On the Second Moment Estimate Involving the $\Lambda $-Primitive

Home , Primitive root modulo n

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1 D (2) P (x)= ALi(x)+ O x log− x . y a a y X≤ In [15], P. J. Stephens introduced a method to widen the range of y to y > exp(c√log x) for some absolute positive constant c without explicitly determining it. Then S. Kim [6] obtained an explicit c = 3.42. For the composite moduli, S. Li [9] proved that for y exp((log x)3/4), ≥ 1 R(n) (3) N (x) . y a ∼ n a y n x X≤ X≤ S. Li and C. Pomerance [10] improved the range of y to: (4) y exp((2 + ǫ) log x log log x), ≥ for any positive ǫ. S. Kim [6] proved (3) with a widerp range of y: If y > exp(3.42√log x), then there exists a positive constant c1 such that 1 R(n) (5) N (x)= + O x exp( c log x) . y a n − 1 a y n x X≤ X≤ p In [6], the author speculated that the following would be true: If y > exp(4.8365√log x), then 2 1 R(n) (6) N (x) x2 exp c log x y  a − n  ≪ − 2 a y n x X≤ X≤ p   for some absolute positive constant c2. This speculation is based on the corresponding normal order result for prime moduli that can be deduced from [6, (4)]: If y > exp(4.8365√log x), then 2 1 R(p) (7) P (x) x2 exp c log x y  a − p  ≪ − 3 a y p x X≤ X≤ p   for some absolute positive constant c3. In this paper, it turns out that the speculated result for composite moduli is not true, even for a wider range of y (3.42 instead of 4.8365): Theorem 1.1. If y > exp(3.42√log x), then 2 1 R(n) x2 (8) N (x) . y  a − n  ≫ (log log log x)2 a y n x X≤ X≤   By applying [13, Theorem 1.1], we are able to obtain the following reﬁned version of [8, Theorem 2.3] which provides the explicit constant 6 0.341326: π2eγ ≈ Theorem 1.2. For large x, we have R(n) 6 x (9) + o(1) . n ≥ π2eγ log log log x n x X≤ Also, we are able to give an explicit constant in Theorem 1.1: Theorem 1.3. If y > exp(3.42√log x), then 2 1 R(n) x2 (10) N (x) (C + o(1)) y  a − n  ≥ (log log log x)2 a y n x X≤ X≤   where 36 1 C = 1+ 1 0.003692. π4e2γ p5 + p4 p3 p2 − ≈ p ! Y − − 2. Preliminaries

Throughout the chapters 2 and 3, the parameter ǫ is allowed to be any positive constant, and c1 is some absolute positive constant which is not necessarily the same at each occurrence. We adopt the notations and deﬁnitions in [10]: If we deﬁne v ∆ (n)=# cyclic factors C v in (Z/nZ)∗ : q λ(n) , q { q || } then 1 (11) R(n)= φ(n) 1 . − q∆q(n) q φ(n) |Y Let rad(m) denote the square-free kernel of m. Let

λ(n) E(n)= a (Z/nZ)∗ : a rad(n) 1 (mod n) , { ∈ ≡ } and we say that χ is an elementary character if χ is trivial on E(n). For each square free h φ(n), let ρn(h) be the number of elementary characters mod n of order h. Then | ρ (h)= (q∆q(n) 1). n − q h Y| For a character χ mod n, let ′ ( 1)ord(χ)R(n) 1 − if χ is elementary, c(χ)= χ(b)= φ(n)ρn(ord(χ)) , φ(n) (0 otherwise. Xb where the sum Σ is over λ-primitive roots in [1,n]. Since R(n) φ(n), ′ ≤ c(χ) c¯(χ), | |≤ where 1 if χ is elementary, c¯(χ)= ρn(ord(χ)) (0 otherwise. For the proof of above, see [10, Proposition 2]. Let X(n) be the set of non-principal elementary characters mod n. In [8], the counting function of λ-primitive roots is deﬁned and denoted by ta(n): 1 if a is a λ-primitive root modulo n, ta(n) := (0 otherwise.

