Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 32 (2016), ISSN

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1 Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 3 06, ISSN A NOTE OF THREE PRIME REPRESENTATION PROBLEMS SHICHUN YANG AND ALAIN TOGBÉ Abstract. In this note, we consider a three prime representation problem asked by Sárközy. We give a negative answer to the upper density case of the problem and obtain a conclusion that the Three Primes Theorem still holds for a general thin subset of primes.. Introduction Let P be the sets of all primes. In 937, I. M. Vinogradov [8] proved the famous three primes theorem. It states that there eists an absolute constant N 0 > 0, such that every odd integer larger than N 0 is the sum of three primes. In 956, Borodzkin [4] showed that N 0 < Since then, this bound has been significantly improved. Most recently, Chen and Wang [5] have established a bound of In [6], Deshouillers, Effinger, Te Riele and Zinoviev outlined a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers. Recently, Helfgott [0] [3] completely solve the problem and proved that the ternary Goldbach conjecture is true. The Three Primes Theorem is one of the most important results studied in analytic number theory. For eample, Pan [], Zhan [30] and Jia [5] studied the Three Primes Theorem in short intervals. Many authors have been looking for thin subsets of primes for which the Three Primes Theorem still holds. In 986, Wirsing [9] showed that there 00 Mathematics Subject Classification. A5, A4, N05, P3. Key words and phrases. Three primes theorem, additive representation, thin subsets of primes. The first author was supported by National Natural Science Foundation of China Grant No.30363, the Innovative Research Team Project of Sichuan Provincial Universities No.4TD0040, and the Applied Basic Research Foundation of Sichuan Provincial Department for Science and Technology No.03JYZ003, and Department for Education No.5ZA0337. The second author was supported in part by Purdue University North Central. 3

2 4 SHICHUN YANG AND ALAIN TOGBÉ eists such a set S with the property that log 3. p<,p S It is also interesting to find more familiar thin sets of primes which serve this purpose. An eample is the set of Piatetski-Shapiro primes of type γ which are of the form n γ, where is the greatest integer not eceeding. Denote this set by P γ. Piatetski-Shapiro [3] showed that for < γ <, P γ = = + o γ log. p<,p = n γ Heath-Brown [9] improved Piatetski-Shapiro s result by etending the range to 66 < γ <. Further improvements were made by Kolesnik [9], Liu and 755 Rivat [], Balog and Friedlander [], Jia [7], [6], [8], et. al. On the other hand, in 00, P. Sárközy asked the following problem see Problem 67 in [5]: Problem A. Is it true that if Q = {q, q, } is an infinite set of primes such that lim inf log {qi : q i, q i Q} >, then every large odd integer n + can be represented in the form with q, q, q 3 Q? q + q + q 3 = n +, Let π denote the number of primes p <. From the prime number theorem, we have π. So Problem A is another type of problem of log thin subsets of primes related to the three primes theorem. For two non-empty sets X and A of positive integers, we define the upper density and lower density of A relative to X by and d X A = lim + A X [, ] sup, X [, ] d X A = lim inf A X [, ]. + X [, ] Thus, Problem A is equivalent to the following problem: If Q P is a subset of primes with d P Q >, could every large odd integer be represented as the sum of three positive integers, each integer composed of primes belonging to Q? In [8], Green obtained a Roth-type generalization of van der Corput s result. Green showed that if P 0 is a subset of P with d P P 0 > 0 then P 0 contains

