Representation Of Integers By Sums Of Fourth Powers Of Primes

For a fixed integer k, the Waring-Goldbach problem is concerned with the solvability of the equation i.e. one tries to obtain the least integer s= s(k) such that all sufficiently large integers n satisfying necessary congruence conditions can be represented as the form of (0.1), where p1,...,ps are prime unknowns. It is conjectured that for s≥k+1, (0.1) is solvable. Obviously, the ternary Goldbach problem is just the liner case of the Waring-Goldbach problem. From this we can see that it is quite difficult to prove this conjecture. Let H(k) denote the least integer such that (0.1) is solvable. The first breakthough came in 1937 when Vinogradov [27] developed a new method for estimating exponential sums over primes and used the circle method to obtain H(1)≤3. Later, Hua [4] showed that which is the best result to date for k≤3. On the other hand, when k≥4, the above result has been improved on greatly. For example, for smaller k≥4, in 1987 Thanigasalam [22] obtained H(6)≤33, H(8)≤63, H(9)≤83 and H(10)≤107; In recent years, Kawada and Wooley [6] showed that H(4)≤14,H(5)≤21, and Kumchev [7] proved thatOn the other hand, if we seek almost all instead of arbitrary sufficiently large n satisfying necessary congruence conditions to make (0.1) be solvable, then the number of s can be reduced further. We defineεk,s(N) as the set of n not exceeding N which can't be represented as (0.1), and write Ek,s(N) =|εk,s(N)|. Then the upper bound of Ek,s(N) is the problem of the exceptional sets. Exploiting the nature of the proofs of the above bounds for H(k), one can show that, along with each estimate of the form H(k)≤s0(k), for any fixed A> 0 and any s≥(?)s0(k).The first improvement in this direction was obtained by Vaughan [26] who showed that for some c> 0. Montgomery and Vaughan [19] proved that there exists an absolute constantθ< 1 such that after that, several authors have given such estimates with explicit values ofθ, and the optimal resultθ= 0.914 at present is due to Li [10]. In the quadratic case, the similar estimate for exceptional set for sums of squares of three primes was obtained by Leung and Liu [11], who showed that with an absolute constantθ'< 1.In 1998 Liu and Zhan [15] found a new approach to treat the enlarged major arcs. Since then, this approach has been applied in many problems of additive number theory. At the same time, the exceptional sets for sums of squares of primes attracted many authors (see [1,13,16,25,12,17,14,21,9,2]); and the best result was obtained just recently by Harman and Kumchev [3]: Due to new estimates for exponential sums and the introduction of sieve ideas, the the exceptional sets of Waring-Goldbach problem for cubes and higer powers has also been improved greatly (see [20,6,24,7,8,18], etc). Take the fourth powers as an example, Kumchev improved the upper bound of (0.2) in [8] and obtained (0.3) (0.4) and whereδ' is a very small positive constant, while Recently Liu and Zhan obtained the largest major arcs to date in [17], where they improved (0.3) further and obtainedIn this paper, we will study the exceptional sets for sums of fourth powers by using the idea of Liu and Zhan combining with the new estimates of exponential sums due to Kumchev. Precisely, our result is as follows: Since one can see that our result improved (0.4).