A Note on Cube-Full Numbers in Arithmetic Progression

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Introduction and Main Results
Let k > 1 be a fixed integer and n be a positive integer. We call n a powerful number (or k-full number) if n � 1 or for a prime p dividing n, p k also divides n. Let P k denote the set of powerful numbers. Suppose k � 2, 3, and this defines square-full numbers and cube-full numbers, respectively. Erdo .. s and Szerkeres [1] first introduced powerful numbers and gave n≤x,n∈P k where c k,m are effective constants and Δ k (x) ≪ x (1/(k+1)) . From then on, many authors have studied the powerful numbers and got a lot of relevant conclusions (see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein). In 2013, Liu and Zhang [19] investigated the distribution of square-full numbers in arithmetic progressions and got an asymptotic formula n≤x,n∈P 2 n≡l(mod q) 1 � α(l, q)x (1/2) + O q (49/141)+ε x (19/47) under the condition of (q, l) � 1. By utilizing the method of exponent pairs, Srichan [20] then obtained n≤x,n∈P 2 n≡l(mod q) 1 � where the error terms had been corrected by Watt [MR3265055]. Recently, Chan [21] got a new asymptotic formula n≤x,n∈P 2 n≡l(mod q) which improved his own result with Tsang [22]. As a critical step, he [21] mainly dealt with a sum in the form of by following closely Montgomery and Vaughan's construction [23]. It is somewhat similar to Dirichlet's hyperbola method shown in Figure 1.
Actually, they divided the above sum into four parts as shown in Figure 2 and then discussed them separately.
Motivated by this idea, we turn to discuss the following sum with three parameters: By extending the construction from Montgomery and Vaughan [23], the summation (5) is divided into eight parts as shown in Figure 3.
en, relying again on Kloosterman-type exponential sum method, an asymptotic formula of (5) is obtained. Finally, an asymptotic formula of cube-full numbers in an arithmetic progression is derived.

Some Lemmas
Before we start the proof, let us give a few lemmas which are needed later.
Proof. e first result can be found in Lemma 3 of [21]. Note that e second and third one can be proved in the same way. e proofs of the last three are slightly different. For example, by orthogonal property of additive characters, we have

Journal of Mathematics 3
where G 3 is the set of all characters χ(mod q) such that χ 3 � χ 0 , the principal character.
□ Lemma 2. For q ≥ 1 and (l, q) � 1, Proof. Let ‖x‖ be the distance from x to the nearest integer; then, we have Proof. Using the trivial estimation of the innermost sum, we have en, by orthogonal property of additive characters, we obtain Interchanging the order of summations and combining Lemma 2 and Eq. (12.48) on page 324 of [24], we have Finally, we get By definition of N 3 (n; q), N 3 (n; q) ≪ ε q ε (see Lemma 2.2 in [25]).

Lemma 4. If we define
Proof. Following much the same way as Lemma 4 of [21], we first suppose q � rs with (r, s) � 1. By the "reciprocity" formula ss + rr ≡ 1(mod q), where ss ≡ 1(mod r) and rr ≡ 1(mod s), and the additive multiplicity of exponential function e an 3 + bn 4 q � e asn 3 + bsn 4 r e arn 3 + brn 4 s , (24) with n � sx + ry, Now we just need to discuss the argument in the following cases: (I) Prime moduli q � p case. Now eorem 2 obtained by Moreno and Moreno [26], which is a special form of the Bombieri-Weil bound [27], implies provided that (( is is impossible if p > 4, by comparing the degrees of both sides of the above. If p ≤ 4, the validity of (26) can be easily checked. (II) Prime power moduli q � p β case with β > 1. Obviously, we only need to consider it with the assumption (a, b, p) � 1. Following the proofs of Lemma 12.2 and 12.3 in [24] with the equation of S(a, b; q), we obtain where g(y) � ay 7 + b y 4 ,

