Additive Problems With Prime Numbers

Additive prime number theory was ushered in by two seminal papers:Vinogradov's [119] celebrated proof of the three primes theorem and Hua's work [36] on nonlinear cases of the Waring-Goldbach problem. In both papers, the circle method of Hardy and Littlewood in combination with the estimates of Vinogradov for exponential sums over primes was exploited to give an affirmative answer to the additive problems considered there. After that, especially in recent years, new ideas in the circle method, sieves, and exponential sums are incorporated into the Waring-Goldbach problem, and hence give a number of remarkable advances.In this paper, we will investigate several additive problems involving prime num-bers by means of circle method, sieve method, and exponential sum estimates.In the first chapter, we outline a brief introduction to the Waring-Goldbach prob-lem, and then devote attention to depicting some basic machineries, especially the new ideas, in the circle method, the sieves, and exponential sum estimates, which will be illustrated by dealing with the Waring-Goldbach problem for lower degrees and closely connected with the argument in our topics of subsequent chapters. That will include, but not limited to, enlarged major arcs in additive problems and Harman's alternative sieves.In additive problems, the quality of arithmetical results obtained usually depends on the size of the major arcs. Traditionally the enlarged major arcs were treated by the Deuring-Heilbronn phenomenon; see, for example, Montgomery and Vaughan [91] and Liu and Tsang [77]. In1998, Liu and Zhan [70] found a new approach to increase the size of the major arcs, in which the possible existence of Siegel zero does not have special influence, and hence the Deuring-Heilbronn phenomenon can be avoided. This approach was improved later in [61]. Very recently, Liu [62,63] injected several new refreshing ingredients different from the prior argument and was able to treat still larger major arcs in additive problems, wherein the number of variables is required at least5. In Chapter2, we will handle, among some other techniques, the "uniform" enlarged major arcs by investigating the Waring-Goldbach problem. In particular, we can improve Liu's results in the fewer variables case, namely that of3and4variables.Let k be a fixed positive integer. The Waring-Goldbach problem of degree k is concerned with the solvability of the equation in primes p1,...,ps for sufficiently large integer n satisfying certain local conditions. It is plain that the Goldbach problem is just a linear example of the Waring-Goldbach problem. To state our results of Chapter2, we first introduce some notations. Assume xk=n, and thus use L to denote log n or logx. Let P, Q be two positive parameters satisfying2P<Q≤n/P. Then by Dirichlet's lemma on rational approximation, each real α∈[1/Q,1+1/Q] can be written in the form α=a/q+λ,|λ|≤1/(qQ) for some integers a, q with1≤a≤q≤Q and (a, q)=1, for which we denote the set of all such α by m(a, q). Then we write the major arcs as Moreover, define the exponential sum In Vinogradov's celebrated proof [119], he actually established the following asymptotic formula for the number of weighted solutions to the problem (0.1) with k=1, s=3on the major arcs provided that where C1, C2are positive constants. Here A>0is arbitrary, and (?)(n) is the singular series in the problem, which satisfies (?)(n)>1/2for odd n. By using ideas from the last decade, we can now take P considerably larger. For convenience, we set where θ is a positive constant, and B>0depends on the given constant A. In [100], Ren obtained the bound θ<6/25. By an argument similar to Liu and Zhan [76, Chapter6], one can actually improve it to θ<1/3. In Chapter2we are able to obtain the following large value of θ.Theorem2.1. Let k=1, and let the major arcs m be defined as in (0.2) with P,Q determined by (0.4). The asymptotic formula (0.3) holds for In the nonlinear case, Liu and Zhan proved the following result: I (Liu and Zhan [76, Chapter6]). Forone has the asymptotic formulae II (Liu [62]). while for one hasHere (?)2,3(n,P) and (?)k,s(n)(for s>4), are the corresponding singular series associated with the additive problem under consideration (whose positivity was established in Hua [38]), and Jk,s(n) is the singular integral which satisfies Jk,s(n)=xs-k. Using the argument similar to that leading to Theorem2.1, we can establish the following uniform estimate for the enlarged major arcs in the Waring-Goldbach