on the pair of forms under consideration), we use the letters p, ℓ, q, ℓ 1 , ℓ 2 , etc. to denote prime numbers throughout the paper.

Preliminaries
We summarize some standard results without proofs which will be used throughout the paper.We closely follow [2] for our exposition.
2A. Chebotarev density theorem.We recall the Chebotarev density theorem which is one of the principal tools needed for proving the main theorems of this paper.
Let K be a finite Galois extension of ‫ޑ‬ with the Galois group G and degree n K .For an unramified prime p, we denote by Frob p , a Frobenius element of K at p in G.For a subset C of G, stable under conjugation, we define π C (X ) := { p ≤ X : p unramified in K and Frob p ∈ C}.
The Chebotarev density theorem states that We will use the following conditional effective version of this theorem which was first obtained by Lagarias and Odlyzko [8] and was subsequently refined by Serre [16].To state this, let d K be the absolute value of the discriminant of K ‫ޑ/‬ and ζ K (s) be the Dedekind zeta function associated with K .
In addition to GRH for ζ K (s), by assuming Artin's holomorphy conjecture (which states that the Artin L-function associated to any nontrivial representation of the Galois group Gal(K ‫)ޑ/‬ has an analytic continuation on the whole complex plane) one can improve the error term in the above asymptotic formula for π C (X ).

2B. mod-h Galois representations.
Let G ‫ޑ‬ = Gal( ‫)ޑ/ޑ‬ be the absolute Galois group of an algebraic closure ‫ޑ‬ of ‫.ޑ‬ Let k ≥ 2, N ≥ 1 and ℓ be a prime.Suppose f ∈ S k (N ) is a newform with integer Fourier coefficients a(n).The work of Eichler, Shimura and Deligne (see [1]) give the existence of a two-dimensional continuous, odd and irreducible Galois representation which is unramified at p ∤ N ℓ.If Frob p denotes a Frobenius element corresponding to such a prime, then the representation ρ f,ℓ has the property that By reduction and semisimplification, we obtain a mod-ℓ Galois representation, where ‫ކ‬ ℓ := ‫.ޚ‪/ℓ‬ޚ‬ Let h = t j=1 ℓ n j j be a positive integer.Using the ℓ j -adic representations attached to f , we consider an h-adic representation given by products of mod-ℓ j 's representations For each 1 ≤ j ≤ t, we have the natural projection ‫ޚ‬ ℓ j ↠ ‫/ޚ‬ℓ n j j ‫,ޚ‬ and hence we obtain a mod-h Galois representation given by ) be newforms having integer Fourier coefficients a 1 (n) and a 2 (n), respectively.Then one can consider the product representation ρh of ρ f 1 ,h and ρ f 2 ,h , defined by Let A h denote the image of G ‫ޑ‬ under ρh .By the fundamental theorem of Galois theory, the fixed field of ker( ρh ), say L h , is a finite Galois extension of ‫ޑ‬ and Let C h be the subset of A h defined by We now define the following function on the set of positive integers which will play an important role throughout the paper.For an integer h > 1, define Since the trace of the image of complex conjugation is always zero, C h ̸ = φ, and hence δ(h) > 0 for every integer h.

