Ivić, Aleksandar
On sums of Hecke series in short intervals
Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 2 , p. 453-468
Zbl 0994.11020 | MR 1879668 | 2 citations dans Numdam
URL stable : http://www.numdam.org/item?id=JTNB_2001__13_2_453_0

On a K-Gk j K+G α j H j 3 (1 2) ϵ GK 1+ϵ pour K ϵ GK,ouα j =ρ j (1) 2 (coshπk j ) -1 ,etρ j (1) est le premier coefficient de Fourier de forme de Maass correspondant à la valeur propre λ j =k j 2 +1 4 à laquelle le série de Hecke H j (s) est attachée. Ce résultat fournit l’estimation nouvelle H j (1 2 ϵ k j 1 3+ϵ .
We have K-Gk j K+G α j H j 3 (1 2) ϵ GK 1+ϵ for K ϵ GK,whereα j =ρ j (1) 2 (coshπk j ) -1 ,andρ j (1) is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue λ j =k j 2 +1 4 to which the Hecke series H j (s) is attached. This result yields the new bound H j (1 2 ϵ k j 1 3+ϵ .

Bibliographie

[1] J.B. Conrey, H. Iwaniec, The cubic moment of central values of automorphic L-functions. Ann. of Math. (2) 151 (2000), 1175-1216. MR 1779567 | Zbl 0973.11056

[2] S.W. Graham, G. Kolesnik, Van der Corput's Method of Exponential Sums. LMS Lecture Note Series 126, Cambridge University Press, Cambridge, 1991. MR 1145488 | Zbl 0713.11001

[3] M.N. Huxley, Area, Lattice Points, and Exponential Sums. London Math. Soc. Monographs 13, Oxford University Press, Oxford, 1996. MR 1420620 | Zbl 0861.11002

[4] A. Ivic, The Riemann zeta-function. John Wiley and Sons, New York, 1985. MR 792089 | Zbl 0556.10026

[5] A. Ivic, Y. Motohashi, On some estimates involving the binary additive divisor problem. Quart. J. Math. (Oxford) 46 (1995), 471-483. MR 1366618 | Zbl 0847.11046

[6] H. Iwaniec, Small eigenvalues of Laplacian for Γ0 (N). Acta Arith. 56 (1990), 65-82. Zbl 0702.11034

[7] H. Iwaniec, The spectral growth of automorphic L-functions. J. Reine Angew. Math. 428 (1992), 139-159. MR 1166510 | Zbl 0746.11024

[8] S. Katok, P. Sarnak, Heegner points, cycles and Maass forms. Israel J. Math. 84 (1993), 193-227. MR 1244668 | Zbl 0787.11016

[9] N.N. Lebedev, Special functions and their applications. Dover Publications, Inc., New York, 1972. MR 350075 | Zbl 0271.33001

[10] W. Luo, Spectral mean-values of automorphic L-functions at special points. Analytic Number Theory: Proc. of a Conference in Honor of H. Halberstam, Vol. 2 (eds. B. C. Berndt et al.), Birkhauser, Boston etc., 1996, 621-632. MR 1409382 | Zbl 0866.11034

[11] Y. Motohashi, Spectral mean values of Maass wave forms. J. Number Theory 42 (1992), 258-284. MR 1189505 | Zbl 0759.11026

[12] Y. Motohashi, The binary additive divisor problem. Ann. Sci. l'École Norm. Sup. 4e série 27 (1994), 529-572. Numdam | MR 1296556 | Zbl 0819.11038

[13] Y. Motohashi, Spectral theory of the Riemann zeta-function. Cambridge University Press, Cambridge, 1997. MR 1489236 | Zbl 0878.11001