Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem

Theorem 1.1

Let j = 5, 6, 7, 8, let 𝓐_j be defined as in (2), and define θ_j by

Then we have

In 2014, Zhao [4] improved the above to θ₇ = 1/2+ε, θ₈ = 1/6+ε for j = 7, 8. In this paper, if j = 7,8, we investigate this problem with p_i taking values in short intervals, i.e.

where y = o(N^1/3) and p_j are primes. Let E_j(N, y) denote the number of integers that N∈Aj,N≤n≤N+N23y, which cannot be represented as in (3). In the case of j = 7, 8, our result is stated as follows.

Theorem 1.2

For y=N13−δwith0<δ≤190, we have E8(N,y)≪N29−23θ21,E7(N,y)≪N29−16θ21, where 0 < θ ≤ 1, δ and θ satisfy 3δ+θ = 1.

Theorem 1.2 is proved by the circle method. When treating the major arcs, we apply the iterative method of Liu [5] and a mean value theorem of Choi and Kumchev [6] to establish the asymptotic formula. It is known that the estimation of exponential sums plays an essential role in this problem. The upper bound of the exponential sums leads to the final result directly. So the new estimate for exceptional sums over primes in short interval in Kumchev [7] plays an important role in treating the minor arcs.

When j = 4, it is a conjecture, which is out of reach at present. For j = 9, Lü and Xu [8] proved that (3) holds unconditionally with δ = 1/198, which is as strong as the result under the Generalized Riemann Hypothesis. In the direction of Waring-Goldbach problem in short intervals, there have been some developments in the last few years. Such as the recent work of Wei and Wooley [9], Huang [10] and Kumchev and Liu [11]. There are other similar problems (see [12, 13] and their references).

Remarks

1. In this series of problems, we could not only focus our attention to the size of y, but also concern with the cardinality of E_j(N, y) for j = 7, 8, such as Theorem 2 in Liu and Sun [14]. In this paper, we give the exact relation formula about the length of short intervals and the size of exceptional sets. Compared with Theorem 2 in [14], a wider range of the length of short intervals is given. At the same time, quantitative relation between size of exceptional sets and length of short intervals is obtained, i.e., 3δ+θ = 1.

2. This paper is focusing on the quantitative relation of short intervals and the exceptional sets. In the reference [15], improved results in shorter intervals are given. Compared with their fixed results for the number of prime variables, we actually obtain the wider range of short interval. Though the number of primes is different, the results are the same by the method in the paper.

3. As the number of variables is becoming smaller, the more difficult the question is. For example, when j = 4, it is a conjecture out of reach at present. Moreover, as the length of short intervals is becoming shorter, the size of exceptional sets is becoming larger. When j = 5, 6, this method does not work for this question, so we could not obtain similar results.

Notation

As usual, φ(n) and Λ(n) stand for the functions of Euler and von Mangoldt, respectively. The letter N is a large integer, and L = log N. The notation A ≍ B means that c₁A ≤ B ≤ c₂ A, r∼ R means R < r ≤ 2R. The letter ε denotes a positive constant, which is arbitrary small, but not the same at different occurrences.

Lemma 2.1

For integer k ≥ 1, let 2 < y ≤ x and α = a/q+ λ be a real number with 1 ≤ a ≤ q and (a, q) = 1. Define

Then for any fixed ε > 0, we have

where the implied constant depends on ε and k only.

Lemma 2.2

Let l be a positive integer, R ≥ 1, T ≥ 1, X ≥ 1 and κ = 1/log X. Then there is an absolute positive constant c such that

where the implied constant is absolute.

Next we bound S(α) on C(𝓜) ∪ 𝓡. We first estimate S(α) on C(𝓜), and this has been done in Kumchev [7] in which Theorem 1.2 states that

Lemma 2.3

Let k ≥ 3 and θ be a real number with (2k+2)/(2k+3) < θ ≤ 1. Suppose that 0 < ρ ≤ ρ_k(θ), where

with σ_k defined by σk−1=min(2k−1,2k(k−2)). Then, for any fixed ε > 0,

Therefore, by Lemma 2.3 for α ∈ C(𝓜), we have

in view of y = x^θ, 8/9 < θ ≤1 and our choice of P in (5).

Now we estimate S(α) on 𝓡. To this end, also by Dirichlet’s lemma on rational approximation, we further write 𝓡 = 𝓡₁ ∪ 𝓡₂, where

and

For α ∈ 𝓡₁, we have |λ|≥1qQ∗≥1N1/3y2, and therefore,

Lemma 2.1 gives, for α ∈ 𝓡₁,

If α ∈ 𝓡₂, then

By Lemma 2.1,

From (13)-(15), we get

Proposition 2.4

Let C(𝓜) and 𝓡 be defined as the above, then we have

For the major arcs, we have the following asymptotic formula, which will be proved in Section 4.

Proposition 2.5

Let 𝓜 be defined as in (7). Then for any sufficiently large n ∈ 𝓐₈ ∩ [N,N+N^2/3y], we have

where C₈ is a positive constant, φ(q) is the Euler function and

In order to prove Theorem 1.2, we also need the following lemma, which can be viewed as a generalization of Hua’s lemma ([16], Lemma 2.5) in short intervals.

