On Exceptional Sets for Waring-Goldbach Problems with Unlike Powers

Abstract

Estimates are obtained for the number of natural numbers n in certain residue classes that do not have a representation of the form

$n = {p_1} + p_2^2 + p_3^k,$

for k = 3 or k = 4 respectively, where p_i are primes. We improve the previous results in [Monatsh. Math., 2019, 188(2): 269–285].

Keywords

Waring-Goldbach problem / circle method / exceptional set / 11P32 / 11P05 / 11P55

References

[1] FengZ, LiuZ, MaJ. On exceptional sets of biquadratic Waring-Goldbach problem. J. Number Theory, 2020, 211: 139-154.

[2] HardyGH, LittlewoodJE. Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes. Acta Math., 1923, 44(1): 1-70.

[3] HarmanG, KumchevAV. On sums of squares of primes II. J. Number Theory, 2010, 130(9): 1969-2002.

[4] HuaL. Some results in the additive prime-number theory. Quart. J. Math. Oxford Ser. (2), 1938, 9(1): 68-80.

[5] HuaLAdditive Theory of Prime Numbers, 1957, Beijing. Science Press. (in Chinese)

[6] KawadaK, WooleyT. Relations between exceptional sets for additive problems. J. Lond. Math. Soc. (2), 2010, 82(2): 437-458.

[7] KawadaK, ZhaoL. The Waring-Goldbach problem for cubes with an almost prime. Proc. Lond. Math. Soc. (3), 2019, 119(4): 867-898.

[8] KumchevA, ZhaoL. On sums of four squares of primes. Mathematika, 2016, 62(2): 348-361.

[9] LeungM, LiuM. On generalized quadratic equations in three prime variables. Monatsh. Math., 1993, 115(1–2): 133-167.

[10] Li J., Liu C., Zhang Z., et al., Exceptional set for sums of symmetric mixed powers of primes. Symmetry, 2020, 12 (3): Paper No. 367

[11] LiT. On sums of squares of primes and a kth power of prime. Rocky Mountain J. Math., 2012, 42(1): 201-222.

[12] LinnikYuV. Hardy-Littlewood problem on the representation as the sum of a prime and two squares. Dokl. Akad. Nauk SSSR, 1959, 124: 29-30

[13] LinnikYuV. An asymptotic formula in an additive problem of Hardy-Littlewood. Izv. Akad. Nauk SSSR Ser. Mat., 1960, 24: 629-706

[14] LiuJ, ZhanT. Sums of five almost equal prime squares, II. Sci. China Ser. A, 1998, 41(7): 710-722.

[15] LiuZ, ZhangR. On sums of squares of primes and a k-th power of prime. Monatsh. Math., 2019, 188(2): 269-285.

[16] RenX. On exponential sums over primes and application in Waring-Goldbach problems. Sci. China Ser. A, 2005, 48(6): 785-797.

[17] SchwarzW. Zur Darstellung von Zahlen durch Summen von Primzahlpotenzen. J. Reine Angew. Math., 1961, 206: 78-112.

[18] WangM. On the sum of a prime and two prime squares. Acta Math. Sinica (Chinese Ser.), 2004, 47(5): 845-858(in Chinese)

[19] WangM. On the sum of a prime, the square of a prime and the k-th power of a prime. Indian J. Pure Appl. Math., 2008, 39(3): 251-271

[20] WangM, MengX. The exceptional set in the two prime squares and a prime problem. Acta Math. Sin. (Engl. Ser.), 2006, 22(5): 1329-1342.

[21] WooleyT. Slim exceptional sets for sums of four squares. Proc. London Math. Soc. (3), 2002, 85(1): 1-21.

[22] ZhangR. Slim exceptional set for sums of two squares, two cubes, and two biquadrates of primes. Front. Math. China, 2019, 14(5): 1017-1035.

[23] ZhaoL. On the Waring-Goldbach problem for fourth and sixth powers. Proc. Lond. Math. Soc. (3), 2014, 108(6): 1593-1622.

[24] ZhaoL. The additive problem with one prime and two squares of primes. J. Number Theory, 2014, 135: 8-27.

RIGHTS & PERMISSIONS

Peking University