On Sums Of Powers Of Almost Equal Primes

The Waring-Goldbach problem is a classic problem in additive number theory. It suggests that whenever there are sufficiently many variables, then all integers satisfying appropriate local conditions should be represented as the sum of powers of prime numbers. With this expectation in mind, consider a natural number k and prime p, take τ=τ(k,p) to be the integer with pτ|k but pτ+1(?)k, and then defineWe then define R=R(k) by putting R(k)=∏ipγ, where the product is taken over primes p with (p-1)|k. In 1938, Hua [11,12] established that whenever s> 2k, and n is a sufficiently large natural number with n= s (mod R), then the equation n=p1k+p2k+...+pks (1.1) is soluble in prime numbers pj. The congruence condition here excludes de-generate situations in which variables might otherwise be forced to be prime divisors of k.An intensively studied refinement of Hua’s theorem is the Waring-Goldbach problem with almost equal primes. In such problems, we constrain the vari-ables in Waring-Goldbach problems to be almost equal. Specifically, writing X=(n/s)1/k, one seeks an analogue of Hua’s theorem in which the variables pj satisfy |pj-X|≤Y, with Y rather smaller than X. That is, we hope all the prime variables pj range in a small interval Y around their average X. Obviously, the smaller Y we choose, the fewer the primes pj range, and hence the more difficult the problem is.The following two facts suggest a reasonable conjecture about the range of Y. First, in 1972, Huxley [13] proved a classic result concerning the distribution of primes in short intervals. Denote π(X) the number of primes not exceeding X. Then Huxley proved that, whenever Y≥X7/12+e, we have the following distribution formula. And when the Generalized Riemann Hypothesis (GRH) is supplied, the range of Y above can be enlarged as Y≥X1/2+ε. Since Huxley’s result represents the effective limit of our knowledge concerning the asymptotic distribution of primes in short intervals, it suggests a reasonable restrict in a number of problems in number theory concerning primes.On the other hand, in 1937, Wright [53] constructed a counterexample to show that, in the corresponding Waring’s problem with almost equal integers, Y can be chosen at most as small as Y= O(X1/2). Consequently, Deamen [6,7] proved that such lower bound can be achieved. Since the solutions of Waring-Goldbach problem form a subset of solutions of the responding War-ing’s problem, the same lower bound applies to the problem respect to primes. Combining these analysis, it seems reasonable to suggest that Y=O(X1/2) is the best possible lower bound of Y.In order to facilitate further discussion, we introduce some additional notations. We say that the exponent Δk,s is admissible when, provided that Δis a positive number with Δ<Δk,s, then for all sufficiently large positive integers n with n=s (mod R), the equation (1.1) has a solution in prime numbers pj satisfying |pj-X|≤X1-Δ (1≤j≤s). According to the upper paragraphs, we have 0≤Δk,s≤1/2.When k≥2, we define the integer tk by putting and define the real number θk by putting The main result of this paper is as following.定理1.1 Let s and k be integers with k≥2 and s>2tk.Suppose that ε>0, that n is a sufficiently large natural number satisfying n= s (mod R), and write X=(n/s)1/k. Then the equation n=p1k+p2k+...+psk has a solution in prime numbers pj with |pj-X|≤Xθk+e (1≤j≤s).This theorem shows that the exponent Δk,s=1/6 is admissible whenever k≥2 and s>2tk.In contrast to the admissible exponents derived in the previous work cited above, this exponent is considerably improved, bounded away from zero as k�'∞ We remark that if the Generalized Riemann Hypothesis (GRH) holds, the result concerning squares in the above theorem can be improved as 1/4 is admissible. Since 1/4=1/2×1/2, this exponent is in some sense half way from the conjectured exponent 1/2. Furthermore, when k is large, this theorem requires much fewer variable. We only need O(k2) primes to support our theorem, instead of O(2k) as before. In fact, whenever k≥5, we can always reduce the number of primes needed.Aficionados of the circle method will anticipate that similar conclusions may be established in problems with fewer variables. This results in the Waring-Goldbach problems with exceptional sets. When Y≥1, denote by Ek,s(N;Y) the number of positive integers n satisfying ⅰ.|n-N|≤kXl-1Y; ⅱ. n≡s (mod R); ⅲ. equation (1.1) does not have a solution satisfying |pj-(n/s)1/k|<Y(1≤j≤s) with primes pj.We say that the exponent Δk,s* is semi-admissible when, provided that Δ* is a positive number with Δ*< Δk,s*, then there exists ε>0, such that Ek,s(N;Y)<<Xk-1-εY.Thus, for almost all positive integers n with n≡s (mod R), the equation (1.1) has a solution in prime numbers pj satisfying|pj-X|≤X1-Δ* (1≤j≤s). We establish the following conclusion.定理1.2 Let s and k be integers with k≥2 and s>tk, and suppose that ε>0. Then for almost all positive integers n with n≡s (mod R) (and, in case k=3 and s=7, satisfying also 9(?)n), the equation n=p1k+p2k+...+Psk has a solution in prime numbers pj with|pj-X|≤Xθk+ε (1≤j≤s), where X=(n/s)1/k..We note that the additional condition 9(?)n in the case k=3 and s=7 is required to ensure the solubility of (1.1) modulo 9. This conclusion also con-siderably improved previous results. Indeed, it follows from our new theorem that whenever k≥4 and s>1/2k(k+1), then the exponent Δk,s*=1/6 is always semi-admissible.The ideas underlying the proof of Theorem 1.1 and 1.2 may be used to good effect in sharpening estimates for exceptional sets underlying in Theorem 1.2. Considering the sum of six squares of primes. We establish the following theorem concerning E2,6(N;Y).定理1.3 Suppose that Y≥X19/24+ε, for some positive number e. Then there is a positive number δ for which E2,6(N;Y)<<Y-1X1-δ.Our proof proceeds via the Hardy-Littlewood circle method. By com-parison with previous treatments, this argument contains two novel features. The first is an estimate for moments of exponential sums over kth powers in short intervals, of order 2s, that achieves essentially optimal estimates as soon as s≥tk. This serves as a substitute for the traditional use of Hua’s lemma, though for problems involving short intervals is considerably sharper. In §3 we explain how this estimate follows from the analogous work of Daemen [6,7], based on his use of the so-called binomial descent method. The second novel feature is a substitute for a Weyl-type estimate for exponential sums over variables in short intervals that delivers non-trivial estimates on the mi-nor arcs in a Hardy-Littlewood dissection even when the corresponding major arcs are rather narrow. This estimate again makes use of Daemen’s estimates via a bilinear form treatment motivated by analogous arguments making use of Vinogradov’s mean value theorem. Both the work makes heavy use of the latest work concerning Vinogradov’s mean value theorem.