On sums of Hecke series in short intervals
Journal de Théorie des Nombres de Bordeaux, Volume 13 (2001) no. 2, pp. 453-468.

We have ∑ K-G≤k j ≤K+G α j H j 3 (1 2)≪ ϵ GK 1+ϵ for K ϵ ≤G≤K,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 .

On a ∑ K-G≤k j ≤K+G α j H j 3 (1 2)≪ ϵ GK 1+ϵ pour K ϵ ≤G≤K,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 . 

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     title = {On sums of {Hecke} series in short intervals},
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Aleksandar Ivić. On sums of Hecke series in short intervals. Journal de Théorie des Nombres de Bordeaux, Volume 13 (2001) no. 2, pp. 453-468. doi : 10.5802/jtnb.333. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.333/

[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

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