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The exceptional set for sums of unlike powers of primes. (English) Zbl 1302.11080

Let \(E(N)\) be the number of positive even integers of size at most \(N\) which cannot be represented in the form \(p_1^2+p_2^3+p_3^4+p_4^5\), where \(p_i\), \(1\leq i\leq 4\), are prime numbers. The author proves an upper bound for \(E(N)\) of the form \(E(N)\ll N^{15/16+\varepsilon}\). This improves on earlier works of Pracher, Ren and Tsang and most recently one of C. Bauer [Rocky Mt. J. Math. 38, No. 4, 1073–1090 (2008; Zbl 1232.11101)].
The proof is based on an application of the Hardy-Littlewood circle method and uses on the minor arcs mean value estimates introduced in earlier work of the author [Proc. London Math. Soc. 108, No. 6, 1593–1622 (2014; Zbl 1370.11116)].

MSC:

11P55 Applications of the Hardy-Littlewood method
11P05 Waring’s problem and variants
11P32 Goldbach-type theorems; other additive questions involving primes
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References:

[1] Bauer, C, An improvement on a theorem of the Goldbach-Waring type, Roky Mountain J. Math., 53, 1-20, (2001)
[2] Bauer, C, A Goldbach-Waring problem for unequal powers of primes, Roky Mountain J. Math., 38, 1073-1090, (2008) · Zbl 1232.11101
[3] Kumchev, A V, On wely sums over prime and almost primes, Michigan Math. J., 54, 243-268, (2006) · Zbl 1137.11054
[4] Prachar, K, Uber ein problem vom Waring-goldbach’schen. typ, Monatsh. Math., 57, 66-74, (1953) · Zbl 0050.04003
[5] Prachar, K, Uber ein problem vom Waring-goldbach’schen. typ. II, Monatsh. Math., 57, 113-116, (1953) · Zbl 0052.27901
[6] Ren, X M, On exponential sum over primes and application in Waring-Goldbach probelm, Sci. China Ser. A, 48, 785-797, (2005) · Zbl 1100.11025
[7] Ren, X M; Tsang, K M, Waring-Goldbach problem for unlike powers, Acta Math. Sin., Engl. Series, 23, 265-280, (2007) · Zbl 1128.11043
[8] Ren, X M; Tsang, K M, Waring-Goldbach problem for unlike powers (II), Acta Math. Sin., Chin. Series, 50, 175-182, (2007) · Zbl 1121.11312
[9] Roth, K F, A problem in additive number theory, Proc. London Math. Soc. (2), 53, 381-395, (1951) · Zbl 0044.03601
[10] Vaughan, R. C.: The Hardy-Littlewood Method, 2nd ed., Cambridge University Press, Cambridge, 1997 · Zbl 0868.11046
[11] Zhao, L, On the Waring-Goldbach problem for fourth and sixth powers, Proc. London Math. Soc., 108, 1593-1622, (2014) · Zbl 1370.11116
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