| The Waring-Goldbach problem is concerned with the representation of posi-tive integers by sums of powers of primes, that is, N= p1k+p2k+…+psk where p1,...,ps are primes. The most optimistic conjecture states that the above equation is solvable for s≥ k+1, which is Goldbach Conjecture when k= 1. This conjecture is out of reach at present. Denote by H(k) the minimum of s for which the above equation is solvable. In 1937, Vinogradov [66] proved that H(1)≤ 3 holds for every sufficiently large odd positive integer N. In 2013, Helfgott [18,19] showed that H(1)< 3 for each positive odd integer N≥ 9. Hua [20] considered the nonlinear cases and proved that H(k)≤ 2k+1 for all k≥ 1. This is still the best result for k< 3. The exceptional sets of this problem have been investigated in [20]. Let Ek,s(N) be the number of integers up to N satisfying some local congruent conditions that can not be represented as sums of skth power of primes. Hua proved that E3,s(N) 《 NL-A with 5≤ s≤ 8 for any A> 0. When s= 5, this result was improved by Ren [57] to E3,5(N) 《N152/153+ε. Kawada, Kumchev and Wooley also made contributions to the improvement of E3,s (See [31,32,70]). In 2014, Zhao [72] obtained the sharpest results so far that p3,5=1/12-ε, p3,6=1/4-ε, p3,7=1/2-ε, P3,8=5/6-ε For k= 4, Davenport [10] concluded that H(4)≤ 15. In 2000, Kawada and Wooley [30] established that H(4)≤ 14. In 2014, Zhao [72] improved this result to H(A)≤ 13. The exceptional set about this problem was considered by Kumchev [32] and the results stated that E4,S(N) 《N1-p4,s with p4,10=3887/56832-ε p4,11= 14543/56832-ε P4,12=15727/56832-ε.When k≥ 5, the optimal results of H(k) are the following:H(5)≤ 21 (Kawada and Wooley [30]), H(6)≤ 32 (Zhao [72]), H(7)≤ 46 (Kumchev [33]), H(8)≤ 63 (Thanigasalam [62]).In addition, many researchers are interested in Waring-Goldbach problems with the variables taken values in small intervals: withθk,s ∈ (0,1/k).For this problem, the existing results are focused on the cases of k≤ 3. When k= 1, Haselgrove [15] firstly proved that (0.3) is solvable whenθ1,3=1/64. Later, many researchers studied this problem (See [8,22,23,24,25,26,27,28,53,55, 71]). The best result at present is due to Baker and Harman [1], who proved that θ1,3=3/7When k= 2, (0.3) was considered firstly by Liu and Zhan [40]. Bauer and Wang [2,3,4], Liu, Lii and Zhan [39,42,43,46,47] also investigated this problem and gave improvements. In 2012, Kumchev and Li [35] proved the sharpest result θ2,5＝1/18-ε.When k= 3, the first breakthrough was made by Meng [50], she established that 03,9=1/198-ε is admissible under the Generalized Riemann Hypothesis. Sub-sequently, Lu and Xu [49] showed the same result unconditionally. In 2012, Li [36] improved this result toθ3,9=1/90-ε. For 4≤ k≤ 10, Sun and Tang [58] proved that θk,2k+1=1/2k(k-1)22k-2+2k-ε.In the first part, we consider (0.3) with ≤k≤10. The main result is the following. Theorem 1 Let 3 ≤k≤ 10. Set Then each sufficiently large positive integer N satisfying some local condition can 6e writte,asIn Theorem 1,we get θ3,9=1/51-ε which improved Li’s result.When 4≤k≤ 10,we refine the results of Sun and Tang.Now we investigate the exceptional sets of(0.3)withk=3 and k=4.For this.we define Ik,s(N,Y)=[N-Y,N+Y]. Let Ek,s(N,Y)denote the number of∈Ik,s,(N,Y)satisfying some local congruent conditions that can not be represented as(0.3),where 2k-1+1≤s≤2k.One needs to show that for arbitrary ε>o,there exists a large θk,s∈(0,1/k),such that Ek,s(N,ks(N/s)I-θk,s)《N1-θk,s-ε. (0.4)Fork=3,Liu and Sun[45]considered this problem and established that θ3,s=s-4/6(s+12)with 5≤s≤8.Wang[64]improved this results toθ3,s=s-4/15s.In 2012,Li[36]proved thatθ3,5=1/48,θ3,6=1/36,θ3,7=1/33.Our consequences are the following:Theorem 2 Lefk=3,5≤s≤8.Then (0.4)D.holds for θ3,5=7/261-2ε,θ3,6=5/159-ε,θ3,7=11/333-ε,θ3,8=19/561-ε.Whenk=4,the only results were due to Tang and Zhao[59],who deduced that θ4,s=s-8/8(s+88) for 9≤s≤13.In fact,one may expect that the least number s such that(0.3)is true for all sufficiently large N is 13.But this can not be established because the diminishing of H(4)depends largely on mean-value estimate developed by Thanigasalam[60,61]which however can not be used in almost equal variable problems like(0.3).Sun and Tang[58]proved that(0.3)is solvable fork=4 and s=17.We consider the exceptional set with 9 ≤s≤16 and prove the following:Theorem 3 Let k=4,9≤s≤16.Then(0.4) holds forThe distribution of primes related to integral vectors in sphere is investigated in this paper. The problem of integral points in sphere is important in analytic number theory. Vinogradov [68] and Chen [7] established independently that Afterwards, the exponent of x in the error term was refined by Chamizo and Iwaniec [6] to 29/44. Heath-Brown [17] further improved this result to 21/32. Friedlander and Iwaniec [12] considered this problem related to primes and proved that Guo and Zhai [14] showed that for any A> 0, where C3 and I3 are the singular series and singular integral. It follows from the above result that Calderon and Velasco [5] studied sums of the divisor of the ternary quadratic form and concluded that Afterwards, Guo and Zhai [14] improved this result to S(x)= 2C1I1x3 logx+(C1I2+C2I1)x3+O(x8/3+ε), where Ci,Ii, (I= 1,2) are constants. Zhao [73] refined the error term of the above formula to x2 log7 x.In the second part of this paper, we firstly consider (0.5) with almost equal variables, that is, where y= xθ with θ∈ (0,1]. For this problem, our purpose is finding a small value of θ to such that S (x, y) has an asymptotic formula. The following are our results.Theorem 4 Assume that θ≥1/2+2ε. Then we have the asymptotic formula whereTheorem 5 Suppose that θ≥6/7+ε. Then we have We next consider the problem related to primes: with y= xδ (0< δ≤ 1). We will establish the following:Theorem 6 Suppose that δ≥26/35+2ε. For any A> 0, π∧(x,y)=8(?)y3＋O(y3L-A), where (?) is the singular series defined as in (1.6).Theorem 7 Suppose thaty= xδ with δ satisfying 26/35+2ε≤ 6≤δ≤ 1. Define For any A> 0, we have where... |
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