Waring–Goldbach Problem with Piatetski-Shapiro Primes
Yıldırım Akbal1; Ahmet M. Güloğlu1
1 Department of Mathematics Bilkent University 06800 Bilkent, Ankara, Turkey
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 449-467.

In this paper, we exhibit an asymptotic formula for the number of representations of a large integer as a sum of a fixed power of Piatetski-Shapiro primes, thereby establishing a variant of Waring–Goldbach problem with primes from a sparse sequence.

Dans cet article nous donnons une formule asymptotique pour le nombre de représentations d’un grand entier comme somme de puissances identiques des nombres premiers de Piatetski-Shapiro, établissant donc une variante du problème de Waring–Goldbach pour des suites clairsemées de nombres premiers.

Received:
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Accepted:
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DOI: 10.5802/jtnb.1033
Classification: 11P32, 11P05, 11P55, 11L03, 11L07, 11L15, 11L20, 11B83
Keywords: Waring–Goldbach Problem, Piatetski-Shapiro Primes, Circle Method, Weyl Sums, Exponential Sums, van der Corput’s Method, Vinogradov’s Mean value theorem
Yıldırım Akbal 1; Ahmet M. Güloğlu 1

1 Department of Mathematics Bilkent University 06800 Bilkent, Ankara, Turkey
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Yıldırım Akbal; Ahmet M. Güloğlu. Waring–Goldbach Problem with Piatetski-Shapiro Primes. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 449-467. doi : 10.5802/jtnb.1033. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1033/

[1] Yıldırım Akbal; Ahmet M. Güloğlu Piatetski-Shapiro meets Chebotarev, Acta Arith., Volume 167 (2015) no. 4, pp. 301-325 | DOI | MR | Zbl

[2] Yıldırım Akbal; Ahmet M. Güloğlu Waring’s Problem with Piatetski Shapiro Numbers, Mathematika, Volume 62 (2016) no. 2, pp. 524-550 | DOI | MR | Zbl

[3] Jean Bourgain On the Vinogradov mean value, Proc. Steklov Inst. Math., Volume 296 (2017), pp. 30-40 | DOI | Zbl

[4] Jean Bourgain; Ciprian Demeter; Larry Guth Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three, Ann. Math., Volume 184 (2016) no. 2, pp. 633-682 | DOI | MR | Zbl

[5] Sidney W. Graham; Grigori Kolesnik van der Corput’s method of exponential sums, London Mathematical Society Lecture Note Series, 126, Cambridge University Press, 1991 | MR | Zbl

[6] D. Roger Heath-Brown A new kth derivative estimate for exponential sums via Vinogradov?s mean value, Proc. Steklov Inst. Math., Volume 296 (2017), pp. 88-103 | DOI | Zbl

[7] Loo-Keng Hua Additive theory of prime numbers, Translations of Mathematical Monographs, 13, American Mathematical Society, 1965, xiii+190 pages | MR | Zbl

[8] Henryk Iwaniec; Emmanuel Kowalski Analytic number theory, Colloquium Publications, 53, American Mathematical Society, 2004, xii+615 pages | MR | Zbl

[9] Angel V. Kumchev On the Piatetski-Shapiro-Vinogradov theorem, J. Théor. Nombres Bordx., Volume 9 (1997) no. 1, pp. 11-23 | DOI | Numdam | MR | Zbl

[10] Angel V. Kumchev; Trevor D. Wooley On the Waring-Goldbach problem for eighth and higher powers, J. Lond. Math. Soc., Volume 93 (2016) no. 3, pp. 811-824 | DOI | MR | Zbl

[11] Angel V. Kumchev; Trevor D. Wooley On the Waring-Goldbach problem for seventh and higher powers, Monatsh. Math., Volume 183 (2017) no. 2, pp. 303-310 | DOI | MR | Zbl

[12] Ilya I. Piatetski-Shapiro On the distribution of prime numbers in sequences of the form [f(n)], Mat. Sb., N. Ser., Volume 33 (1953), pp. 559-566 | MR | Zbl

[13] Robert C. Vaughan The Hardy-Littlewood method, Cambridge Tracts in Mathematics, 125, Cambridge University Press, 1997, xiv+232 pages | MR | Zbl

[14] Trevor D. Wooley The asymptotic formula in Waring’s problem, Int. Math. Res. Not., Volume 2012 (2012) no. 7, pp. 1485-1504 | DOI | MR | Zbl

[15] Deyu Zhang; Wenguang Zhai The Waring-Goldbach problem in thin sets of primes. II, Acta Math., Volume 48 (2005) no. 4, pp. 809-816 | MR | Zbl

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