Diophantine inequalities over Piatetski-Shapiro primes

doi:10.1007/s11464-021-0916-7

Front. Math. China

2021, Vol. 16

Issue (3) : 749-770 https://doi.org/10.1007/s11464-021-0916-7

RESEARCH ARTICLE

Diophantine inequalities over Piatetski-Shapiro primes

Jing HUANG¹, Wenguang ZHAI², Deyu ZHANG¹(

)

¹. School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
². Department of Mathematics, China University of Mining and Technology, Beijing 100083, China

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Abstract

Let c>1 and $0 < γ < 1$ : We study the solubility of the Diophantine inequality $| p 1 c + p 2 c + ⋯ + p s c − N | < (l o g N) − 1$ in Piatetski-Shapiro primes p₁,p₂, .., p_s of the form $p j = [m γ]$ for some $m ∈ ℕ$ , and improve the previous results in the cases s = 2, 3, 4.

Keywords Diophantine inequalities Piatetski-Shapiro primes exponential sums additive problems

Corresponding Author(s): Deyu ZHANG

Issue Date: 14 July 2021

Cite this article:

Jing HUANG,Wenguang ZHAI,Deyu ZHANG. Diophantine inequalities over Piatetski-Shapiro primes[J]. Front. Math. China, 2021, 16(3): 749-770.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-021-0916-7
https://academic.hep.com.cn/fmc/EN/Y2021/V16/I3/749

