Almost Convergent 0-1-Sequences and Primes

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/p9825-1385-3019-c Almost Convergent 0-1-Sequences and Primes Avdeev N. N. Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 4. Abstract: This paper is devoted to 0-1-sequences. We establish the connection between values that Sucheston functional can take on 0-1-sequence and multiplicative structure of the support of the sequence. If~the set of all the divisors of support elements is finite, then the sequence is almost convergent to zero. Then we consider characteristic sequences of sets of multiples and establish necessary and sufficient conditions for the upper Sucheston functional to be 1 on such sequence. We prove that there are infinitely many sets of pairwise relative prime numbers such that the lower Sucheston functional evaluates to 1 on the corresponding set of multiples (and lower Sucheston functional never evaluates to 0 on a set of multiples). Keywords: space of bounded sequences, Banach limit, Sucheston functional, almost convergent sequence, 0-1-sequence, integer factorization, sets of multiples Language: Russian Download the full text For citation: Avdeev, N. N. Almost Convergent 0-1-Sequences and Primes, Vladikavkaz Math. J., 2021, vol. 23, no. 4, pp. 5-14 (in Russian). DOI 10.46698/p9825-1385-3019-c + References 1. Mazur, S. O Metodach Sumowalnosci, Ann. Soc. Polon. Math. (Supplement), 1929, pp. 102-107. 2. Banach, S. Theorie des Operations Lineaires, Sceaux, Editions Jacques Gabay, 1993, iv+128 p. 3. Lorentz, G. G. A Contribution to the Theory of Divergent Sequences, Acta Mathematica, 1948, vol. 80, no. 1, pp. 167-190. DOI: 10.1007/BF02393648. 4. Sucheston, L. Banach Limits, Amer. Math. Monthly, 1967, vol. 74, pp. 308-311. DOI: 10.2307/2316038. 5. Semenov, E. M., Sukochev, F. A. and Usachev, A. S. Geometry of Banach Limits and their Applications, Russian Mathematical Surveys, 2020, vol. 75, no. 4, pp. 725. DOI: 10.1070/RM9901. 6. Avdeev, N. On the Space of Almost Convergent Sequences, Mathematical Notes, 2019, vol. 105, no. 3/4, pp. 464-468. DOI: 10.1134/S0001434619030179. 7. Hall, R. R. and Tenenbaum, G. On Behrend Sequences, Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 1992, vol. 112, pp. 467-482. DOI: 10.1017/S0305004100071140. 8. Davenport, H. and Erdos, P. On Sequences of Positive Integers, Acta Arithm, 1936, vol. 2, pp. 147-151. DOI: 10.4064/aa-2-1-147-151. 9. Davenport, H. and Erdos, P. On Sequences of Positive Integers, J. Indian Math. Soc., New Series, 1951, vol. 15, pp. 19-24. DOI: 10.18311/JIMS/1951/17063. 10. Besicovitch, A. On the Density of Certain Sequences of Integers, Mathematische Annalen, 1935, vol. 110, no. 1, pp. 336-341. DOI: 10.1007/BF01448032. 11. Hall, R. R. Sets of Multiples, Cambridge Tracts in Mathematics, Cambridge, Cambridge University Press, 1996. DOI: 10.1017/CBO9780511566011. 12. Euler, L. Variae Observationes Circa Series Infinitas, Commentarii Academiae Scientiarum Imperialis Petropolitanae, 1737, vol. 9, no. 1737, pp. 160-188. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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