Journal of Mathematics
Note that g ′ (y) � ((3ay 10 − 4by 3 )/y 8 ) and h(y) � ((3ay 11 + 10by 4 )/y 10 ). Now we concentrate on the number of solutions of congruence equation β � 2α with α ≥ 1. en, the congruence equation is Relying on the properties of indices, we deduce that (32) has no solution when p > 3 and one solution when p � 3. Next, we assume (b, p) � 1. If p � 2 and 4‖a, then (32) has at most seven solutions. And if p > 3 and (a, p) � 1, (32) also has at most seven solutions. en, we have where β � 2α + 1 with α ⩾ 1. Firstly, in the same way, if (b, p) � p, by the analysis in the case β � 2α, the sum in (30) is empty unless p � 3 in which case one has In any case, we have Combining (25), (26), (33), and (34), we finally obtain which completes the proof of Lemma 4. By applying Lemma 4, we have the following.
e remaining part of the proof is similar to eorem 4 in [21].

Proof of Theorem 2
Consider three positive parameters λ, μ, and ]. By extending the construction from Montgomery and Vaughan [23] as shown in Figure 3, we have Journal of Mathematics First, we estimate T 3 .
For T 33 , we have by Lemma 3. en, we estimate T 31 and T 32 . For T 31 , we know e first term in the above formula is In order to simplify our final result, by using Euler's summation formula which can be found in eorem 3.2 in [28], the constant ∞ b�1 (1/b (4/3) ) can be rewritten as us, we obtain the asymptotic formula of T 31 as Next, we deal with T 32 . 8

Journal of Mathematics
If we let F c (μ) � a≤μ N 4 (lac 3 ; q) − ((ϕ(q))/q)μ, then the first term in the above formula is − x (1/4) q c≤x ] N 4 lc 3 ; q − ((ϕ(q))/q) And the second term is So, we can get And for T 34 , following closely Chan [21] as shown in Figure 2 instead of Dirichlet's hyperbola method shown in Figure 1, we just need to divide the interval (a, b): { a > x n , b > x m , a 3 b 4 ≤ x} in the same way. Note that the sum Journal of Mathematics 9 can be estimated with the help of asymptotic formula given at the end of page 101 in [21]; then, using Lemma 5, we can get where 2 J 1 satisfies (X μ 1 /(2 J 1 q (1/2) )) ≪ 1.
Picking λ 1 to have the same size as μ 1 and combining the previous results, we have For T 2 , in the same way as T 3 , we have

10
Journal of Mathematics And we can obtain For T 22 , again in the same way as the proof of T 3 , we have (54) If we let F b (μ) � a≤μ N 5 (la 2 b; q) − ((ϕ(q))/q)μ, then the first term in the above formula is Journal of Mathematics and the second term is en, we can obtain 12
Similar to the proofs of T 2 and T 3 , we can get the following: Journal of Mathematics 13 Finally, we discuss T 8 as follows: dividing interval Let a 0 be the intersection of the plane b � x μ , c � x ] + n((x ((1− 3λ− 4μ)/5) − x ] )/m), and curved surface a 3 b 4 c 5 � x, which is and then we obtain Similarly, we have We define the intersection of a 3 b 4 c 5 � x and x ] + n((x ((1− 3λ− 4μ)/5) − x ] )/m) as 14 Journal of Mathematics Now we divide the interval in the nth part and follow the construction in [23]; firstly, we have rectangles In the remaining regions we place additional rectangles R ijk . If we let 1 ≤ k ≤ 2 j− 1 , in the same way we place a further rectangles and so on, then we can get R ijk which is e remaining regions are In this part, expanding R i and R ijk to a cube with height ((x ((1− 3λ− 4μ)/5) − x ] )/m), we can get R ni , R nijk , S ni , S nijk S nijk ′ , and the remaining region S n correspondingly. en, we have the asymptotic formula en, we further obtain in which the main term is Now we deal with the error term of T 8 . Note that us, the area of S niJk is at most which implies that the estimation of the first error term is Area of S niJk q ≪ 1

Conflicts of Interest
e authors declare that they have no conflicts of interest.