On Sums Of A Prime And Five Cubes Of Primes In Short Intervals

The aim of the Waring-Goldbach problem is to seek to represent natural number N which satisfy some necessary congruence conditions by sums of k-th powers of primes, i.e. N=p1k+p2k+…+psk, where s depends on k.The circle method of Hardy and Littlewood is a basic way to solve the Waring-Goldbach problem. If E3j(z) denote the number of positive integers N up to z that satisfy some congruence conditions which can’t be written as sums of j cubes of some primes, then we can get E3j(z)<<z(log z)-A, j=5,6,7,8, where A is an arbitary positive constant. The result can be found in Hua [10].In order to gain a better upper bound of the exceptional set Ej, one idea is that we assume the Generalized Riemann Hypothesis(GRH), another is the method of enlarged major arcs. For details, please see Jianya Liu and Tao Zhan [7], Montgomery and Vaughan [6], Bauer [2] and Jianya Liu and Tao Zhan [8] respectively.We concern the Waring-Goldbach problem in short intervals. In this pa-per, we will prove Theorem1. For each sufficiently large positive integer N≡0(mod2), 65/84<δ≤1and y=(N/6)θ/3.The equation is always solvable for In particular,we can take y=N5/16+εwhenδ=1.It is obvious that we can use circle method to deal with it.First,we set P=N1/21+ε,Q=N8/9+εDefine the major arcs m as m={α=a/q+λ:1≤a≤q≤p,(a,q)=1,|λ|≤1/qQ} and the minor arcs is m=[1/Q,1+1/Q]\m.Thus we only need to show Here,s(α)and T(α)are the corresponding exponential sums over primes variables,whose definitions can be found in Chapter1.About the major arcs,we have the following asymptotic formula: Lemma2.Let m be defined as above.then for (?)A>0, where It’s clear that the result on the major arcs is similar to the ones of previous. So the key is that we need a satisfactory result on the minor arcs. That depends on the estimate of exponential sums over primes variables and some other technologies such as Hua’s Lemma.In the paper, we will apply the following estimate, it belongs to Kumchev, see [1].First define exponential sum of the form where y=xθ with θ<1, k≥2.For a given P, m(P) denote the set of real a which have rational approx-imations of the form l≤a≤P.(a,q)=l,|qa-a|≤x-k+2[l-θ)P, and let m(F) denote the complement of m(P).Lemma3. Let θ be a real number with4/5<θ≤1and suppose0<ρp≤ρ3(θ), where Then for any fixed ε>0,...