In the Waring-Goldbach problem, one studies the representation of positive integers by powers of j primes, where j is a positive integer. One of the most famous results of Hua  in 1938 states that each sufficiently large odd integer n can be written as the sum of nine cubes of primes.
When the number of variables j is becoming smaller, such as, 5≤ j ≤ 8, we could consider the exceptional sets of these problems. Denote by Ej(N) the set of integers n ∈ 𝓐j, not exceeding N with j = 5, 6, 7, 8 such that
Let j = 5, 6, 7, 8, let 𝓐j be defined as in (2), and define θj by
Then we have
In 2014, Zhao  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 pi taking values in short intervals, i.e.
where y = o(N1/3) and pj are primes. Let Ej(N, y) denote the number of integers that
Theorem 1.2 is proved by the circle method. When treating the major arcs, we apply the iterative method of Liu  and a mean value theorem of Choi and Kumchev  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  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  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 , Huang  and Kumchev and Liu . There are other similar problems (see [12, 13] and their references).
- In this series of problems, we could not only focus our attention to the size of y, but also concern with the cardinality of Ej(N, y) for j = 7, 8, such as Theorem 2 in Liu and Sun . 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 , 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.
- This paper is focusing on the quantitative relation of short intervals and the exceptional sets. In the reference , 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.
- 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.
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.
2 Outline of the method
In this section, we give an outline of the proof of Theorem 1.2 (take j = 8 for example). In order to apply the circle method, for some δ > 0, we set
for some integers a, q with 1 ≤ a ≤ q ≤ Q∗ and (a, q) = 1. Denote by 𝓜(a, q) the set of α satisfying (6), and define the major arcs 𝓜 as follows:
Again by Dirichlet’s lemma, each number α ∈ [1/Q∗, 1+1/Q∗]∖𝓜 can be written as
with (a, q) = 1, 1 ≤ a ≤ q ≤ Q. Define the minor arcs C(𝓜) to be the set of α ∈ [1/Q∗, 1+1/Q∗]∖ 𝓜 satisfying (8) with P ≤ q ≤ Q.
Obviously, 𝓜 and C(𝓜) are disjoint. Let 𝓡 be the complement of 𝓜 and C(𝓜) in [1/Q∗, 1+1/Q∗], so that
Let N be a sufficiently large integer and n ∈ 𝓐8 satisfying
where e(t) = e2πit and
Then we can write
Clearly, in order to prove Theorem 1.2, it is sufficient to show that r(n) > 0 for almost all integers n ∈ 𝓐8 ∩ [N,N+N2/3y]. The tools that we need are an estimate for exponential sums over primes in short intervals of Liu, Lü and Zhan , Kumchev  and a mean value theorem of Choi and Kumchev , which are stated as follows.
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.
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.
Let k ≥ 3 and θ be a real number with (2k+2)/(2k+3) < θ ≤ 1. Suppose that 0 < ρ ≤ ρk(θ), where
with σk defined by
Therefore, by Lemma 2.3 for α ∈ C(𝓜), we have
in view of y = xθ, 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
For α ∈ 𝓡1, we have
Lemma 2.1 gives, for α ∈ 𝓡1,
If α ∈ 𝓡2, then
By Lemma 2.1,
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.
Let 𝓜 be defined as in (7). Then for any sufficiently large n ∈ 𝓐8 ∩ [N,N+N2/3y], we have
where C8 is a positive constant, φ(q) is the Euler function and
Let k be a positive integer, X ≥ Y ≥ 2 and
Then for any ε > 0 and 1 ≤ s ≤ k, we have
This is Lemma 4.1 in Li and Wu . □
Introduce the function
Clearly, we have
and write Card(
By Proposition 2.5, we have
And Lemma 2.6 implies
If ρ is such that y2 ≫ x2−ρ+ε, this leads to the bound
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. □
3 Preliminaries for Proposition 2.5
For χ mod q, define
If χ1, χ2,…, χ8 are characters mod q, then we write
The following lemma is important for proving Proposition 2.5.
Let χi mod ri with i = 1,…, 8 be primitive characters, r0 = [r1,…, r8], and χ0 be the principal character mod q. Then
It is similar to that of Lemma 7 in , so we omit the details. □
Recall the definition of x, y as in (4), and define
where δχ = 1 or 0 according as χ is principal or not. We also set
Let P∗, Q∗ be as in (2.2). We have
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.