Technical results
Let f 1 and f 2 be newforms as before.The main aim of this section is to obtain an asymptotic size of δ(ℓ n ) for n = 1, 2 and this requires the computation of the cardinalities of A ℓ n and C ℓ n .Building on the work of Ribet [15] and Momose [10], Loeffler [9] has determined the image A ℓ n of the product Galois representations.
More precisely, Loeffler [9, Theorem 3.2.2]has proved that that there exists a positive constant M( f 1 , f 2 ) such that, for all primes ℓ ≥ M( f 1 , f 2 ) and n ≥ 1, In other words, the mod-ℓ n representations of two newforms (that are not character twists of each other) are as independent as possible.In the rest of the paper, we denote the constant M( f 1 , f 2 ) by M and without loss of generality, we assume that M ≥ 3. Clearly, for ℓ ≥ M, 3A. Combinatorial lemmas.Here we obtain results about cardinalities of A ℓ n and C ℓ n for any ℓ ≥ M. We first assume that Recall that ℓ ≥ 3. We now consider the group homomorphism Since φ is surjective and its kernel {v ∈ (‫/ޚ‬ℓ n ‫)ޚ‬ × : v λ n = 1} is a cyclic subgroup of (‫/ޚ‬ℓ n ‫)ޚ‬ × of order λ n , we obtain We first recall the following result proved in [2, Lemma 3.3].
Lemma 3.2.For any prime ℓ ≥ M and integer n ≥ 1, Proof.Let ψ : A ℓ n → A ℓ be the natural map defined by Since it is a surjective group homomorphism, we have Therefore, in view of Lemma 3.1, to evaluate |A ℓ n | it is sufficient to compute |ker(ψ)|.For that we first compute the cardinality of the set where d ∈ (‫/ޚ‬ℓ n ‫)ޚ‬ × such that d ≡ 1 (mod ℓ) is fixed and Id is the identity element in GL 2 ‫ކ(‬ ℓ ).Any general element of the above set will be of the form 1+xℓ yℓ zℓ 1+wℓ , where 0 ≤ x, y, z, w < ℓ n−1 with the condition that As d ≡ 1 (mod ℓ), the above equation reduces to Since 1 + wℓ ∈ (Z /ℓ n ‫)ޚ‬ × for such w, for any choices of 0 ≤ y, z, w < ℓ n−1 the above equation gives a unique x.Therefore . Now, we note that ker(ψ) = {(A, B) ∈ A ℓ n : (A, B) ≡ (Id, Id) (mod ℓ)} ; therefore from (3-1) In the above, congruence conditions on A and B compel that d 1 ≡ d 2 ≡ 1 (mod ℓ), and hence using (3)(4)(5)(6) gives Since the sum appearing on the right side of the above equation is the cardinality of the kernel of the natural (surjective) reduction map n → 1 given in (3-5), we have 1) .Now using (3)(4) in the above yields the desired result. □ Our next aim is to compute the cardinalities of C ℓ and C ℓ 2 .Though an explicit computation is possible, we only obtain asymptotic formulas here and that is enough for our purpose.To simplify our notation, we denote the set of quadratic and nonquadratic residue elements in (‫/ޚ‬ℓ n ‫)ޚ‬ × by Q n and Q c n , respectively.Lemma 3.3.For any prime ℓ ≥ M, Proof.From the definition of C ℓ Split the sum over 1 into three parts, namely and we denote the corresponding sums by S 1 , S 2 and S 3 , respectively.Thus To proceed further, note that for given d ∈ ‫ކ‬ × ℓ and t ∈ ‫ކ‬ ℓ one can obtain the following result by employing an elementary counting argument: Again using (3-8) Collecting the terms containing ℓ 4 gives Similarly, we have Combining all together, we have, from (3-7), and now using (3-4) completes the proof.□ To compute |C ℓ 2 |, we first prove the following result which is a generalization of (3-8) for the ring ‫/ޚ‬ℓ 2 ‫.ޚ‬ Lemma 3.4.For any d ∈ (‫/ޚ‬ℓ 2 ‫)ޚ‬ × and t ∈ ‫/ޚ‬ℓ 2 ‫,ޚ‬ we have where N := {(a, b, c) ∈ (‫/ޚ‬ℓ 2 ‫)ޚ‬ 3  : a 2 −at +bc = −d}.To compute |N | we divide the set N into three disjoint subsets N 1 , N 2 and N 3 based on the following three cases, respectively.Hence Case (i): a = 0. Then the condition bc = −d forces that b and c both have to be units and for any b there exists a unique c.Hence Case (ii): a ̸ = 0 and bc = 0.The latter condition implies that either b or c is 0, or both are (nonzero) zero-divisors of ‫/ޚ‬ℓ 2 ‫.ޚ‬The total number of such pairs is We now claim that To prove this, we see that if t 2 − 4d = 0, then any a ≡ t 2 (mod ℓ) is a solution of a 2 − at + d = 0 and there are ℓ such choices for a. Next, assume that 0 ̸ = t 2 − 4d ≡ 0 (mod ℓ).If a 2 − at + d = 0 has solutions, say x and y, then Therefore, x − y ≡ 0 (mod ℓ) = ⇒ t 2 − 4d = (x − y) 2 = 0, which is a contradiction.The last two cases are clear.
Case (iii): a ̸ = 0 and bc ̸ = 0.In this case, bc can be either a (nonzero) zero-divisor or a unit.Clearly, the number of choices for b and c such that bc is a given nonzero zero-divisor is 2ℓ(ℓ − 1) and for a given unit the number of such choices is ℓ 2 − ℓ.Therefore, we have Note that there exists a unique m ∈ ‫ކ‬ × ℓ such that t 2 − 4(d − mℓ) = 0, and in that case 0 ̸ = t 2 − 4d ≡ 0 (mod ℓ).Therefore
Since the fixed field of ker( ρh ) is contained in the compositum of fixed fields of ker( ρℓ n i i ), from (2-1) and hence a simple counting argument gives Therefore now it is clear that, for any integer h ≥ 1, 3B. Asymptotic size of δ(ℓ).Recall that, for any positive integer h > 1, An immediate consequence of the results in the previous section is the following.
Proposition 3.6.If ℓ varies over primes then, for n = 1, 2, Finally, we state the following multiplicative property which plays an important role to prove our main result.

