Some Applications Of The Hardy-Littlewood Method

In the two letters to Euler in 1742, Goldbach raising the famous Goldbach Conjecture which can be stated as follows:(1) Every even number which is greater than or equal to 6 is the sum of two odd primes; (2) Every odd number which is greater than or equal to 9 is the sum of three odd primes. The former one is known as the binary Goldbach Conjecture and the latter one is regarded as the ternary Goldbach Conjecture. In 1937, Vinogradov almost solved (2) in [64]. By applying the Hardy-Littlewood method together with the estimates of exponential sums over primes, he proved that every sufficiently large odd positive integer is the sum of three odd primes. In 2013, the ternary Goldbach Conjecture was completely solved by Helfgott [22,23]. The nonlinear case of Goldbach problem, which is also known as Waring-Goldbach problem, attracts many authors’attentions. These problems are concerned with the representation of positive integers by sums of powers of primes, that is, N=p1k+pk2+…+pks, (0.8) where p1,...,ps are primes. Denote by H(k) the minimum of s for which all the suf-ficiently large positive integers N satisfying some necessary congruence conditions can be represented as the form (0.8). For fixed k, we concern the upper bound of H(k). In 1938, Hua [26] investigated Waring-Goldbach problem and gave a number of remarkable advances. He proved that H(k)≤2k+1 for all k≥1, by means of the Hardy-Littlewood method in combination with the estimate of Vinogradov for exponential sums over primes. For k≤3, this is still the best result. For 4≤k≤1, the upper bound of H(k) has been improved dramatically. The optimal results of H(k) at present are the following:H(4)≤13 (Zhao [72]);H(5)≤21 (Kawada and Wooley [32]); H(6)≤32 (Zhao [72]); H(7)≤45 (Kumchev and Wooley [36]). For k≥8, the bound of H(k) can be found in [36].Most of the above results are obtained by the Hardy-Littlewood method. According to Hardy[14],the Hardy-Littlewood method can be used in solving the classical additive problems including the Goldbach Conjecture and Waring’s problem. Although the binary Goldbach Conjecture is out of reach at present, the Hardy-Littlewood method is indeed one of the powerful tools in solving the additive problems.In this paper, we consider some applications of the Hardy-Littlewood method to several additive problems. Our first problem is to investigate the representation of positive integers by sums of unlike powers of primes, that is, n=p22+p33+p44+p55, (0.9) where p2,...,P5 are primes. The earliest result for this problem was proved by Prachar [51] in 1953 who stated that (0.9) is expressible for almost all positive even integers n. Precisely, let E(N) denote the number of positive even integers n up to N that cannot be written in the form (0.9). Prachar [51]proved that E(N)<<N(log N)-30/47+ε. Bauer[1,2], Ren and Tsang [57,58] also investigated this problem and gave improvements. The sharpest result so far was proved by Zhao [73],who showed that E(N)<<N15/16+ε.In this paper, we study (0.9) with almost equal variables, that is, where U=N1-θ+ε with θ>0 hoped to be as large as possible. Let E(N, U) denote the number of all positive even integers n satisfying N-4U≤n≤N+4U which cannot be written as (0.10). One wants to show that there exists 0<θ<1 such that E(N,U)<<U1-ε for U=N1-θ+ε,(0.11) where ε>0 is arbitrary. This problem was first considered by Li and Tang [37] in 2012 who showed that (0.11) holds for θ=1/264. In this paper, we study (0.10) and improve the result in [37].We will establish the following result.Theorem 1 Let notations be as above. Then (0.11)holds for θ=4/325.In another paper [52], Prachar considered the expression N= P1+P22+p33+P44+P55 (0.12) and showed that all large odd integers n can be written in this way. In [37],Li and Tang considered (0.12) with almost equal variables, that is, where U=N1-θ+ε. They proved that (0.13) is solvable for 9= 1/264. By Theorem 1, we obtain the following theorem.Theorem 2 For every sufficiently large positive odd integer N, (0.13) is solvable for U=N1-4/325+e.In this paper, we also investigate mean-value estimate related to divisors. Let d(n) be the number of divisors of n and k a positive integer. For X>1, consider the sums of divisors of the form For the estimate of T(k, s; X), the existing results have been focused on the case of k=2. The first breakthrough was given by Gafurov [10,11] who studied the divisors of the quadratic form with s= 2 in (0.14) and obtained T(2,2; X)= A1X2 log X+A2X2+O(X5/3 log9 X), where A1 and A2 are certain constants. The above error term was improved to O(X3/2+ε) by Yu in [69]. In 2000, C. Calderon and M. J. de Velasco [6] investigated (0.14) in the case of s=3 and established the asymptotic formula This result was improved by Guo and Zhai[13] where they showed that where Ci,Uj, (i, j=1,2) are constants. The error term above was refined to O(X2 log7 X) by Zhao [71] in 2014. In addition, Hu [24] investigated the case of s=4 and ob-tained the asymptotic formula for T(2,4; X) T(2,4; X)=2C’1I’1X4 log X+(C’1I’2+C2I’1[)X4+O(X7/2+e), where C’i,I’j(i,j=1,2) are constants. Recently, Hu and Liu [25] improved the above error term to O(X3 log7 X).In this paper, we consider the asymptotic formulas for T(k, s; X) when k≥2. For k=2, we can state our main results as follows.Theorem 3 Let T(k, s; X) be defined as in (0.14) and Ci,s, Ij,s (i, j=1,2) defined as in (1.14). Then for k=2 and s≥3, we have T(2,s;X)=2C1,sI1,sXslog X+(C1,sI2,s+C2,sI1,s)Xs+Os(Xs+1/2 logs+4 X+Xs-2 log X), where ε>0 is arbitrary and the singular series Ci,s (i=1,2) are absolutely convergent and satisfy Ci,s>>1.Note that for s=3,the result in Theorem 3 implies the result of Zhao in [71].For s=4, the error term in Theorem 1 is O(X5/2 log8 X) which improved the result of Hu and Liu in [25].For k≥3, we have the following result:Theorem 4 Let T(k,s;X) be defined as in (0.14) and k≥ 3. Then for s> min{2k-1,k2+k-2},we haveT(k, s;X)=kC1,k,sI1,k,sXs log X+(C1,k,sI2,k,s+C2,k,sI1,k,s)Xs+O(XS-θ+ε), where and Ci,k,s,Ij,k,s (i,j=1,2) are defined in (1-14) and (1.15), respectively. Moreover, the singular series Ci,k,s (i=1,2) are absolutely convergent and satisfy Ci,k,s>> 1.