Abstract
The well-known three primes theorem says that, for every sufficiently large odd integer N, the equation \(N=p_1+p_2+p_3\) is solvable for prime variables \(p_1, p_2, p_3\). In this paper we shall prove that the three primes theorem still holds if each of the three primes is in the intersection of two Piatetski--Shapiro sets.
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References
R. C. Baker, The intersection of Piatetski-Shapiro sequences, Mathematika, 60 (2014), 347–362.
A. Balog and J. Friedlander, A hybrid of theorems of Vinogradov and Piatetski-Shapiro, Pacific J. Math., 156 (1992), 45–62.
S. W. Graham and G. Kolesnik, Van der Corput's Method of Exponential Sums, Cambridge University Press (1991).
D. R. Heath-Brown, The Pjatechkiĭ-S̆apiro prime number theorem, J. Number Theory, 16 (1983), 242–266.
G. Kolesnik, The distribution of primes in sequences of the form \([n^c]\), Mat. Zametki, 2 (1967), 117–128 (in Russian).
G. Kolesnik, Primes of the form \([n^c]\), Pacific J. Math., 118 (1985), 437–447.
A. Kumchev, On the Piatetski-Shapiro-Vinogradov theorem, J. Thèor. Nombres Bordeaux, 9 (1997), 11–23.
D. Leitmann, Durchschnitte von Pjateckij-Shapiro-Folgen, Monatsh. Math., 94 (1982), 33–44.
H. Q. Liu and J. Rivat, On the Pjateckiĭ-S̆apiro prime number theorem, Bull. London Math. Soc., 24 (1992), 143–147.
I. I. Pyateckiĭ-S̆apiro, On the distribution of prime numbers in sequences of the form \([f(n)]\), Mat. Sbornik N.S., 33 (1953), 559–566 (in Russian).
J. Rivat and P. Sargos, Nombres premiers de la forme \(\lfloor n^c \rfloor\), Canad. J. Math., 53 (2001), 414–433.
E. R. Sirota, Laws for the distribution of primes of the form \(P=[t^c]=[n^d]\) in arithmetic progressions, Studies in number theory, 8, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), 121 (1983), 94–102 (in Russian).
I. M. Vinogradov, Representation of an odd number as the sum of three primes, Dokl. Akad. Nauk SSSR, 15 (1937), 169–172.
E. Wirsing, Thin subbases, Analysis, 6 (1986), 285–308.
W. G. Zhai and X. T. Li, Primes in the intersection of two Piatetski-Shapiro sets (submitted).
W. G. Zhai, On the k-dimensional Piatetski-Shapiro prime number theorem, Sci. China Ser. A, 42 (1999), 1173–1183.
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This work was supported by National Natural Science Foundation of China (Grant No. 11971476).
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Li, X., Zhai, W. The three primes theorem with primes in the intersection of two Piatetski--Shapiro sets. Acta Math. Hungar. 168, 228–245 (2022). https://doi.org/10.1007/s10474-022-01264-9
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DOI: https://doi.org/10.1007/s10474-022-01264-9