Then it is possible to write ta(n) as a character sum:

ta(n)= c(χ)χ(a). χ modX n Distinguishing the character by principal and non-principal, the sum a y Na(x) can be decomposed into three parts: ≤ P R(n) (12) N (x)= y + B(x,y)+ O(x log x), a n a y n x X≤ X≤ where (13) B(x,y)= c(χ) χ(a). n x χ X(n) a y X≤ ∈X X≤ Following the proof in [10], (14) B(x,y) µ(d) S , | |≤ | | d d x X≤ where

∗ (15) Sd = c(χχ0,dkm1m2 ) χ(a) , x x x | | y k m1 m2 χ mod k a ≤ d ≤ dk ≤ dkm1 ≤ d (k,dX)=1 rad(Xm ) k X X X 1 (m2,k)=1 | and Σ∗ is over non-principal primitive characters. Let χ0,n be the principal character modulo n. For positive integer k and reals w, z, deﬁne

1 ∗ F (k, z)= c¯(χχ ) χ(a) , m 0,km rad(m) k χ(mod k) a z X | X X≤

T (w, z)= F (k, z), k w X≤ 1 w 1 S(w, z)= w F (k, z)= T (w, z)+ w T (u, z)du, k u2 k w 1 X≤ Z and x y S S , . d ≤ d d We exhibit here a series of lemmas in [6] that lead to the estim ation of B(x,y). The function f is deﬁned by: 1 K2 f(K)= log + 1 + 1 , K 2 K which is the same function as f1(K)+ 4 as in [6]. Lemma 2.1. (1) If z > exp(4.18√log w), then

13 (16) T (w, z) wz 16 exp log w (f(4.18) + ǫ) . ≪ (2) If exp(3.419906√log w) < z exp(16√log w), thenp ≤ 3 (17) T (w, z) wz 4 exp log w (f(3.419906) + ǫ) . ≪ w 1 p By S(w, z)= T (w, z)+ w 1 u2 T (u, z)du, Lemma 2.2. (1) If z > exp(4R .18√log w), then

13 (18) S(w, z) wz 16 exp log w (f(4.18) + ǫ) . ≪ (2) If exp(3.419906√log w) < z exp(16√log w), thenp ≤ 3 7 (19) S(w, z) wz 4 exp log w (f(3.419906) + ǫ) + w log w z 8 . ≪ · We also have by [11, Theorem 1] and partialp summation, Lemma 2.3. log w S(w, z) wz exp 3 . ≪ slog log w !

Combining these, we obtain for y > exp(3.42√log x), there is a positive absolute constant c1 such that (20) B(x,y) xy exp( c log x). ≪ − 1 p 3. Proof of Theorems 3.1. Basic Set Up. By (12), we have 2 2 R(n) R(n) R(n) N (x) = N (x)2 2N (x) +  a − n   a − a n  n   a y n x a y n x n x X≤ X≤ X≤ X≤ X≤    2    R(n) = N (x)2 y + O(x2 log x)+ O (xB(x,y)) . a −  n  a y n x X≤ X≤ 2   We treat the sum a y Na(x) ﬁrst: ≤ P 2 N (x)2 = t (n) = t (n )t (n )= t (n )t (n ) a  a  a 1 a 2 a 1 a 2 a y a y n x a y n1 x n2 x n1 x n2 x a y X≤ X≤ X≤ X≤ X≤ X≤ X≤ X≤ X≤   = c(χ1)χ1(a) c(χ2)χ2(a)

n1 x n2 x a y χ mod n χ mod n X≤ X≤ X≤ 1 X 1 2 X 2 R(n1) R(n2) φ(n1n2) • 2 2 = y + c(χ1)c(χ2)χ1χ2(a)+ O(x log x) φ(n1) φ(n2) n1n2 n1 x n2 x n1 x n2 x a y χ mod n ≤ ≤ ≤ ≤ ≤ 1 1 X X X X X χ2 modX n2 where Σ• is for at least one of χ1 or χ2 being non-principal. Then by (20), 2 R(n) R(n1)R(n2) (n1,n2) Na(x) = y 1  − n  n1n2 φ((n1,n2)) − a y n x n1 x n2 x X≤ X≤ X≤ X≤   • + c(χ1)c(χ2)χ1χ2(a)

n1 x n2 x a y χ mod n ≤ ≤ ≤ 1 1 X X X χ2 modX n2 + O(x2 log2 x)+ O (xB(x,y))

