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On the Exceptional Set of the Sum of a Prime Number and a Fixed Degree of a Prime Number

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Abstract

Let X be a sufficiently great real number and M denote the set of natural numbers not exceeding X which cannot be written as a sum of a prime and a fixed degree of a prime number from the arithmetical progression with difference d. Let Ed(X) = cardM. We obtain a new numerical degree estimate for the set Ed(X) and an estimate from below for the number of presentations of n ∉ M in the specified type. The proven estimates refine the generalization for an arithmetical progression of results earlier got by V.A. Plaksin.

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References

  1. Plaksin, V.A. “On a Problem of Hua Loo Keng”, Math. Notes 47 (3–4), 278–286 (1990).

    Article  MathSciNet  Google Scholar 

  2. Plaksin, V.A. “An exceptional set of the sum of a prime and a fixed degree of a prime”, Petrozavodsk. 1984. Dep. VINITI №7010-84, 33 p.

    Google Scholar 

  3. Montgomery, H.L., Vaughan, R.C. “The Exceptional Set in Goldbach’s Problem”, Acta Arith. 27, 353–370 (1975).

    Article  MathSciNet  Google Scholar 

  4. Allakov, I. The Study of the Density of Numerical Sequences by Analytical Methods, Doctoral Dissertation in Mathematics and Physics (Tashkent, 2008).

    Google Scholar 

  5. Vinogradov, A.I. “The Binary Hardy-Littlewood Problem”, Acta Arith. 46 (1), 33–56 (1985).

    Article  MathSciNet  Google Scholar 

  6. Vaughan, R.C. “A New Iterative Method in Warings Problems”, Acta Math. 162 (1–2), 1–71 (1989).

    Article  MathSciNet  Google Scholar 

  7. Allakov, I.A. An Exceptional Set of the Sum of Two Primes, Candidate’s Dissertation (Leningrad, 1983).

    Google Scholar 

  8. Allakov, I.A., Khamzaev, E. “Generalization of a Theorem of Gallagher for the Primes of an Arithmetic Progression”, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 1, 13–18 (1987).

    MathSciNet  MATH  Google Scholar 

  9. Karatsuba, A.A. Fundamentals of Analytical Number Theory (Nauka, Moscow, 1983) [in Russian].

    Google Scholar 

  10. Montgomery, H.L., Vaughan, R.C. Multiplicative Number Theory. I. Cassical theory (Cambridge Univ. Press, Cambridge, 2007).

    MATH  Google Scholar 

  11. Davenport, R.H. Multiplicative Number Theory, Second edition (Springer, New York-Berlin, 1980).

    Book  Google Scholar 

  12. Gallagher, P.X. “A large sieve density estimate near σ =1”, Invent. Math. 11 (4), 329–339 (1970).

    Article  MathSciNet  Google Scholar 

  13. Vinogradov, A.I. Method of Trigonometric Sums in Number Theory (Nauka, Moscow, 1980) [in Russian].

    MATH  Google Scholar 

  14. Allakov, I.A. “An Estimate for Trigonometric Sums in Powers of Prime Numbers in an Arithmetic Progression”, Dokl. Akad. Nauk UzSSR 9, 3–4 (1990).

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Termez State University, 43 Hodzhaeva str., Termez, 190111, Republic of Uzbekistan

    I. Allakov & A. Sh. Safarov

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  1. I. Allakov
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  2. A. Sh. Safarov
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Correspondence to I. Allakov or A. Sh. Safarov.

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Russian Text © The Author(s), 2020, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2020, No. 1, pp. 11–25.

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Allakov, I., Safarov, A.S. On the Exceptional Set of the Sum of a Prime Number and a Fixed Degree of a Prime Number. Russ Math. 64, 8–21 (2020). https://doi.org/10.3103/S1066369X20010028

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  • DOI: https://doi.org/10.3103/S1066369X20010028

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