4 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
We will prove that I0 produces the main term, and the other Ik (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 Ik (1 ≤ k ≤ 8). We shall only treat I8, the most complicated one. The treatment for Ik (1 ≤ k ≤ 7) are similar.
for any fixed A > 0.
Following a similar procedure to treat I8, we can show that
5 Estimation of K(d)
The proofs of Lemmas 3.2 and 3.3 are rather similar to those of Proposition 2.2 in . In order to use Choi and Kumchev’s mean value theorem effectively, we need a preliminary lemma in  as follows.
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.
Then we have
The contribution of O(N1/6(r/Q*)1/2) to the left-hand side of (29 ) is
for any R ≤ P*.
By Gallagher’s lemma (, Lemma 1), we have
If R = 1, we have
which implies, in view of Q* < x2y,
For R ≥ 2 and r ∼ R, we have δχ = 0. Thus, we can apply (37) to write
6 Estimation of J(d)
Estimation of J(d). Replacing W(χ, λ) by W͠(χ, λ) as in §5, we get that the resulting error is
Hence, Lemma 3.3 is a consequence of the estimate
where R ≤ P* and c > 0 is some constant.
The case R < 1 contributes to d−3+εyL which is obviously acceptable. For R ≥ 1, we have δχ = 0. Thus,
By partial summation and Perron’s summation formula, we get
where 0 < b < L−1 and T = (1 + | λ | N)yL2. Using Lemmas 4.3, 4.5 of  and a trivial estimate, we have
Then for b → 0, W͠(χ,λ) is bounded by
Thus, it suffices to show that the estimates
holds for R ≤ P* and 0 < T1 ≤ T̂;
holds for R ≤ P* and T̂ < T2 ≤ T*; and
holds for R ≤ P* and T* < T3 ≤ T.
Estimation of J(1). The result is the same as that of J(d) except for the saving of L−A on its right hand side. To order to get this saving, we have to distinguish two cases LC < R ≤ P and R ≤ LC, where C is a constant depending on A. The proof of the first case is the same as that of J(d), so we omit the details.
Now we prove the second case R ≤ LC. We use the well-known explicit formula
where ρ = β + iγ is a non-trivial zero of the function L(s, χ), and 2 ≤ T ≤ u is a parameter. Then by inserting (44) into Ŵ(χ, λ), and applying partial summation formula, we get
Now let η(T) = c2 log−4/5T. By Prachar ,
provided that Q* = N31/36+2ε. Then the lemma follows.
This work is supported by the National Natural Science Foundation of China (Grant No. 11371122, 11471112). The author would like to express thanks to the referees for reading the manuscript carefully and giving suggestions professionally. Theorem 1.2 is actually improved by the valuable advices. The author also express thanks to Doctor Ge Wenxu, because of the helpful discussion during writing the paper.
Hua L.K., Some results in the additive prime number theory, Quart. J. Math. (Oxford) 1938, 9, 68-80.
Kumchev A.V., On the Waring-Goldbach problem: exceptionalal sets for cubes and higher powers, Cana. J. Math., 2005, Vol. 57 (2), 298-327.
Liu J.Y., On Lagrange’s theorem with prime varibles, Quart. J. Math. (Oxford) 2003, 54, 454-462.
Choi S.K.K., Kumchev A.V., Mean valus of Dirichlet polynimials and applications to linear equations with prime varibles, Acta Arith. 2006, 123, 125-142.
Kumchev A.V., On Weyl sums over primes in short intervals,"Number Theory: Arithmetic in Shangri-La", World Scientific, 2013, 116-131.
Wei B., Wooley T.D., On sums of powers of almost equal primes, Proc. London Math. Soc., 2015, 111 (3), 1130-1162.
Huang B.R., Exponential sums over primes in short intervals and an application to the Waring-Goldbach Problem, Mathematika, 2016, 62, 508-523.
Lü G.S., On sums of nine almost equal prime cubes, Acta Math. Sin., 2006, 49, 195-204 (in Chinese).
Vaughan R.C., The Hardy-Littlewood Method, 2nd edition, Cambridge University Press, 1997.
Liu J.Y., Lü G.S., Zhan T., Exceptional sums over prime varibles in short intervals, Sci. China Ser. 2006, A 49, no. 4, 448-457.
Titchmarsh E.C., The Theory of the Riemann zeta-function. New York: Oxford University press, 1986.
Prachar K., Primzahlverteilung. Springer, Berlin, 1957.
Davenport H., Multiplicative Number Theory. Springer, Berlin, 1980.
Huxley M.N., Large values of Dirichlet polynomials III, Acta Arith., 1974, 26, 435-444.