= yΣ1 +Σ2 + E where E = O(x2 log2 x)+ O x2y exp( c √log x) if y > exp(3.42√log x). − 1 3.2. Estimation of Σ1. Let d = (n1,n2), n1 = dk 1, and n2 = dk2, we may rewrite Σ1 as

R(n1)R(n2) (n1,n2) Σ1 = 1 n1n2 φ((n1,n2)) − n1 x n2 x X≤ X≤ R(dk1)R(dk2) 1 1 = 2 x x k1k2 dφ(d) − d d x k1 k2 X≤ X≤ d X≤ d (k1,k2)=1

φ(φ(dk1))φ(φ(dk2)) 1 1 2 ≥ x x k1k2 dφ(d) − d d √x k1 k2 X≤ X≤ d X≤ d (k1,k2)=1

φ(φ(k1))φ(φ(k2)) 1 1 2 ≥ x x k1k2 dφ(d) − d d √x k1 k2 X≤ X≤ d X≤ d (k1,k2)=1 As treated in [8, Theorem 2.3], we apply the normal order result for ω(φ(n)) in [3], and the existence of limiting density function of φ(n)/n in [16]. By [3], the set of numbers n which has at most (log log n)2 distinct prime factors has asymptotic density 1. Let B be the set of k x such that φ(k )/k 2/3, 1 1 ≤ d 1 1 ≥ x and B2 be the set of k2 d such that φ(k2)/k2 ǫ2 > 0 which has asymptotic density 2/3. Then there x≤ ≥ exists a set B2 of k2 d of asymptotic density at least 1/3 satisfying the three conditions: ≤ 2 1. k2 has at most (log log k2) distinct prime factors. 2. (k1, k2)f = 1. 3. φ(k )/k ǫ > 0. 2 2 ≥ 2 Summing over B2, it follows that φ(φ(k )) x f 2 . x k2 ≫ d log log log x k2 X≤ d (k1,k2)=1

The remaining sum over k , we restrict the sum over k B . Then we have a set B of k x of 1 1 ∈ 1 1 1 ≤ d asymptotic density ǫ1 > 0 satisfying the conditions: 2 1. k1 has at most (log log k1) distinct prime factors. f 2. φ(k )/k 2/3. 1 1 ≥ Summing over B1, it follows that φ(φ(k )) x f 1 . x k1 ≫ d log log log x k1 X≤ d Thus, we obtain x2 1 1 1 x2 Σ . 1 ≫ (log log log x)2 d2 dφ(d) − d2 ≫ (log log log x)2 d √x X≤

3.3. Estimation of Σ2. Recall that

• Σ2 = c(χ1)c(χ2)χ1χ2(a)

n1 x n2 x a y χ mod n ≤ ≤ ≤ 1 1 X X X χ2 modX n2 where Σ• is for at least one of χ1 or χ2 is non-principal. Suppose that one of the characters, say χ1 = χ0,n1 is principal. Then χ2 is non-principal and induced by a non-principal primitive character. Thus, as in [10], the contribution in this case is

∗ µ(d) c(χ2χ0,km1 ) χ0,n1 χ2(a) ≤ | | x x x y n1 x d x k m1 m2 χ2 mod k a X≤ X≤ X≤ d X≤ dk ≤Xdkm1 X X≤ d rad(m1) k (m2,k)=1 |

x ∗ µ(d) c(χ2χ0,km1 ) χ0,n1 χ2(a) . ≤ | | x dkm1 y n1 x d x k rad(m1) k χ2 mod k a X≤ X≤ X≤ d X | X X≤ d

By Polya-Vinogradov inequality for imprimitive characters (see [12, p. 307]), H¨older inequality, and the large sieve inequality, we have