Lemma 2.4

Let k be a positive integer, X ≥ Y ≥ 2 and

Then for any ε > 0 and 1 ≤ s ≤ k, we have

Proof

This is Lemma 4.1 in Li and Wu [18]. □

Proof of Theorem 1.2

To prove Theorem 1.2, we apply the method introduced by Wooley [19]. We only give detailed proof for j = 8. Denote by E8∗ (N, y) the set of integers n ∈ 𝓐₈ ∩ [N,N+N^2/3y] such that

Introduce the function

Clearly, we have

and write Card( E8∗ (N, y)) = Z

By Proposition 2.5, we have

From (17) and (18), we deduce that

where I1=∫01|S(α)|6|Z(α)|2dα,I2=∫01|S(α)|8dα. One could find that (see Lemma 6.2 in [19])

And Lemma 2.6 implies

Collecting (16), (19), (20) and noting that x = (N/8)^1/3, we obtain

If ρ is such that y² ≫ x^2−ρ+ε, this leads to the bound

which implies Z≪N29−23θ21, by the choice of ρ in Lemma 2.3.

In the case of j = 7, we obtain the following asymptotic formula on major arcs by similar argument as described in Section 4,

For j = 7, estimations on C(𝓜) ∪ 𝓡 are also similar to the case of j = 8. The other treatment is quite similar, so we omit the details. This completes the proof of Theorem 1.2. □

Lemma 3.1

Let χ_i mod r_i with i = 1,…, 8 be primitive characters, r₀ = [r₁,…, r₈], and χ⁰ be the principal character mod q. Then

Proof

It is similar to that of Lemma 7 in [20], so we omit the details. □

Recall the definition of x, y as in (4), and define

and

where δ_χ = 1 or 0 according as χ is principal or not. We also set

Define further

where the sum ∑χmodr∗ denotes summation for all primitive characters modulo r. The proof of Proposition 2.5 depends on the following two lemmas, which will be proved in Section 5.

Lemma 3.2

Let P_∗, Q^∗ be as in (2.2). We have

Lemma 3.3

Let P_∗, Q^∗ be as in (2.2). We have

Further if d = 1, the estimate can be improved to

where A > 0 is arbitrary.

Proof of Proposition 2.5

Since q ≤ P^*, we have (p, q) = 1 for p ∈ (x − y, x + y]. Using the orthogonality relation, we can write

where S₀(λ) and W(χ, λ) are as in (25). By (32), we can write

where

We will prove that I₀ produces the main term, and the other I_k (1 ≤ k ≤ 8) contribute to error term.

The computation of I₀ is standard, and we can prove

where C₈ and 𝔖₈(n) are defined in Proposition 2.5.

It remains to estimate I_k (1 ≤ k ≤ 8). We shall only treat I₈, the most complicated one. The treatment for I_k (1 ≤ k ≤ 7) are similar.

Suppose that χk∗ (mod r_k) with r_k|q being the primitive character inducing χ_k. Thus we may write χ_k = χk∗ χ⁰, where χ⁰ is the principal character modulo q, r₀ = [r₁, …, r₈]. It is easy to see that W(χ_k, λ) = W( χk∗ , λ). Since r₀ = [r₁, …, r₈] = [[r₁, …, r₇], r₈], by Lemma 3.1 and Cauchy’s inequality, we have

Now we introduce an iterative procedure to bound the above sums over r₈, …, r₁ consecutively. Since r₀ = [r₁, …, r₈] = [[r₁, …, r₇], r₈], we use (29) two times, (30) five times and (31) once to get

for any fixed A > 0.

Following a similar procedure to treat I₈, we can show that

Now the required asymptotic formula follows from (33), (34), (35) and (36).□

Lemma 5.1

Let χ be a Dirichlet character modulo r. Let 2 ≤ X < Y ≤ 2X, T₀ = (log(Y/X))⁻¹, T = X⁶ and κ = 1/log X. Define

Then we have

The implied constant is absolute.

Proof of Lemma 3.2

Introduce

Then we have

which implies

The contribution of O(N^1/6(r/Q^*)^1/2) to the left-hand side of (29 ) is

where we have used [d, r](d, r) = dr, l = (d, r), (4) and (5). Thus in order to prove (29), it suffices to show that

for any R ≤ P_*.

By Gallagher’s lemma ([21], Lemma 1), we have

where

If R = 1, we have

which implies, in view of Q^* < x²y,

For R ≥ 2 and r ∼ R, we have δ_χ = 0. Thus, we can apply (37) to write

Since

and

Therefore, the contribution of the first term of (41) to the left-hand side of (38) is

Introducing

The contribution of the second term of (41) to the left-hand side of (38) is

Finally, the contribution of the last term of (41) to the left-hand side of (38) is

Now the inequality (38) follows from (40), (42), (43) and (44). This completes the proof of Lemma 3.2.□