3 THREE PRIME REPRESENTATION PROBLEMS 5 infinitely many non-trivial 3-term arithmetic progressions. In [0], Li and Pan used Green s idea to etend Vinogradov s theorem as follows. Theorem.. Suppose that P, P, P 3 are three subsets of P with d P P + d P P + d P P 3 >. Then for every sufficiently large odd integer n, there eist p P, p P and p 3 P 3 such that p + p + p 3 = n. By Theorem., we can derive that if Q P is a subset of primes such that d P Q >, then every large odd integer can be represented as the sum of three 3 primes belonging to Q. Recently, Shao [6] obtained that can be improved 3 to 5. And he pointed out that the constant 5 cannot be improved. In fact, 8 8 we may take Q = {p P, p,, 4, 7, 3 mod 5}, then d P Q = 5. It is 8 not hard to see that if N 4 mod 5 then N cannot be written as sum of three elements of Q. Therefore, the answer to Problem A is negative. In this note, we consider Problem A. First, we give a negative answer to the upper density case of Problem A. Second, we obtain a conclusion that the Three Primes Theorem still holds for a general thin subset of primes. The organization of this paper is as follows. In Section, we give a proof of Theorem. which is the negative answer to Problem A and we remark that the Problem 68 in [5] has a similar conclusion. In Section 3, we use Vinogradov s Three Primes Theorem and a theorem of Bombieri and Davenport [3] who showed that a large even number is the sum of two primes, we obtain Theorem 3. which states that the Three Primes Theorem still holds for a thin subset of primes. We ask an interesting research question in the last section.. The upper density case of problem A About Problem A, we have the following result. Theorem.. For any small positive constant ε, there eists an infinite set of primes Q = {q, q,...} such that d P Q = ε, and there are infinitely many odd integers n that cannot be represented in the form q + q + q 3 = n, with q, q, q 3 Q. Proof. Let p i be the i-th prime and s, t two large numbers. We construct an infinite set Q of primes by Q = {p, p,..., p s, p 3s, p 3s+,..., p 3ts, p 9ts, p 9ts+,..., p 9t s,..., p t r 3 r s, p t r 3 r s+,..., p t r 3 r s, p t r 3 r+ s, p t r 3 r+ s+,..., p t r+ 3 r+ s,...}. If > 35599, then from the prime number theorem See Theorem.0 in [7], we have + log log +.8 log π + log log log.

4 6 SHICHUN YANG AND ALAIN TOGBÉ Hence, we have lim = lim If t, then t = lim t log sup {q i : q i, q i Q} t tr 3 r+ + + t t r 3 r + + t r+ 3 r+ + 3t + 3 t +. t + t 3t + 3 t + = 3t 3t = ε, where ε is a small positive constant. So formula holds. For k >, we have see page of [7] 4 klog k + log log k < p k < klog k + log log k. We can take an odd integer n such that 5 3p tr 3 r s < n < p tr 3 r+ s. If n has the form q + q + q 3 = n with q, q, q 3 Q, then p q, q, q 3 p tr 3 r s, and then q + q + q 3 3 p tr 3 r s < n. This is impossible. So the integer n cannot have the form q + q + q 3 = n. Therefore, from 4 we obtain 6 p tr 3 r+ s 3 p tr 3 r s > t r 3 r+ s logt r 3 r+ s + log logt r 3 r+ s 3 t r 3 r s logt r 3 r s + log logt r 3 r s = t r 3 r+ logtr 3 r+ s s log 3 log. logt r 3 r s If r > 0, s > 0, and t >, then we have logtr 3 r+ s logt r 3 r s <.0. So inequality 6 implies p r 3 r+ s 3p r 3 r s > t r 3 r+ s log 3 log.0 > t r 3 r+ s. Thus, there are infinitely many odd integers n that cannot have the form q +q +q 3 = n with q, q, q 3 Q. This completes the proof of Theorem.. In [5], Sárközy set the following similar problem see Problem 68 in [5]. Problem B. Is it true that if Q is an infinite set of primes such that 7 lim inf log log q >, q Q,q

5 THREE PRIME REPRESENTATION PROBLEMS 7 then every large odd integer can be represented as the sum of three positive integers each composed of primes belonging to Q? From [6], the answer to Question B is negative also, and must be replaced by 5. Similarly, using the same method, according to Mertens formula see 8 page 34 of [4]: p = log log + β + O, log p where β is a constant, we get the following Theorem. Theorem.. For any small positive constant ε, there eists an infinite set of primes Q = {q, q,...} such that 8 lim sup log log q = ε q Q,q and there are infinitely many odd integers n that cannot be represented in the form q + q + q 3 = n with q, q, q 3 Q. Remark.3. The set of primes Q = {q, q, } in Theorem. satisfy 9 d P Q = 3, d PQ = ε. A natural question is: Does there eist a set of primes Q such that d P Q = 5 8, d P Q = ε and that there are infinitely many odd integers n which cannot be represented in the form q + q + q 3 = n, with q, q, q 3 Q? 3. The main conclusion In this section, we will proved our main result. First, we state Vinogradov s Three Primes Theorem. Lemma 3.. For every sufficiently large odd integer N, let rn denote the number of the solutions of the equation N = p + p + p 3, where p < p < p 3 are primes. Then rn = N 0 + o CN log 3 N, with CN = p N Proof. See [8] or [7]. p Now we will prove the following result. p N +. p 3

6 8 SHICHUN YANG AND ALAIN TOGBÉ Theorem 3.. Let Q = {q, q,...} be an infinite set of primes such that {qi : q i, q i Q} >, log 3 log and r n denote the number of the solutions of the equation n = q + q + q 3 with q, q, q 3 Q. Then r n > n Proof. Let P = Q S and Q S =. Assume that is a large integer. We write S = {q i : q i, q i S}. According to the Prime Number Theorem, to calculate the maimum number of elements in the set S, we have S + log log +.5 log 3 log 3 <.00 log 3 log.5. On the other hand, a lager odd integer n can be represented by n = q + q + q 3, q, q, q 3 P. From Lemma 3., we see that the number of solutions of the above equation is 4 As Cn = p n then 5 p rn > p n + rn > n Cn > 3 p n > 0.585, n If q, q, q 3 S and as q, q, q 3 < n, then using inequalities 3 and 3, one can determine the possible number of the values of q. This is at most.00n. 3 log.5 n Similarly, the possible number of the values of q is also at most.00n. Since 3 log.5 n q 3 = n q q, then the number of equations n = q + q + q 3 is at most.00n 3 log.5 n.00n 3 log.5 n <.003n 9 log 3 n n 6 < 0. n If q, q S, q 3 Q, using the same method, one can see that the number of equations n = q + q + q 3 is at most 0. n

7 THREE PRIME REPRESENTATION PROBLEMS 9 If q S, and q, q 3 Q, then q + q 3 is even. According to Bombieri and Davenport theorem [3] a large even number is the sum of two primes, if p, p are two primes such that p < p and N is even, we have 6 θn = N < 8 DN log N, where Then, we get 7 DN = p> q +q 3 =n q p +p =N p p N, p > < 8n q log n q < p p <. 8n log n..00n Since the possible number of the values of q is at most, then the 3 log.5 n number of equations n = q + q + q 3 is at most 8n log n.00n 3 log.5 n = 8.008n 3 log 3.5 n < 0.00n So from 5 we obtain n 0. n 0. n 0.00 n n r n > log 3 n log 3 n log 3 n log 3 = n Therefore, there are infinitely many odd integers n that can be represented into the form q + q + q 3 = n with q, q, q 3 Q. This completes the proof of Theorem 3.. A natural question is the following. 4. A question Problem C. Let r n denote the number of the solutions of n = q + q + q 3 with q, q, q 3 Q. Determine the minimum value of µ such that if Q is an infinite set of primes and 9 then r n n. log 3 n {q : q, q Q} > log log µ, Obviously, Theorem 3. shows that µ. We conjecture that µ 0. We also conjecture that when Q is an infinite set of primes such that {q : q, q Q} >, log log log then Vinogradov s Three Primes Theorem holds and r n n. log 3 n

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9 THREE PRIME REPRESENTATION PROBLEMS 3 [5] A. Sárközy. Unsolved problems in number theory. Period. Math. Hungar., 4-:7 35, 00. [6] X. Shao. A density version of the Vinogradov three primes theorem. Duke Math. J., 633:489 5, 04. [7] I. M. Vinogradov. The method of trigonometrical sums in the theory of numbers. Interscience Publishers, London and New York., 954. Translated, revised and annotated by K. F. Roth and Anne Davenport. [8] I. M. Vinogradow. Representation of an odd number as a sum of three primes. C. R. Dokl. Acad. Sci. URSS, n. Ser., 5:69 7, 937. [9] E. Wirsing. Thin subbases. Analysis, 6-3:85 308, 986. [30] T. Zhan. On the representation of large odd integer as a sum of three almost equal primes. Acta Math. Sinica N.S., 73:59 7, 99. A Chinese summary appears in Acta Math. Sinica 35 99, no. 4, 575. Received December 3, 04. Shichun Yang corresponding author, Department of Mathematics, Aba Teachers University, Wenchuan, Sichuan, 63000, P. R. China address: ysc00@sina.com Alain Togbé, Mathematics Department, Purdue University North Central, 40 S, U.S. 4, Westville IN 4639 USA address: atogbe@pnc.edu