Analytic results on primes
Recall that f 1 and f 2 are non-CM newforms with integer Fourier coefficients which are not character twists of each other.For a positive integer h ≥ 1 and a real number X ≥ 2, consider the function 1.
The representation ρh , defined in Section 2, is unramified outside h N .Also, it is ramified at all the primes ℓ | h because its determinant contains a nontrivial power of the mod-ℓ cyclotomic character which is ramified at ℓ.However, there may exist some primes dividing N at which ρh is unramified.It follows that a prime p is unramified in L h only if either ( p, h N ) = 1 or p | N .Since the image of Frobenius elements under ρh generate A h , we can write where the error term is due to the possible primes divisors of N which are unramified in L h .Since the trace and the determinant maps are stable under conjugation, the group A h and the set C h are also stable under conjugation.Now applying the Chebotarev density theorem (see Proposition 2.1) for the field L h , we obtain the following.
Proposition 4.1.Let f 1 ∈ S k 1 (N 1 ) and f 2 ∈ S k 2 (N 2 ) be non-CM newforms with rational integral coefficients a 1 (n) and a 2 (n), respectively.Assume that f 1 and f 2 are not character twists of each other.Let N = lcm(N 1 , N 2 ) and h ≥ 1 be an integer.If GRH holds for the field L h , then We remark that to establish Proposition 4.1, we need to use (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) and the following variation of a result of Hensel (see [16,Proposition 5,p. 129]): For our purpose, we now use Proposition 4.1 to obtain the following result giving an upper bound for the set of primes p with a 1 ( p) = a 2 ( p).This may be also of independent interest.Proposition 4.2.Let f 1 and f 2 be newforms as before.Then under GRH Proof.Clearly, for any prime ℓ, Hence using Proposition 4.1, for a large prime ℓ, Now by Bertrand's postulate, we choose a prime ℓ between X 1/14 /log X and 2(X 1/14 / log X ) and this proves the result.□ Note that in Proposition 4.2, GRH is used for the field L ℓ , for all but finitely many primes ℓ.
We remark that for newforms of weight 2 and by making use of various abelian extensions, in [13, Theorem 10], a better estimate in Proposition 4.2 is obtained.
We now define 1.
Using Propositions 4.1 and 4.2 we deduce the following.
Proposition 4.3.Let f 1 and f 2 be newforms as in Proposition 4.1 and h ≥ 1 be an integer.If GRH holds for the field L h , then 13/14 ).Remark 4.5.In the above propositions, if one assumes Artin's holomorphy conjecture in addition to GRH, then an improved error term can be obtained.More precisely, in Propositions 4.1 and 4.3, we have O(h 3 X 1/2 log(h N X )) instead of O(h 6 X 1/2 log(h N X )) which gives the estimate for Proposition 4.2 5. Sieving tool: Richert's weighted one-dimensional sieve form We will prove Theorem 1.1 by using a suitably weighted sieve due to Richert [4].
The sieve problem we encounter here is a one-dimensional sieve problem in the parlance of "sieve methods".We will use notation and conventions from [4].
Let A be a finite set of integers not necessarily positive or distinct.Let P be an infinite set of prime numbers.For each prime ℓ ∈ P, let A ℓ := {a ∈ A : a ≡ 0 (mod ℓ)}.We write where X (resp.δ(ℓ) X ) and r 1 (resp.r ℓ ) are a close approximation and remainder to A (resp.A ℓ ), respectively.For a square free integer d composed of primes of P, let 1 : There exists a constant A 1 > 0 such that where A 2 ≥ 1 and L ≥ 1 are some constants independent of z and w.
R(1, α): There exist 0 < α < 1 and A 3 , A 4 ≥ 1 such that, for X ≥ 2, For A and P as above and for real numbers u, v and λ with u ≤ v, define the weighted sum We now state the following form of Richert's weighted one-dimensional sieve.
where c is a constant depends at most on u and v (as well as on the A i 's and α) and Here γ is Euler's constant and X is the approximation of A given in (5-1).

