Geodesic
A new $k$th derivative estimate for exponential sums via Vinogradov's mean value
Informatics and Automation, Analytic and combinatorial number theory, Tome 296 (2017), pp. 95-110.

Voir la notice de l'article provenant de la source Math-Net.Ru

We give a slight refinement to the process by which estimates for exponential sums are extracted from bounds for Vinogradov's mean value. Coupling this with the recent works of Wooley, and of Bourgain, Demeter and Guth, providing optimal bounds for the Vinogradov mean value, we produce a powerful new $k$th derivative estimate. Roughly speaking, this improves the van der Corput estimate for $k\ge 4$. Various corollaries are given, showing for example that $\zeta (\sigma +it)\ll _{\varepsilon }t^{(1-\sigma )^{3/2}/2+\varepsilon }$ for $t\ge 2$ and $0\le \sigma \le 1$, for any fixed $\varepsilon >0$. 
@article{TRSPY_2017_296_a6,
     author = {D. R. Heath-Brown},
     title = {A new $k$th derivative estimate for exponential sums via {Vinogradov's} mean value},
     journal = {Informatics and Automation},
     pages = {95--110},
     publisher = {mathdoc},
     volume = {296},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2017_296_a6/}
}
TY  - JOUR
AU  - D. R. Heath-Brown
TI  - A new $k$th derivative estimate for exponential sums via Vinogradov's mean value
JO  - Informatics and Automation
PY  - 2017
SP  - 95
EP  - 110
VL  - 296
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2017_296_a6/
LA  - ru
ID  - TRSPY_2017_296_a6
ER  - 
%0 Journal Article
%A D. R. Heath-Brown
%T A new $k$th derivative estimate for exponential sums via Vinogradov's mean value
%J Informatics and Automation
%D 2017
%P 95-110
%V 296
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2017_296_a6/
%G ru
%F TRSPY_2017_296_a6
D. R. Heath-Brown. A new $k$th derivative estimate for exponential sums via Vinogradov's mean value. Informatics and Automation, Analytic and combinatorial number theory, Tome 296 (2017), pp. 95-110. http://geodesic.mathdoc.fr/item/TRSPY_2017_296_a6/

[1] Bourgain J., Demeter C., Guth L., “Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three”, Ann. Math. Ser. 2, 184:2 (2016), 633–682 ; arXiv: 1512.01565[math.NT] | DOI | MR | Zbl

[2] Ford K., “Vinogradov's integral and bounds for the Riemann zeta function”, Proc. London Math. Soc. Ser. 3, 85:3 (2002), 565–633 | DOI | MR | Zbl

[3] Gritsenko S. A., “Ob otsenkakh trigonometricheskikh summ po tretei proizvodnoi”, Mat. zametki, 60:3 (1996), 383–389 | DOI | MR | Zbl

[4] Korobov N. M., “Otsenki trigonometricheskikh summ i ikh prilozheniya”, UMN, 13:4 (1958), 185–192 | MR | Zbl

[5] Montgomery H. L., Topics in multiplicative number theory, Lect. Notes Math., 227, Springer, Berlin, 1971 | DOI | MR | Zbl

[6] Robert O., “On van der Corput's $k$-th derivative test for exponential sums”, Indag. math. New Ser., 27:2 (2016), 559–589 | DOI | MR | Zbl

[7] Robert O., Sargos P., “A fourth derivative test for exponential sums”, Compos. Math., 130:3 (2002), 275–292 | DOI | MR | Zbl

[8] Sargos P., “Points entiers au voisinage d'une courbe, sommes trigonométriques courtes et paires d'exposants”, Proc. London Math. Soc. Ser. 3, 70:2 (1995), 285–312 | DOI | MR | Zbl

[9] Sargos P., “An analog of van der Corput's $A^4$-process for exponential sums”, Acta arith., 110:3 (2003), 219–231 | DOI | MR | Zbl

[10] Titchmarsh E. C., The theory of the Riemann zeta-function, 2nd ed., Clarendon Press, Oxford, 1986 | MR | Zbl

[11] Vinogradov I. M., “Novye otsenki summ Veilya”, DAN SSSR, 3:5 (1935), 195–198 | Zbl

[12] Vinogradov I. M., “Novaya otsenka funktsii $\zeta(1+it)$”, Izv. AN SSSR. Ser. mat., 22:2 (1958), 161–164 | MR | Zbl

[13] Wooley T. D., “The cubic case of the main conjecture in Vinogradov's mean value theorem”, Adv. Math., 294 (2016), 532–561 ; arXiv: 1401.3150[math.NT] | DOI | MR | Zbl