∗ x w(n1) x y c(χ2χ0,k2m2 ) χ0,n1 χ2(a) 2 S2 , x dkm1 y ≪ d d k rad(m1) k χ2 mod k a X≤ d X | X X≤ d

where S2(w, z) satisﬁes the same set of inequalities (Lemma 2.2 and 2.3) as S(w, z). Here, we regard χ1χ2 as an imprimitive character induced by the primitive character χ2. The contribution of this case is x y 2w(n1) µ(d) S , ≪ | | 2 d d n1 x d x X≤ X≤ y x Since S2 satisﬁes Lemma 2.2, the contribution of d > exp 3.419906 log d is xy (x log x) µ(d) exp( c plog x) ≪ | |dα − 1 d x X≤ p 2 for some α > 1 and c1 > 0 if y > exp(3.42√log x). The contribution is O(x y exp( c1√log x)). For y x 2 − exp 3.419906 log , Lemma 2.3 for S2 implies that the contribution is O(x y exp( c1√log x)) if d ≤ d2 − y > exp(3.42√log xq). Let χ1 and χ2 are both non-principal. As in [10], we begin with

∗∗ (21) Σ2 = c(χ1χ0,n1 )c(χ2χ0,n2 ) χ1χ2(a)χ0,n1 (a)χ0,n2 (a)

n1 x k1 n1 χ1 mod k1 a y X≤ X| X X≤ n2 x k2 n2 χ2 mod k2 ≤ | ∗∗ (22) = c(χ1χ0,n1 )c(χ2χ0,n2 ) χ1χ2(d)µ(d) χ1χ2(a) y n1 x k1 n1 χ1 mod k1 d n1n2 a X≤ X| X |X X≤ d n2 x k2 n2 χ2 mod k2 ≤ | where ∗∗ is for the pairs (χ1,χ2) with both χi are non-principal primitive. We use (21) for the case χ1χ2 is principal, use (22) otherwise. P Suppose that χ1χ2 is principal. In this case, k1 = k2 since χi are primitive, and χ2 = χ1. Then we have ord(χ ) ( 1) 1 R(n1) − if χ1χ0,n1 is elementary modulo n1, φ(n1)ρn1 (ord(χ1)) c(χ1χ0,n1 )= (0 otherwise, and ord(χ ) ( 1) 1 R(n2) − if χ2χ0,n2 is elementary modulo n2, φ(n2)ρn2 (ord(χ1)) c(χ2χ0,n2 )= c(χ1χ0,n2 )= (0 otherwise.

ord(χ1) Since the terms ( 1) cancel out, we see that c(χ1χ0,n1 )c(χ1χ0,n2 ) 0. Thus, the contribution of this case is − ≥ ∗ c(χ χ )c(χ χ ) χ (a) 0. 1 0,n1 1 0,n2 0,n1n2 ≥ n1 x k1 n1 χ1 mod k1 a y X≤ X| X X≤ n2 x k1 n2 ≤ | Suppose now that χ1χ2 is non-principal. Since µ(d) = 0 only when d is square-free, we may use d = d1d2 2 6 with d1 n1 and d2 n2. Moreover, d x and we can impose (d, k1k2) = 1 because of χ1χ2(d). Thus, as in [10], | | ≤

∗∗ Σ µ(d) c(χ χ )c(χ χ ) χ χ (a) . 2 1 0,d1k1m1n1 2 0,d2k2m2n2 1 2 | |≤ 2 | | x x x | | y d x k1 m1 n1 χ1 mod k1 a X≤ X≤ d1 X≤ d1k1 ≤Xd1k1m1 X X≤ d d=d1d2 x χ2 mod k2 k2 rad(m1) k1 (n1,k1)=1 ≤ d2 x| x m2 n2 (d,k1k2)=1 ≤ d2k2 ≤ d2k2m2 rad(m2) k2 (n2,k2)=1 | By [10, Proposition 2, 3],

1 ∗∗ F (k1, k2, z)= c(χ1χ0,k1m1 )c(χ2χ0,k2m2 ) χ1χ2(a) , m1m2 rad(m1) k1 χ1 mod k1 a z X | χ Xmod k X≤ rad(m2) k2 2 2 |