Proof of Theorem 1.1
We shall closely follow the arguments of [6].The idea is to apply Theorem 5.2 to the situation where M = M( f 1 , f 2 ) is the constant in Section 3. It is clear that, for any ℓ ∈ P, Applying Proposition 4.3, under GRH, we obtain where r ℓ = O(ℓ 6 X 1/2 log(ℓN X )) + O(X 13/14 ).If d is a square free integer composed of primes from P, then from Propositions 3. To apply Theorem 5.2, we now verify that hypotheses 1 , 2 (L , 1) and R(1, α), given in Hypothesis 5.1, hold for our choice of A and P.
Lemma 6.1.Let f 1 and f 2 be newforms as before.Then we have the following: (1) Hypothesis 1 holds with a suitable A 1 .
We now see that, for any α < 1  14 , and this completes the proof.□ Next we need to choose sieve parameters α, u, v, λ satisfying conditions in Theorem 5.2.For k ≥ 2 we take α and 0 < λ < 1.This shows that these parameters satisfy the conditions required for applying Theorem 5.2, and hence for our choices of A and P, we obtain Note that here we have used the fact that |A| ≫ X/log X and W (X ) ≫ 1/log X for X ≫ 0 which follows immediately by using Proposition 3.6.Also for the choices of sieve parameters α, u, v, λ given in (6-2), the function F(α, v, u, λ), defined by (5-3), can be computed explicitly and is given by e γ 14k 6/5 log 3+log 42k −(1+14k) log 56k 14k+1 28k 6/5  .
There are at least X/(log X ) 2 many primes p ≤ X which make a positive contribution to the left-hand side of .Therefore to complete the proof of the first part of Theorem 1.1 it is sufficient to show that, for any such prime p, ω(a 1 ( p) − a 2 ( p)) ≤ 7k + 1 2 + k 1/5 .
Therefore, we write 1.Now to estimate the first sum on the right of (6-5) we use  and obtain log q log X .
For the second sum we observe that if q ≥ X 1/u then log q/log X ≥ 1 u that gives Substituting the values of u and λ from (6-2) and choosing X large enough completes the proof of the first part of the theorem.Finally, the last assertion of the theorem when k ≥ 6 can be achieved by taking λ = 1/ √ log k instead of λ = 1/k 1/5 in the above proof and then following the same arguments.

Proof of Theorem 1.6
The idea of the proof is similar to the proof of Theorem 1.1 with minor modifications.We shall apply Theorem 5.2 with the same setting as in Section 6.For k ≥ 2, we choose the sieve parameters as Again, these parameters satisfy the conditions required for Theorem 5.2 and the corresponding function F(α, v, u, λ) > 0 for k > 1.006.Hence as in the proof of Theorem 1.1, the corresponding weighted sum satisfies (7-1) W(A, P, v, u, λ) ≫ X (log X ) 2 .
Next we observe that p ≤ X : where the error term is due to the presence of those primes p such that p | ℓN and ℓ 2 | (a 1 ( p) − a 2 ( p)).Applying Proposition 4.1 gives that the left side of the above equality is equal to π(X ) Since u > 26, we have  p ≤ X : We conclude, by combining (7-1) and (7-2), that there are at least X/(log X ) 2 many primes p ≤ X such that

Remark 4 . 4 .
Indeed, the estimates given in Propositions 4.1 and 4.3 are also valid for the set of primes p ≤ X with h | (a 1 ( p) + a 2 ( p)).This can be achieved by considering the set C ′ h = {(A, B) ∈ A h : tr(A) = −tr(B)} instead of C h in Section 3 and following the same arguments.