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:
Revised:
Accepted:
Published online:
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 
@article{JTNB_2018__30_2_449_0,
     author = {Y{\i}ld{\i}r{\i}m Akbal and Ahmet M. G\"ulo\u{g}lu},
     title = {Waring{\textendash}Goldbach {Problem} with {Piatetski-Shapiro} {Primes}},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {449--467},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {2},
     year = {2018},
     doi = {10.5802/jtnb.1033},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1033/}
}
TY  - JOUR
TI  - Waring–Goldbach Problem with Piatetski-Shapiro Primes
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2018
DA  - 2018///
SP  - 449
EP  - 467
VL  - 30
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1033/
UR  - https://doi.org/10.5802/jtnb.1033
DO  - 10.5802/jtnb.1033
LA  - en
ID  - JTNB_2018__30_2_449_0
ER  - 
%0 Journal Article
%T Waring–Goldbach Problem with Piatetski-Shapiro Primes
%J Journal de Théorie des Nombres de Bordeaux
%D 2018
%P 449-467
%V 30
%N 2
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1033
%R 10.5802/jtnb.1033
%G en
%F JTNB_2018__30_2_449_0
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 | Zbl: 1317.11081

[2] Yıldırım Akbal; Ahmet M. Güloğlu Waring’s Problem with Piatetski Shapiro Numbers, Mathematika, Volume 62 (2016) no. 2, pp. 524-550 | Zbl: 1346.11054

[3] Jean Bourgain On the Vinogradov mean value, Proc. Steklov Inst. Math., Volume 296 (2017), pp. 30-40 | Article | Zbl: 1371.11138

[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 | Zbl: 06662221

[5] Sidney W. Graham; Grigori Kolesnik van der Corput’s method of exponential sums, London Mathematical Society Lecture Note Series, Volume 126, Cambridge University Press, 1991 | Zbl: 0713.11001

[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 | Article | Zbl: 06740899

[7] Loo-Keng Hua Additive theory of prime numbers, Translations of Mathematical Monographs, Volume 13, American Mathematical Society, 1965, xiii+190 pages | Zbl: 0192.39304

[8] Henryk Iwaniec; Emmanuel Kowalski Analytic number theory, Colloquium Publications, Volume 53, American Mathematical Society, 2004, xii+615 pages | Zbl: 1059.11001

[9] Angel V. Kumchev On the Piatetski-Shapiro-Vinogradov theorem, J. Théor. Nombres Bordx., Volume 9 (1997) no. 1, pp. 11-23 | Zbl: 0890.11029

[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 | Zbl: 1358.11117

[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 | Zbl: 1381.11099

[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 | Zbl: 0053.02702

[13] Robert C. Vaughan The Hardy-Littlewood method, Cambridge Tracts in Mathematics, Volume 125, Cambridge University Press, 1997, xiv+232 pages | Zbl: 0868.11046

[14] Trevor D. Wooley The asymptotic formula in Waring’s problem, Int. Math. Res. Not., Volume 2012 (2012) no. 7, pp. 1485-1504 | Zbl: 1267.11104

[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 | Zbl: 1124.11317

Cited by Sources: