Frontiers of Mathematics in China >
Diophantine inequalities over Piatetski-Shapiro primes
Received date: 25 Jun 2020
Accepted date: 28 Feb 2021
Published date: 15 Jun 2021
Copyright
Let c>1 and : We study the solubility of the Diophantine inequality in Piatetski-Shapiro primes p1,p2, .., ps of the form for some , and improve the previous results in the cases s = 2, 3, 4.
Jing HUANG , Wenguang ZHAI , Deyu ZHANG . Diophantine inequalities over Piatetski-Shapiro primes[J]. Frontiers of Mathematics in China, 2021 , 16(3) : 749 -770 . DOI: 10.1007/s11464-021-0916-7
1 |
Baker R C, Harman G, Rivat J. Primes of the form [nc]. J Number Theory, 1995, 50: 261–277
|
2 |
Baker R C,Weingartner A. Some applications of the double large sieve. Monatsh Math, 2013, 170: 261–304
|
3 |
Baker R C, Weingartner A. A ternary Diophantine inequality over primes. Acta Arith, 2014, 162: 159–196
|
4 |
Cai Y C. A Diophantine inequality with prime variables. Acta Math Sinica (Chin Ser), 1996, 39: 733–742 (in Chinese)
|
5 |
Cai Y C. On a Diophantine inequality involving prime numbers III. Acta Math Sin (Engl Ser), 1999, 15: 387–394
|
6 |
Cai Y C. A ternary Diophantine inequality involving primes. Int J Number Theory, 2018, 14: 2257–2268
|
7 |
Cao X D, Zhai W G. A Diophantine inequality with prime numbers. Acta Math Sinica (Chin Ser), 2002, 45: 361–370 (in Chinese)
|
8 |
Garaev M Z. On the Waring-Goldbach problem with small non-integer exponent. Acta Arith, 2003, 108: 297–302
|
9 |
Graham S W, Kolesnik G. Van der Corput's Method of Exponential Sums. London Math Soc Lecture Note Ser, 126. Cambridge: Cambridge Univ Press, 1991
|
10 |
Han X, Liu H F, Zhang D Y. A system of two Diophantine inequalities with primes. J Math, 2021, Art ID: 6613947 (12 pp)
|
11 |
Harman G. The values of ternary quadratic forms at prime arguments. Mathematika, 2004, 51: 83–96
|
12 |
Heath-Brown D R. The Piatetski-Shapiro prime number theorem. J Number Theory, 1983, 16: 242–266
|
13 |
Jia C H. The distribution of square-free number (II). Sci China Ser A, 1992, 22(8): 812–827 (in Chinese)
|
14 |
Jia C H. On Piatetski-Shapiro prime number theorem II. Sci China Ser A, 1993, 36: 913–926
|
15 |
Jia C H. On Piatetski-Shapiro prime number theorem. Chin Ann Math Ser B, 1994, 15: 9–22
|
16 |
Kolesnik G A. The distribution of primes in sequences of the form [nc]: Mat Zametki, 1967, 2: 117–128
|
17 |
Kolesnik G A. Primes of the form [nc]: Pacific J Math, 1985, 118: 437–447
|
18 |
Kumchev A. A Diophantine inequality involving prime powers. Acta Arith, 1999, 89: 311–330
|
19 |
Kumchev A. On the distribution of prime numbers of the form [nc]: Glasg Math J, 1999, 41: 85–102
|
20 |
Kumchev A, Laporta M B S. 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 |
Kumchev A, Nedeva T. On an equation with prime numbers. Acta Arith, 1998, 83: 117–126
|
22 |
Kumchev A, Petrov Z. A hybrid of two theorems of Piatetski-Shapiro. Monatsh Math, 2019, 189: 355–376
|
23 |
Li S H, Cai Y C. On a Diophantine inequality involving prime numbers. Ramanujan J, 2020, 52: 163–174
|
24 |
Li T Y, Liu H F. Diophantine approximation over Piatetski-Shapiro primes. J Number Theory, 2020, 211: 184–198
|
25 |
Liu H F. Diophantine approximation with one prime, two squares of primes and powers of two. Ramanujan J, 2020, 51: 85–97
|
26 |
Liu H F, Huang J. Diophantine approximation with mixed powers of primes. Taiwanese J Math, 2019, 23: 1073–1090
|
27 |
Liu H Q. On the number of abelian groups of a given order. Acta Arith, 1991, 59: 261–277
|
28 |
Liu H Q, Rivat J. On the Piatetski-Shapiro prime number theorem. Bull Lond Math Soc, 1992, 24: 143–147
|
29 |
Piatetski-Shapiro I I. On a variant of the Waring-Goldbach problem. Mat Sbornik N S, 1952, 30: 105–120
|
30 |
Piatetski-Shapiro I I. On the distribution of prime numbers in sequences of the form [f(n)]: Mat Sbornik N S, 1953, 33: 559–566
|
31 |
Rivat J, Sargos P. Nombres premiers de la forme [nc]: Canad J Math, 2001, 53: 414–433
|
32 |
Rivat J, Wu J. Prime numbers of the form [nc]: Glasg Math J, 2001, 43: 237–254
|
33 |
Sargos P. Points entiers au voisinage d'une courbe, sommes trigonometriques courtes et paires d'exposants. Proc Lond Math Soc, 1995, 70: 285–312
|
34 |
Tolev D I. On a Diophantine inequality involving prime numbers. Acta Arith, 1992, 61: 289–306
|
35 |
Vaughan R C. Diophantine inequality by prime numbers I. Proc Lond Math Soc, 1974, 28: 373–384
|
36 |
Vaughan R C. Diophantine inequality by prime numbers II. Proc Lond Math Soc, 1974, 28: 385–401
|
37 |
Vaughan R C. Diophantine inequality by prime numbers III. Proc Lond Math Soc, 1976, 33: 177–192
|
38 |
Zhai W G, Cao X D. On a Diophantine inequality over primes. Adv Math (China), 2003, 32: 63–73
|
39 |
Zhai W G, Cao X D. On a Diophantine inequality over primes II. Monatsh Math, 2007, 150: 173–179
|
/
〈 | 〉 |