An improvement on a theorem of the Goldbach-Waring type. (English) Zbl 1035.11047

In this paper the author shows, using the circle method, that all but \(O (x^{19193/19200 +\varepsilon})\) positive even integers up to \(x\) are representable in the form \(p^2_1 + p^3_2 + p^4_3 + p^5_4\) where the \(p_i\)’s are prime numbers. This result sharpens an earlier result of the author in which the exceptional set was of the form \(O(x^{1-\delta})\) with an unspecified positive (small) \(\delta\). The basic structure of the proof here is largely the one used by many authors in this area, which originates from the paper [M. C. Liu and K. M. Tsang, Théorie des Nombres, C. R. Conf. Int., Québec/Can. 1987, 595–624 (1989; Zbl 0682.10043)]. The exponent \(19193/19200\) obtained here is a consequence of using in a main step, the inequality \([r_2, r_3, r_4, r_5] \geq (r_2 r_3)^{1/7} (r_4 r_5)^{5/14}\) where \([\cdots]\) denotes the least common multiple. Recently a more efficient iterative procedure has been developed which essentially yields \([r_2 , r_3 , r_4, r_5] \approx r_2 r_3 r_4 r_5\). Thus the exponent \(19193/19200\) has now been reduced to \(65/66\) by X. Ren and the reviewer.


11P32 Goldbach-type theorems; other additive questions involving primes
11P55 Applications of the Hardy-Littlewood method


Zbl 0682.10043
Full Text: DOI Link


[1] C. Bauer, On a problem of the Goldbach-Waring type , Acta Math. Sinica New Ser. 14 (1998), 223-234. · Zbl 0901.11030
[2] C. Bauer, M.C. Liu and T. Zhan, On sums of three primes , J. Number Theory, · Zbl 0961.11034
[3] H. Davenport, Multiplicative number theory , 2nd ed., Springer-Verlag, Chicago, 1980. · Zbl 0453.10002
[4] P.X. Gallagher, A large sieve density estimate near \(\s=1\) , Invent. Math. 11 (1970), 329-339. · Zbl 0219.10048
[5] G. Harman, Trigonometric sums over primes , Mathematika 28 (1981), 249-254. · Zbl 0465.10029
[6] D.R. Heath Brown, Prime numbers in sort intervals and a generalized Vaughan’s identity , Canadian J. Math. 34 (1982), 1365-1377. · Zbl 0478.10024
[7] M.C. Leung and M.C. Liu, On generalized quadratic equations in three prime variables , Monatsh. Math. 115 (1993), 133-169. · Zbl 0779.11045
[8] M.C. Liu and K.M. Tsang, Small prime solutions of linear equations , in Number theory (J.-M. De Konick and C. Levesque, eds.), W. de Gruyter, Berlin, 1989. · Zbl 0682.10043
[9] H.L. Montgomery and R.C. Vaughan, On the exceptional set in Goldbach’s problem , Acta Arith. 27 (1975), 353-370. · Zbl 0301.10043
[10] Chengdong Pan and Chengbiao Pan, Goldbach’s conjecture , Science Press, Beijing, 1992.
[11] K. Prachar, Über ein Problem von Waring-Goldbach’schen Typ. , Monatsh. Math. 57 (1953), 66-74. · Zbl 0050.04003
[12] ——–, Primzahlverteilung , Springer-Verlag, Berlin, New York, 1978.
[13] E.C. Titchmarsh, The theory of the Riemann zeta function , 2nd ed., Clarendon Press, Oxford, 1986. · Zbl 0601.10026
[14] I.M. Vinogradov, Estimation of certain trigonometric sums with prime variables , Izv. Akad. Nauk SSSR Ser. Mat. 3 (1939), 371-398.
[15] T. Zhan, On the representation of a large odd integer as a sum of three almost equal primes , Acta Math. Sinica 7 (1991), 259-272. · Zbl 0742.11048
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