## 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

### Keywords:

Goldbach-Waring problems; unequal powers; prime numbers

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
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