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A logarithmic inequality involving prime numbers

S. I. Dimitrov
장전수학회 (Jangjeon Mathematical Society)|2021.07|Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집) (Proceedings of the Jangjeon Mathematical Society)|제24권 제3호|pp. 403-416|DOI 10.17777/pjms.2021.24.3.011

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키워드

Diophantine inequalityLogarithmic inequalityPrime numbers
주제분류수학

초록

Assume that N is a sufficiently large positive number. In this paper we show that for a small constant ε > 0, the logarithmic inequality |p1 log p1 + p2 log p2 + p3 log p3 - N | < ε has a solution in prime numbers p1, p2, p3.

Abstract

Assume that N is a sufficiently large positive number. In this paper we show that for a small constant ε > 0, the logarithmic inequality |p1 log p1 + p2 log p2 + p3 log p3 - N | < ε has a solution in prime numbers p1, p2, p3.

참고문헌 (14)

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  2. R. Baker, "A ternary diophantine inequality over primes", Acta Arith, 162, pp.159~196 (2014)
  3. Y. Cai, "On a diophantine inequality involving prime numbers", Acta Math. Sinica, Chinese Series, 39, pp.733~742 (1996)
  4. Y. Cai, "On a diophantine inequality involving prime numbers III", Acta Math. Sinica, English Series, 15, pp.387~394 (1999)
  5. Y. Cai, "A ternary Diophantine inequality involving primes", Int. J. Number Theory, 14, pp.2257~2268 (2018)
  6. X. Cao, "A Diophantine inequality with prime numbers", Acta Math. Sinica, Chinese Series, 45, pp.361~370 (2002)
  7. S. W. Graham, "Van der Corput’s Method of Exponential Sums" (1991), Cambridge University Press
  8. H. Iwaniec, "Analytic number theory, Colloquium Publications, 53" (2004), Amer. Math. Soc
  9. A. Kumchev, "On an equation with prime numbers", Acta Arith, 83, pp.117~126 (1998)
  10. A. Kumchev, "A diophantine inequality involving prime powers", Acta Arith, 89, pp.311~330 (1999)
  11. I. Piatetski-Shapiro, "On a variant of the Waring-Goldbach problem", Mat. Sb, 30, pp.105~120 (1952)
  12. E. Titchmarsh, "The Theory of the Riemann Zeta-function" (1986), Clarendon Press
  13. D. Tolev, "On a diophantine inequality involving prime numbers", Acta Arith, 61, pp.289~306 (1992)
  14. R. C. Vaughan, "An elementary method in prime number theory", Acta Arithmetica, 37, pp.111~115 (1980)

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