Some exponent pairs by Vinogradov’s method. (Quelques paires d’exposants par la méthode de Vinogradov.) (French) Zbl 1021.11025

Estimates for exponential sums \(\sum_{m\sim M}e(f(m))\) depend on conditions on the \(k\)th derivative of \(f\), which should not be too large. If the relevant values of \(k\) are small, then one uses van der Corput’s method, whereas Vinogradov’s method is more powerful for large values of \(k\). The author studies the “junction” of these methods for the values \(k=9, 10, 11\). Using recent advances in both methods, he obtains the following new exponent pairs: \((1/615, 1- 9/615)\) and \((1/915, 1-10/915)\). These are applied to the estimation of Riemann’s zeta-function near the 1-line.


11L15 Weyl sums
11L07 Estimates on exponential sums
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[1] Graham, S.W., Kolesnik, G., Van der Corput’s method for exponential sums. Series 126, Cambridge University Press, 1991. · Zbl 0713.11001
[2] Hua, L.K., On a theorem due to Vinogradov. Quarterly Journal of Maths, Oxford Series 11, 161-176 (1940). This paper is included in Loo-Keng Hua Selected Papers, edited by H. Halberstam, Springer Verlag, 1983. · JFM 66.0165.02
[3] Huxley, M.N., Area, lattice points and exponential sums. Clarendon Press, Oxford, 1996. · Zbl 0861.11002
[4] Karatsuba, A.A., Estimates for trigonometric sums by Vinogradov’s method and some applications. Proc. Steklov Inst. Math.112 (1971), 251-265. · Zbl 0259.10040
[5] Robert, O., Application des systèmes diophantiens aux sommes d’exponentielles. Thèse Université Henri Poincaré - Nancy I, 2001.
[6] Sargos, P., An analog of van der Corput’s A4-process for exponential sums, manuscrit. · Zbl 1069.11034
[7] Wooley, T.D., On Vinogradov’s mean value theorem. Mathematika39 (1992), 379-399. · Zbl 0769.11036
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