Autors: Dimitrov, S. I., Lazarova, M. D. Title: On the Distribution of Modulo One over Special Primes Keywords: 511.3, distribution modulo one, Piatetski-Shapiro primesAbstract: 2026.Let denote the integer part of, and let denote the distance from to the nearest integer.In this paper, we prove that if is irrational and is an arbitrary real, then for every fixed, there exist infinitely many primes of the form satisfying the inequality. References - Vinogradov IM The method of trigonometrical sums in the theory of numbers Trudy Mat. Inst. Steklov. 1947 23 3 109 29417
- Matomäki K The distribution of modulo one Math. Proc. Cambridge Philos. Soc. 2009 147 2 267 283 2525926 10.1017/S030500410900245X
- Ghosh A The distribution of modulo one Proc. London Math. Soc. (3) 1981 42 2 252 269 607303 10.1112/plms/s3-42.2.252
- Baker RC Harman G On the distribution of modulo one Mathematika 1991 38 1 170 184 1116693 10.1112/S0025579300006525
- Harman G On the distribution of modulo one. II Proc. London Math. Soc. (3) 1996 72 2 241 260 1367078 10.1112/plms/s3-72.2.241
- Harman G Trigonometric sums over primes. II Glasgow Math. J. 1983 24 1 23 37 685480 10.1017/S0017089500005024
- Harman G On the distribution of modulo one J. London Math. Soc. (2) 1983 27 1 9 18 686496 10.1112/jlms/s2-27.1.9
- Wong KC On the distribution of modulo 1 Glasgow Math. J. 1997 39 2 121 130 1460628 10.1017/S0017089500032018
- Piatetski-Shapiro II On the distribution of prime numbers in sequences of the form Mat. Sb. (N.S.) 1953 33(75) 3 559 566 59302
- Rivat J Wu J Prime numbers of the form Glasgow Math. J. 2001 43 2 237 254 1838628 10.1017/S0017089501020080
- Maier H Rassias MT The ternary Goldbach problem with two Piatetski-Shapiro primes and a prime with a missing digit Commun. Contemp. Math. 2023 25 2 4546880 10.1142/S0219199721501017 2150101 45 pp.
- Maier H Rassias MT Prime avoidance property of th powers of Piatetski-Shapiro primes J. Théor. Nombres Bordeaux 2025 37 2 715 725 4963753 10.5802/jtnb.1339
- Dimitrov SI On the distribution of modulo one over Piatetski-Shapiro primes Indian J. Pure Appl. Math. 2023 54 3 858 867 4626038 10.1007/s13226-022-00307-9
- Li X Li J Zhang M On the distribution of modulo one in the intersection of two Piatetski-Shapiro sets Ramanujan J. 2024 65 2 743 758 4800989 10.1007/s11139-024-00914-6
- Baier S Rahaman H Diophantine Approximation with Piatetski-Shapiro Primes 2024 arXiv:2408.01314
- Dimitrov SI Lazarova MD On the distribution of modulo one over primes of the form Ramanujan J. 2025 68 3 4973087 10.1007/s11139-025-01210-7 79 13 pp.
- Heath-Brown DR The Pjateckii-Šapiro prime number theorem J. Number Theory 1983 16 2 242 266 698168 10.1016/0022-314X(83)90044-6
- Li J Zhang M Hua’s theorem with the primes in Piatetski-Shapiro prime sets Int. J. Number Theory 2018 14 1 193 220 3726249 10.1142/S1793042118500124
- Karatsuba AA Principles of Analytic Number Theory 1983 Moscow Nauka Second edition; Language: Russian
- Vaughan RC On the distribution of modulo 1 Mathematika 1977 24 2 135 141 472731 10.1112/S0025579300009025
- Heath-Brown DR A new th derivative estimate for exponential sums via Vinogradov’s mean value Proc. Steklov Inst. Math. 2017 296 1 88 103 10.1134/S0081543817010072
- Iwaniec H Kowalski E Analytic Number Theory 2004 Providence Amer. Math. Soc. 10.1090/coll/053 Amer. Math. Soc. Colloq. Publ., 53
- Harman G Trigonometric sums over primes. I Mathematika 1981 28 2 249 254 645105 10.1112/S0025579300010305
- Vaughan RC An elementary method in prime number theory Acta Arith. 1980 37 111 115 598869 10.4064/aa-37-1-111-115
Issue
| Siberian Mathematical Journal, vol. 67, pp. 922-932, 2026, Russia, https://doi.org/10.1134/S0037446626040166 |
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