1	R C Baker, G Harman, J Rivat. Primes of the form [nc]. J Number Theory, 1995, 50: 261–277 https://doi.org/10.1006/jnth.1995.1020
2	R C Baker, A Weingartner. Some applications of the double large sieve. Monatsh Math, 2013, 170: 261–304 https://doi.org/10.1007/s00605-012-0447-0
3	R C Baker, A Weingartner. A ternary Diophantine inequality over primes. Acta Arith, 2014, 162: 159–196 https://doi.org/10.4064/aa162-2-3
4	Y C Cai. A Diophantine inequality with prime variables. Acta Math Sinica (Chin Ser), 1996, 39: 733–742 (in Chinese)
5	Y C Cai. On a Diophantine inequality involving prime numbers III. Acta Math Sin (Engl Ser), 1999, 15: 387–394 https://doi.org/10.1007/BF02650733
6	Y C Cai. A ternary Diophantine inequality involving primes. Int J Number Theory, 2018, 14: 2257–2268 https://doi.org/10.1142/S1793042118501361
7	X D Cao, W G Zhai. A Diophantine inequality with prime numbers. Acta Math Sinica (Chin Ser), 2002, 45: 361–370 (in Chinese)
8	M Z Garaev. On the Waring-Goldbach problem with small non-integer exponent. Acta Arith, 2003, 108: 297–302 https://doi.org/10.4064/aa108-3-8
9	S W Graham, G Kolesnik. Van der Corput's Method of Exponential Sums. London Math Soc Lecture Note Ser, 126. Cambridge: Cambridge Univ Press, 1991 https://doi.org/10.1017/CBO9780511661976
10	X Han, H F Liu, D Y Zhang. A system of two Diophantine inequalities with primes. J Math, 2021, Art ID: 6613947 (12 pp) https://doi.org/10.1155/2021/6613947
11	G Harman. The values of ternary quadratic forms at prime arguments. Mathematika, 2004, 51: 83–96 https://doi.org/10.1112/S0025579300015527
12	D R Heath-Brown. The Piatetski-Shapiro prime number theorem. J Number Theory, 1983, 16: 242–266 https://doi.org/10.1016/0022-314X(83)90044-6
13	C H Jia. The distribution of square-free number (II). Sci China Ser A, 1992, 22(8): 812–827 (in Chinese) https://doi.org/10.1360/za1992-22-8-812
14	C H Jia. On Piatetski-Shapiro prime number theorem II. Sci China Ser A, 1993, 36: 913–926
15	C H Jia. On Piatetski-Shapiro prime number theorem. Chin Ann Math Ser B, 1994, 15: 9–22
16	G A Kolesnik. The distribution of primes in sequences of the form [nc]: Mat Zametki, 1967, 2: 117–128 https://doi.org/10.1007/BF01094244
17	G A Kolesnik. Primes of the form [nc]: Pacific J Math, 1985, 118: 437–447 https://doi.org/10.2140/pjm.1985.118.437
18	A Kumchev. A Diophantine inequality involving prime powers. Acta Arith, 1999, 89: 311–330 https://doi.org/10.4064/aa-89-4-311-330
19	A Kumchev. On the distribution of prime numbers of the form [nc]: Glasg Math J, 1999, 41: 85–102 https://doi.org/10.1017/S0017089599970477
20	A Kumchev, M B S Laporta. On a binary Diophantine inequality involving prime pow-ers. In: Number Theory for the Millennium II (Urbana, IL, 2000). Wellesley: A K Peters, 2002, 307–329
21	A Kumchev, T Nedeva. On an equation with prime numbers. Acta Arith, 1998, 83: 117–126 https://doi.org/10.4064/aa-83-2-117-126
22	A Kumchev, Z Petrov. A hybrid of two theorems of Piatetski-Shapiro. Monatsh Math, 2019, 189: 355–376 https://doi.org/10.1007/s00605-018-1217-4
23	S H Li, Y C Cai. On a Diophantine inequality involving prime numbers. Ramanujan J, 2020, 52: 163–174 https://doi.org/10.1007/s11139-018-0132-6
24	T Y Li, H F Liu. Diophantine approximation over Piatetski-Shapiro primes. J Number Theory, 2020, 211: 184–198 https://doi.org/10.1016/j.jnt.2019.10.002
25	H F Liu. Diophantine approximation with one prime, two squares of primes and powers of two. Ramanujan J, 2020, 51: 85–97 https://doi.org/10.1007/s11139-018-0121-9
26	H F Liu, J Huang. Diophantine approximation with mixed powers of primes. Taiwanese J Math, 2019, 23: 1073–1090 https://doi.org/10.11650/tjm/190201
27	H Q Liu. On the number of abelian groups of a given order. Acta Arith, 1991, 59: 261–277 https://doi.org/10.4064/aa-59-3-261-277
28	H Q Liu, J Rivat. On the Piatetski-Shapiro prime number theorem. Bull Lond Math Soc, 1992, 24: 143–147 https://doi.org/10.1112/blms/24.2.143
29	I I Piatetski-Shapiro. On a variant of the Waring-Goldbach problem. Mat Sbornik N S, 1952, 30: 105–120
30	I I Piatetski-Shapiro. On the distribution of prime numbers in sequences of the form [f(n)]: Mat Sbornik N S, 1953, 33: 559–566
31	J Rivat, P Sargos. Nombres premiers de la forme [nc]: Canad J Math, 2001, 53: 414–433 https://doi.org/10.4153/CJM-2001-017-0
32	J Rivat, J Wu. Prime numbers of the form [nc]: Glasg Math J, 2001, 43: 237–254 https://doi.org/10.1017/S0017089501020080
33	P Sargos . Points entiers au voisinage d'une courbe, sommes trigonometriques courtes et paires d'exposants. Proc Lond Math Soc, 1995, 70: 285–312 https://doi.org/10.1112/plms/s3-70.2.285
34	D I Tolev. On a Diophantine inequality involving prime numbers. Acta Arith, 1992, 61: 289–306 https://doi.org/10.4064/aa-61-3-289-306
35	R C Vaughan. Diophantine inequality by prime numbers I. Proc Lond Math Soc, 1974, 28: 373–384 https://doi.org/10.1112/plms/s3-28.2.373
36	R C Vaughan. Diophantine inequality by prime numbers II. Proc Lond Math Soc, 1974, 28: 385–401 https://doi.org/10.1112/plms/s3-28.3.385
37	R C Vaughan. Diophantine inequality by prime numbers III. Proc Lond Math Soc, 1976, 33: 177–192 https://doi.org/10.1112/plms/s3-33.1.177
38	W G Zhai, X D Cao. On a Diophantine inequality over primes. Adv Math (China), 2003, 32: 63–73
39	W G Zhai, X D Cao. On a Diophantine inequality over primes II. Monatsh Math, 2007, 150: 173–179 https://doi.org/10.1007/s00605-005-0390-4

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