Goldbach-Linnik type problems on eight cubes of primes. (English) Zbl 1505.11129

Zhu, Li

Rocky Mt. J. Math. 52, No. 3, 1127-1139 (2022).

A classical result of L. K. Hua [Additive theory of prime numbers. Providence, RI: American Mathematical Society (AMS) (1965; Zbl 0192.39304)] states that if \(N\) is a sufficiently large odd integer, then the equation \(N = \sum_{1\leq i\leq 9} p_i^3\) is soluble in primes \(p_1,\dots,p_9\). One may also conjecture that every sufficiently large even integer is a sum of eight cubes of primes. This statement remains open, but good progress has been made on similar problems with an additional \(\epsilon\) of freedom.
One strand of such results, replacing one of the primes with another type of integer (e.g. a natural number or an almost-prime), can be found in the works [K. F. Roth, Proc. Lond. Math. Soc. (2) 53, 268–279 (1951; Zbl 0043.27303); J. Brüdern, Ann. Sci. Éc. Norm. Supér. (4) 28, No. 4, 461–476 (1995; Zbl 0839.11045); K. Kawada, Arch. Math. 69, No. 1, 13–19 (1997; Zbl 0882.11057); K. Kawada and L. Zhao, Proc. Lond. Math. Soc. (3) 119, No. 4, 867–898 (2019; Zbl 1443.11208)].
The author contributes to a different strand, where one includes powers of \(2\) as additional summands, in the spirit of [Y. V. Linnik, Trudy Nat. Inst. Steklov. 38, 152–169 (1951); P. X. Gallagher, Invent. Math. 29, 125–142 (1975; Zbl 0305.10044)]. See [J. Liu and M. C. Liu, Acta Math. Hung. 91, No. 3, 217–243 (2001; Zbl 0980.11045); J. Liu and G. Lü, Acta Arith. 114, No. 1, 55–70 (2004; Zbl 1050.11084); Z. Liu, J. Number Theory 132, No. 4, 735–747 (2012; Zbl 1287.11117); D. J. Platt and T. S. Trudgian, J. Number Theory 153, 54–62 (2015; Zbl 1328.11102); X. Zhao and W. Ge, Int. J. Number Theory 16, No. 7, 1547–1555 (2020; Zbl 1470.11257); X. Zhao, Ramanujan J. 57, No. 1, 239–251 (2022; Zbl 1498.11203)], the most recent of which shows that every sufficiently large even integer can be written in the form \(p_1^3+\dots+p_8^3+2^{\nu_1}+\dots+2^{\nu_{169}}\) with \(p_1,\dots,p_8\) prime and \(\nu_1,\dots,\nu_{169}\geq 1\).
In the present paper, the author decreases \(169\) by a large factor, down to \(30\). The improvement comes from sharpened integral estimates in the circle method. One of the cleanest innovations may be Lemma 4.7, an estimate for the ninth moment of a cubic exponential sum over primes, over certain minor arcs.

Reviewer: Victor Wang (Princeton)

MSC:

11P05	Waring’s problem and variants
11P55	Applications of the Hardy-Littlewood method

Keywords:

Waring-Goldbach problem; Hardy-Littlewood method

Citations:

Zbl 0192.39304; Zbl 0043.27303; Zbl 0839.11045; Zbl 0882.11057; Zbl 1443.11208; Zbl 0305.10044; Zbl 0980.11045; Zbl 1050.11084; Zbl 1287.11117; Zbl 1328.11102; Zbl 1470.11257; Zbl 1498.11203

References:

