Abstract
In this paper, we treat the estimate on exponential sums over cubes of primes in short intervals, and improve a previous bound of Kumchev (in: Number Theory: Arithmetic in Shangri-La (Proc. China-Japan Seminar Number Theory), pp. 116–131, World Scientific, Singapore, 2013). Moreover, we present some applications to the cubic Waring–Goldbach problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baker, R.C., Harman, G.: On the distribution of \(\alpha p^k\) modulo one. Mathematika 38, 170–184 (1991)
Heath-Brown, D.R.: Prime numbers in short intervals and a generalized Vaughan identity. Can. J. Math. 34, 1365–1377 (1982)
Kawada, K., Wooley, T.D.: On the Waring-Goldbach problem for fourth and fifth powers. Proc. Lond. Math. Soc. 83, 1–50 (2001)
Kumchev, A.V.: On Weyl sums over primes and almost primes. Mich. Math. J. 54, 243–268 (2006)
Kumchev, A.V.: On Weyl sums over primes in short intervals. In: Number Theory: Arithmetic in Shangri-La (Proc. China-Japan Seminar Number Theory), pp. 116–131. World Scientific, Singapore (2013)
Liu, J.Y., Lü, G.S., Zhan, T.: Exponential sums over primes in short intervals. Sci. China Ser. A 49, 611–619 (2006)
Liu, Z.X., Sun, Q.F.: Sums of cubes of primes in short intervals. Ramanujan J. 28, 309–321 (2012)
Lü, G.S., Xu, Y.F.: Hua’s theorem with nine almost equal prime variables. Acta Math. Hungar. 116, 309–326 (2007)
Meng, X.M.: The Waring-Goldbach problem in short intervals (in Chinese). J. Shandong Univ. 32, 255–264 (1997)
Ren, X.M.: On exponential sums over primes and application in Waring-Goldbach problem. Sci. China Ser. A 48, 785–797 (2005)
Ren, X.M., Yao, Y.J.: Exceptional set for sums of almost equal prime cubes (in Chinese). Sci. Sin. Math. 45, 23–30 (2015)
Wooley, T.D.: Slim exceptional sets for sums of cubes. Can. J. Math. 54, 417–448 (2002)
Yao, Y.J.: Sums of nine almost equal prime cubes. Front. Math. China 9, 611–619 (2014)
Zhao, L.L.: On the Waring-Goldbach problem for fourth and sixth powers. Proc. Lond. Math. Soc. 108, 1593–1622 (2014)
Acknowledgements
The authors wish to express their sincere appreciation to the referees for their careful reading and wise advice of the manuscript. This work is supported by Natural Science Foundation of China (Grant No. 11871307, 11701344 and 11401344).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Li, T., Yao, Y. Exponential sums over cubes of primes in short intervals and its applications. Math. Z. 299, 83–99 (2021). https://doi.org/10.1007/s00209-020-02649-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-020-02649-8