1 ∗ T (w1, w2, z)= F (k1, k2, z)= c(χ1χ0,k1m1 )T2(χ1, w2, z), m1 k1 w1 k1 w1 rad(m1) k1 χ1 mod k1 k2X≤w2 X≤ X | X ≤ 1 1 S(w1, w2, z)= w1w2 F (k1, k2, z)= w1 S2(k1, w2, z). k1k2 k1 k1 w1 k1 w1 ≤ ≤ k2Xw2 X ≤ By partial summation applied two times, we have w1 1 w2 1 S(w , w , z)= T (w , w , z)+ w T (u , w , z)du + w T (w , u , z)du 1 2 1 2 1 u2 1 2 1 2 u2 1 2 2 Z1 1 Z1 2 w1 w2 1 + w w T (u , u , z)du du . 1 2 u2u2 1 2 2 1 Z1 Z1 1 2 By Polya-Vinogradov inequality,

3 3 2 2 log w1 log w2 (23) T (w1, w2, z) w1 w2 exp 3 + 3 . ≪ slog log w1 slog log w2 !

3 If w = w w z 2 then 1 2 ≤

3 log w (24) T (w1, w2, z) wz 4 exp 6 . ≪ slog log w !

3 3 3 If w > z 2 then one of w1, or w2 is greater than z 4 , say w2 > z 4 . By H¨older inequality, we have 2r 2r 1 (25) T (χ , w , z) A − B, 2 1 2 ≤ where

1 ∗ 2r 2r−1 (26) A = c¯(χ2χ0,k2m2 ) , m2 k2 w2 χ (mod k ) ≤ 2 2 rad(Xm2) k2 X | and 2r 2r 1 ∗ k2 ∗ (27) B = χ1χ2(a) = χ1χ2(a) . m2 φ(k2) k2 w2 χ2(mod k2) a z k2 w2 χ2(mod k2) a z ≤ ≤ ≤ ≤ rad(Xm2) k2 X X X X X |

Then by 0 c¯(χ χ ) 1, ≤ 2 0,k2m2 ≤

log w2 (28) A w2 exp 3 . ≪ slog log w2 ! By large sieve inequality,

2 r 2 2 r 2 (29) B (w + z ) τ ′ (a)χ (a) (w + z ) τ ′ (a) . ≪ 2 | r 1 | ≤ 2 r a zr a zr X≤ X≤ 3 1 3 2r If w2 > z 4 , then w2 > z 20 . We apply the following lemma [6, Corollary 2.2]: Lemma 3.1. Let c> 0. If z 1 and r 1 c log z, then ≥ − ≤ r 1 r 2 (1 + c) − r 1 (30) (τ ′ (a)) z log − z , r ≤ (r 1)! a zr X≤ − with 2 log w 2 z = exp(K log w ), r = 2 , c = . 2 log z K2 By Stirling’s formula, we obtain that p

1 (31) B 2r z exp log w (f(K)+ ǫ) . ≪ 2 Therefore, p

1 1 17 (32) T (χ , w , z) w − 2r z exp log w (f(K)+ ǫ) w z 20 exp log w (f(K)+ ǫ) . 2 1 2 ≪ 2 2 ≪ 2 2 This yields p p

4 3 Lemma 3.2. (1) If exp 4.87√log w2 < z < w2 , then

17 (33) T (χ , w , z) w z 20 exp log w (f(4.87) + ǫ) . 2 1 2 ≪ 2 2 (2) If exp 3.419906√log w < z exp 20√log w , thenp 2 ≤ 2 3 (34) T (χ , w , z) w z 4 exp log w (f(3.419906) + ǫ) . 2 1 2 ≪ 2 2 By [11, Theorem 1], p

Lemma 3.3. (1) If z > exp 4.87√log w2 , then 17 log w1 (35) T (w1, w2, z) wz 20 exp log w2 (f(4.87) + ǫ) + 3 . ≪ slog log w1 ! p (2) If exp 3.419906√log w < z exp 20√log w , then 2 ≤ 2 3 log w1 (36) T (w1, w2, z) wz 4 exp log w2 (f(3.419906) + ǫ) + 3 . ≪ slog log w1 ! p Finally, we have the estimate for S(w1, w2, z):

