Trigonometric sums over primes III
Journal de Théorie des Nombres de Bordeaux, Tome 15 (2003) no. 3, pp. 727-740.

On donne une majoration nouvelle de la somme trigonométrique Pp<2P e(αp k )k5,p désigne un nombre premier et e(x)=exp(2πix).

New bounds are given for the exponential sum Pp<2P e(αp k ) were k5,p denotes a prime and e(x)=exp(2πix).

@article{JTNB_2003__15_3_727_0,
     author = {Harman, Glyn},
     title = {Trigonometric sums over primes {III}},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {727--740},
     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {3},
     year = {2003},
     doi = {10.5802/jtnb.423},
     mrnumber = {2142233},
     zbl = {02184621},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.423/}
}
Glyn Harman. Trigonometric sums over primes III. Journal de Théorie des Nombres de Bordeaux, Tome 15 (2003) no. 3, pp. 727-740. doi : 10.5802/jtnb.423. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.423/

[1] R.C. Baker, G. Harman, On the distribution of αpk modulo one. Mathematika 38 (1991), 170-184. | Zbl 0751.11037

[2] E. Fouvry, P. Michel, Sur certaines sommes d'exponentielles sur les nombres premiers. Ann. Sci. Ec. Norm. Sup. IV Ser. 31 (1998), 93-130. | Numdam | MR 1604298 | Zbl 0915.11045

[3] A. Ghosh, The distribution of αp2 modulo one. Proc. London Math. Soc. (3) 42 (1981), 252-269. | Zbl 0447.10035

[4] G. Harman, Trigonometric sums over primes I. Mathematika 28 (1981), 249-254. | MR 645105 | Zbl 0465.10029

[5] G. Harman, Trigonometric sums over primes II. Glasgow Math. J. 24 (1983), 23-37. | MR 685480 | Zbl 0504.10017

[6] D.R. Heath-Brown, Prime numbers in short intervals and a generalized Vaughan identity. Can. J. Math. 34 (1982), 1365-1377. | MR 678676 | Zbl 0478.10024

[7] K. Kawada, T.D. Wooley, On the Waring-Goldbach problem for fourth and fifth powers. Proc. London Math. Soc. (3) 83 (2001), 1-50. | MR 1829558 | Zbl 1016.11046

[8] R.C. Vaughan, Mean value theorems in prime number theory. J. London Math. Soc. (2) 10 (1975), 153-162. | MR 376567 | Zbl 0314.10028

[9] R.C. Vaughan, The Hardy-Littlewood Method second edition. Cambridge University Press, 1997. | MR 1435742 | Zbl 0868.11046

[10] I.M. Vinogradov, Some theorems concerning the theory of primes. Rec. Math. Moscow N.S. 2 (1937), 179-194. | JFM 63.0131.05 | Zbl 0017.19803

[11] I.M. Vinogradov, A new estimation of a trigonometric sum containing primes. Bull. Acad. Sci. URSS Ser. Math. 2 (1938), 1-13. | JFM 64.0983.01 | Zbl 0018.39002

[12] K.C. Wong, On the distribution of αpk modulo one. Glasgow Math. J. 39 (1997), 121-130. | MR 1460628 | Zbl 0880.11052

[13] T.D. Wooley, New estimates for Weyl sums. Quart. J. Math. Oxford (2) 46 (1995), 119-127. | MR 1326136 | Zbl 0855.11043