A Goldbach-Waring problem for unequal powers of primes.(English)Zbl 1232.11101

The author proves that all even numbers $$n\leq x$$ having the representation $n=p_1^2+p_2^3+p_3^4+p_4^5\;,$
where the $$p_i$$ are prime numbers, with at most $$\ll x^{1633/1680+\varepsilon}$$ exceptions (i.e., the cardinality of the, say, “exceptional set”, is bounded in this way, $$\varepsilon>0$$ arbitrarily small, as usual). This is, in fact, an instance of the Goldbach-Waring problem (i.e., representation of natural numbers $$n$$ as sums of powers of primes, with congruence conditions for $$n$$, here $$n\equiv 0\pmod 2$$, of course), with unequal powers (because, without further specification, we assume all powers to be the same). See that the abstract contains a serious misprint (exchanging up & down indices for the fourth prime-power!).
The paper contains an extensive literature of the subject and, as the author explains clearly, his improvement of the previous record for the exponent above (i.e., $$1633/1680$$ for the exceptional set) stems from an exponential sums estimate due to A. V. Kumchev [Mich. Math. J. 54, 243–268 (2006; Zbl 1137.11054)] for his new treatment of the minor arcs (of course, he works in the usual environment of the circle method, distinguishing major and minor arcs).

MSC:

 11P32 Goldbach-type theorems; other additive questions involving primes 11L07 Estimates on exponential sums

Zbl 1137.11054
Full Text:

References:

 [1] C. Bauer, On a problem of the Goldbach-Waring type , Acta Math. Sinica, New Series, 14 (1998), 223-234. · Zbl 0901.11030 [2] ——–, An improvement on a theorem of the Goldbach-Waring type , Rocky Mountain J. Mathematics 31 (2001), 1151-1170. · Zbl 1035.11047 [3] ——–, A remark on a theorem of the GoldbachWaring type , Studia Sci. Math. Hungar. 41 (2004), 309-324. · Zbl 1064.11065 [4] S. Choi and A. Kumchev, Mean values of Dirichlet polynomials and applications to linear equations with prime variables , Acta Arith. 123 (2006), 125-142. · Zbl 1182.11048 [5] P.X. Gallagher, A large sieve density estimate near $$\sigma=1$$ , Invent. Math. 11 (1970), 329-339. · Zbl 0219.10048 [6] N.M. Huxley, Large values of Dirichlet polynomials III, Acta Arithmet. 26 (1975), 435-444. · Zbl 0268.10026 [7] K. Kawada and T.D. Wooley, On the Waring-Goldbach problem for fourth and fifth powers , Proc. London Math. Soc. 83 (2001), 1-50. · Zbl 1016.11046 [8] A. Kumchev, On Weyl sums over primes and almost primes , Michigan Math. J. 54 (2006), 243-268. · Zbl 1137.11054 [9] J.Y. Liu, On Lagrange’s theorem with prime variables , Quart. J. Math. 54 (2003), 453-462. · Zbl 1080.11071 [10] J.Y. Liu and M.C. Liu, The exceptional set in the four prime squares problem , Illinois J. Math. 44 (2000), 272-293. · Zbl 0942.11044 [11] K. Prachar, Über ein Problem vom Waring-Goldbach‘schen Typ. , Monatshefte Math. 57 (1953), 66-74. · Zbl 0050.04003 [12] ——–, Primzahlverteilung. , Springer Verlag, Berlin, 1978. · Zbl 0080.25901 [13] X. Ren and K.M. Tsang, The Goldbach-Waring problem for unlike powers , preprint, http://hkumath.hku.hk/$$\sim$$imr/IMRPreprintSeries/IM R2004-006.pdf. · Zbl 1121.11312 [14] ——–, Waring Goldbach problem for unlike powers , Acta Math. Sinica 23 (2007), 265-280. · Zbl 1128.11043 [15] K.F. Roth, A problem in additive number theory , Proc. London Math. Soc. 53 (1951), 381-395. · Zbl 0044.03601 [16] E.C. Titchmarsh, The theory of the Riemann zeta-function , second edition, Clarendon Press, Oxford, 1986. · Zbl 0601.10026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.