Lemma 3.4. (1) If z > exp 4.87√log w2 , then 17 log w1 (37) S(w1, w2, z) wz 20 exp log w2 (f(4.87) + ǫ) + 3 . ≪ slog log w1 ! p (2) If exp 3.419906√log w2 < z exp 20√log w2 , then (38) ≤ 3 log w1 2 log w1 9 S(w1, w2, z) wz 4 exp log w2 (f(3.419906) + ǫ) + 3 +w(log w) exp 3 z 10 . ≪ slog log w1 ! slog log w1 ! p By [11, Theorem 1] and partial summation, Lemma 3.5. log w (39) S(w1, w2, z) wz exp 6 . ≪ slog log w !

Thus, the contribution of non-principal primitive χi’s with χ1χ2 being non-principal, is x x y µ(d) S , , . ≪ 2 | | d1 d2 d dXx d=≤d1d2 By Lemma 3.4, the contribution of y > exp 3.419906 log x is d d2 q log x x2y exp 6 µ(d) τ(d) exp( c log x) ≪ log log x | | dα − 1 s ! d x2 X≤ p for some α > 1 and c > 0 if y > exp(3.42√log x). The contribution is O x2y exp c √log x . By 1 − 1 y x 2 Lemma 3.5, the contribution of exp 3.419906 log is also O x y exp c1√log x provided that d ≤ d2 − y > exp(3.42√log x). By the estimate of Σ1, Theoremq 1.1 follows.

3.4. Proof of Theorem 1.2. This is an easy consequence of [13, Theorem 1.1(i)] and partial summation. In fact, [13, Theorem 1.1(i)] states that:

Theorem 3.1 ((P. Pollack)). 3 x2 φ(φ(n)) . ∼ π2eγ log log log x n x X≤ Note that R(n) φ(φ(n)). Thus, ≥ R(n) φ(φ(n)) . n ≥ n n x n x X≤ X≤

Let A(t)= n t φ(φ(n)). Then by partial summation, ≤ P φ(φ(n)) 1 x A(t) 6 x = A(x)+ dt . n x t2 ∼ π2eγ log log log x n x 1 X≤ Z − This completes the proof of Theorem 1.2. Indeed, this provides an alternative proof of [8, Theorem 2.3] 6 with explicit constant π2eγ 0.341326. Theorem 3.2 and its proof by P. Pollack will be greatly helpful toward the proof of Theorem≈ 1.3.

3.5. Proof of Theorem 1.3. By Theorem 1.1, it is enough to consider Σ1:

R(n1)R(n2) (n1,n2) R(dk1)R(dk2) 1 1 (40) Σ1 = 1 = 2 . n1n2 φ((n1,n2)) − x x k1k2 dφ(d) − d n1 x n2 x d x k1 k2 X≤ X≤ X≤ X≤ d X≤ d (k1,k2)=1

Since R(n ) x, we have i ≤

R(dk1)R(dk2) 1 1 1 3 2 2 2 2 (x log x) x (log x) . x x k1k2 dφ(d) − d ≪ dφ(d) ≪ √x d x k1 k2 √x d x X≤ ≤ X≤ d X≤ d X≤ ≤ (k1,k2)=1

Thus, we may consider only

R(dk1)R(dk2) 1 1 φ(φ(dk1))φ(φ(dk2)) 1 1 2 2 . x x k1k2 dφ(d) − d ≥ x x k1k2 dφ(d) − d d √x k1 k2 d √x k1 k2 X≤ X≤ d X≤ d X≤ X≤ d X≤ d (k1,k2)=1 (k1,k2)=1 Then