problem, which improves the bound of θ above in the quadratic case. Theorem2.2. Let s≥3and let the major arcs m be defined as in (0.2) with P, Q determined by (0.4). The asymptotic formulae (0.5),(0.6) and (0.7) hold for θ<9/20. Now let K≥2be a fixed positive integer, and suppose that where some Xj may be equal to x1, and others arc of lower order of magnitude. In view of (0.8), one has log xj=logn, and we use L to denote log n. Instead of the problem in (0.1), we now consider the representation of n by k-th powers of primes: where the variables pj may lie in intervals of different lengths. Set where θ is a positive constant, B>0is a constant sufficiently large in terms of a given constant A. Note that P in (0.9) is defined in term of xs, instead of x. Define the major arcs m=m(P) as in (0.2). For j=1...., s, setThe argument leading to Theorems2.1and2.2can be modified slightly to give the following estimate for the enlarged major arcs in the Waring-Goldbach problem of the the diminishing-ranges situations.Theorem2.3. Let s≥3, x and xj be as in (0.8). Also let the major arcs m be defined as in (0.2) with P,Q determined by (0.9), and θ<9/20Then where A>0is arbitrary,(?)k,s(n) is the singular series (?)2,3(n, P) or (?)k,s(n) given in Theorem2.2according as s=3or s≥4, and J(n) is the singular integral which satisfiesActually, when Liu and Zhan found the technique to enlarge the major arcs they investigated the problem on sums of almost equal squares of primes. Define the sets In Chapter3, we study the additive representations of a large integer n as the sum of three, four or five almost equal squares of primes. Given a large integer n∈Hs, s≥3, we arc interested in its representations of the form where H=o(n1/2). Many authors have worked on this problem for the case s=5(charting the developments in [69,72,9,82,12,84]), culminating in Liu, Lu and Zhan's demonstration [67] that (0.10) holds with s=5and H=n9/20+ε. By injecting Harman's alternative sieve into this problem, we arc able to obtain still better results. This is our first theorem in this direction.Theorem3.1. All sufficiently large integers n∈H5can be represented in the form (0.10) with s=5and H=n4/9+ε for any fixed ε>0.We also obtain bounds for the number of integers n∈Hs, s=3,4, without representations as sums of s almost equal squares of primes. For H=o(X1/2) and s=3,4, we define Our interest in Es(X; H) is twofold. First, a non-trivial bound of the form for some fixed△>0, implies that almost all integers n€Hs are represcntable in the form (0.10) with H=nθ/2. Thus, we are interested in bounds of the form (0.11) with θ as small as possible. Furthermore, given a value of θ for which we can achieve a bound of the above form, we want to maximize the value of△.When s=4, Lii and Zhai [87] obtained results in both directions outlined above. First, they proved that (0.11) holds with s=4,θ>0.84and some△=△(θ)>0. Moreover, Lii and Zhai showed that, for θ>0.9and some η=η(θ)>0, The next theorem sharpens and generalizes (0.12).Theorem3.2. For any fixed θ with8/9<θ<1and for any fixed ε>0, one hasWe also obtain a small improvement on the first result of Lii and Zhai [87]. Our next theorem extends that result in two ways:it reduces the lower bound on θ to θ>0.82, and it gives an explicit expression for△.Theorem3.3. For any fixed θ with0.82<θ<1and for any fixed ε>0, one has where σ=σ{θ)=min(θ-31/40,(2θ-1)/8).The methods used to establish Theorems3.2and3.3can also be used to establish the following estimate for E3(X;Xθ/2).Theorem3.4. For any fixed θ with0.85<θ<1and for any fixed ε>0, one has where σ=σ-(θ) is the function from Theorem3.3. Furthermore, when8/9<θ<1, one hasUnlike the classical Waring-Goldbach problem, the problem considered in Chapter3remains of interest even when the number of variables exceeds five. Particularly, we also obtain the following theorem. Theorem3.5. Let s≥6and define All sufficiently large integers n∈Hs can be represented in the form (0.10) with H=nθs/2+εfor any fixed ε>0.In the last chapter, we investigate several other topics related to additive problems involving prime numbers. That will include our work on sums of squares of primes and a k-th power of prime, sums of almost equal cubes of primes, and sums of unlike powers of primes. We apply the circle method in combination with the sharp estimates for exponentials sums to the problems and thus are able to improve the previous results considerably in the former two topics and establish a stronger version of Prachar's classical theorem on unlike powers of primes.