[1]	J. Brüdern, “A sieve approach to the Waring-Goldbach problem, I: Sums of four cubes”, Ann. Sci. École Norm. Sup. (4) 28:4 (1995), 461-476. · Zbl 0839.11045 · doi:10.24033/asens.1721
[2]	P. X. Gallagher, “Primes and powers of \[2\]”, Invent. Math. 29:2 (1975), 125-142. · Zbl 0305.10044 · doi:10.1007/BF01390190
[3]	G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 6th ed., Oxford University Press, 2008.
[4]	L. K. Hua, Additive theory of prime numbers, Translations of Mathematical Monographs 13, American Mathematical Society, Providence, R.I., 1965. · Zbl 0192.39304
[5]	K. Ireland and M. Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics 84, Springer, 1990. · Zbl 0712.11001 · doi:10.1007/978-1-4757-2103-4
[6]	K. Kawada, “Note on the sum of cubes of primes and an almost prime”, Arch. Math. (Basel) 69:1 (1997), 13-19. · Zbl 0882.11057 · doi:10.1007/s000130050088
[7]	K. Kawada and L. Zhao, “The Waring-Goldbach problem for cubes with an almost prime”, Proc. Lond. Math. Soc. (3) 119:4 (2019), 867-898. · Zbl 1443.11208 · doi:10.1112/plms.12244
[8]	Y. V. Linnik, “Prime numbers and powers of two”, Trudy Nat. Inst. Steklov. 38 (1951), 152-169. · Zbl 0049.31402
[9]	Y. V. Linnik, “Addition of prime numbers with powers of one and the same number”, Mat. Sbornik N.S. 32(74) (1953), 3-60. · Zbl 0051.03402 · doi:10.1016/0370-1573(77)90003-5
[10]	Z. Liu, “Density of the sums of four cubes of primes”, J. Number Theory 132:4 (2012), 735-747. · Zbl 1287.11117 · doi:10.1016/j.jnt.2011.12.003
[11]	J. Liu and M.-C. Liu, “Representation of even integers by cubes of primes and powers of 2”, Acta Math. Hungar. 91:3 (2001), 217-243. · Zbl 0980.11045
[12]	J. Liu and G. Lü, “Four squares of primes and 165 powers of 2”, Acta Arith. 114:1 (2004), 55-70. · Zbl 1050.11084 · doi:10.4064/aa114-1-4
[13]	Z. Liu and G. Lü, “Eight cubes of primes and powers of 2”, Acta Arith. 145:2 (2010), 171-192. · Zbl 1239.11109 · doi:10.4064/aa145-2-6
[14]	J. Pintz and I. Z. Ruzsa, “On Linnik’s approximation to Goldbach’s problem, I”, Acta Arith. 109:2 (2003), 169-194. · Zbl 1031.11060 · doi:10.4064/aa109-2-6
[15]	D. J. Platt and T. S. Trudgian, “Linnik’s approximation to Goldbach’s conjecture, and other problems”, J. Number Theory 153 (2015), 54-62. · Zbl 1328.11102 · doi:10.1016/j.jnt.2015.01.008
[16]	X. Ren, “Sums of four cubes of primes”, J. Number Theory 98:1 (2003), 156-171. · Zbl 1100.11033 · doi:10.1016/S0022-314X(02)00022-7
[17]	K. F. Roth, “On Waring’s problem for cubes”, Proc. London Math. Soc. (2) 53 (1951), 268-279. · Zbl 0043.27303 · doi:10.1112/plms/s2-53.4.268
[18]	L. Zhao, “On the Waring-Goldbach problem for fourth and sixth powers”, Proc. Lond. Math. Soc. (3) 108:6 (2014), 1593-1622. · Zbl 1370.11116 · doi:10.1112/plms/pdt072
[19]	X. Zhao, “Goldbach-Linnik type problems on cubes of primes”, Ramanujan J. 57:1 (2022), 239-251. · Zbl 1498.11203 · doi:10.1007/s11139-020-00303-9
[20]	X. Zhao and W. Ge, “Eight cubes of primes and 204 powers of 2”, Int. J. Number Theory 16:7 (2020), 1547-1555. · Zbl 1470.11257 · doi:10.1142/S1793042120500803