φ(φ(dk1))φ(φ(dk2)) 1 1 2 x x k1k2 dφ(d) − d d √x k1 k2 X≤ X≤ d X≤ d (k1,k2)=1

1 1 µ(k) φ(φ(dkk1))φ(φ(dkk2)) = 2 2 dφ(d) − d x k x x k1k2 d √x k k1 k2 X≤ X≤ d X≤ dk X≤ dk 2 1 1 µ(k) φ(φ(dkj)) = dφ(d) − d2 k2  j  d √x k x j x X≤ X≤ d X≤ dk   2 1 1 µ(k) φ(φ(dkj)) 7 2 = + O(x 4 (log x) ). dφ(d) − d2 k2  j  d √x 1 j x X≤ kXx 4 X≤ dk ≤   Let A be the set of n x of asymptotic density 1 with the property: ≤ φ(n) (41) eγ log log log n, along n in A. φ(φ(n)) ∼ →∞ Let B be the complement of A in Z [1,x]. In the proof of Theorem 3.2, P. Pollack proved that ∩ x2 (42) φ(φ(n)) = o . log log log x n x, n B ≤X∈ 3 It follows that uniformly for 1 d x 4 , ≤ ≤ Corollary 3.1. φ(dn) x2 3x2 1 x2 φ(φ(dn)) = + o = + o . eγ log log log dn log log log x dπ2eγ log log log x 1+ 1 log log log x n x n x p d p X≤ d X≤ d Y| Therefore, we now have 1 1 µ(k) 36x2 1 x2 Σ1 2 2 2 2 4 2γ 2 2 + o 2 ≥ dφ(d) − d k d k π e (log log log x) 1 (log log log x) 1 p dk d √x k x 4 1+ p X≤ X≤ Y| x2 = (C + o(1)) (log log log x)2 where 2 1 36 1 − 1 1 − 36 1 C = 1+ 1+ 1 1 = 1+ 1 . π4e2γ p p5 − p − π4e2γ p5 + p4 p3 p2 − p ! ! p ! Y Y − − This completes the proof of Theorem 1.3.

4. Remarks 1. We observed that there is a significant difference in the problems involving prime moduli and composite moduli. This difference becomes evident in the normal order results which is mainly due to the following: For prime moduli, (p,q)=1if p and q are distinct primes, but for composite moduli, (m,n) is not necessary equal to 1 for distinct m and n. 2. The classical large sieve inequality (see [5, Theorem 7.13]) is as follows: 2 q ∗ a χ(n) (N + Q2) a 2. φ(q) n ≤ | n| q Q χ mod q n N n N X≤ X X≤ X≤

The key to bringing down the constant 4.8365 to 3.42 in [6], is by applying the large sieve inequality of the following form: For any Dirichlet character χ1, we have 2 q ∗ a χχ (n) (N + Q2) a χ (n) 2 (N + Q2) a 2. φ(q) n 1 ≤ | n 1 | ≤ | n| q Q χ mod q n N n N n N X≤ X X≤ X≤ X≤

This allows us to treat the character double sum one by one rather than combining two characters into one as done in [14], [15], and [6]. Consequently, the constant 4.8365 in [6] can be now improved to 3.42. We summarize those improvement of results in [6]: Theorem 4.1. If y > exp(3.42√log x), then for any D,E > 0, the following statements hold: 1 x (43) Pa(x)= Aπ(x)+ O , y logD x a y X≤ 1 x2 (44) (P (x) Aπ(x))2 , y a − ≪ E a y log x X≤ 1 where A = p 1 p(p 1) is the Artin’s constant, and − − Q 1 ℓ (p) x (45) a = CLi(x)+ O , y p 1 logD x a y p x X≤ X≤ − 2 1 ℓ (p) x2 (46) a CLi(x) , y p 1 − ≪ E a y p

1 x2 (48)  1 CLi(x) , y2 − ≪ logE x a y b y  p x  X≤ X≤  p Xa≤n b   | −  for some n    where C = 1 p is the Stephens’ constant. p − p3 1 3. Recall the function− f deﬁned by: Q 1 K2 f(K)= log + 1 + 1 . K 2 As in [6], the number 3.42 in Theorem 1.1 and 1.3, also 4.1 can be replaced by any number greater than ρ where ρ 3.4199057 is the unique positive root of the equation K = f(K). 1 1 ≈ 4 5. Further Developments In [7], the author proved the following: 1 δ There exists δ > 0 such that if x − = o(y), then 1 x2 log log x (49) ℓ (n)= exp B (1 + o(1)) y a log x log log log x a exp(3.42√log x). References

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