Diophantine inequalities over Piatetski-Shapiro primes

Abstract

Let c>1 and : We study the solubility of the Diophantine inequality in Piatetski-Shapiro primes p₁,p₂, .., p_s of the form p_j=[m^{\gamma }] for some m\in \mathbb{N}, and improve the previous results in the cases s = 2, 3, 4.

Key words： Diophantine inequalities; Piatetski-Shapiro primes; exponential sums; additive problems

References

1 Baker R C, Harman G, Rivat J. Primes of the form [n^c]. 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 [n^c]: Mat Zametki, 1967, 2: 117–128

17 Kolesnik G A. Primes of the form [n^c]: 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 [n^c]: 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 [n^c]: Canad J Math, 2001, 53: 414–433

32 Rivat J, Wu J. Prime numbers of the form [n^c]: 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