On sums of Hecke series in short intervals

Aleksandar Ivić

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 2, page 453-468
  • ISSN: 1246-7405

Abstract

top
We have K - G k j K + G α j H j 3 ( 1 2 ) ϵ G K 1 + ϵ for K ϵ G K , where α j = ρ j ( 1 ) 2 ( cosh π k j ) - 1 , and ρ 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 .

How to cite

top

Ivić, Aleksandar. "On sums of Hecke series in short intervals." Journal de théorie des nombres de Bordeaux 13.2 (2001): 453-468. <http://eudml.org/doc/248725>.

@article{Ivić2001,
abstract = {We have $\sum _\{K-G \le k_\{j\} \le K + G\} \alpha _j H^3_j (\frac\{1\}\{2\}) \ll _\epsilon GK^\{1 + \epsilon \}$ for $K^\epsilon \le G \le K, \text\{ where \} \alpha _j = \left|\rho _j (1) \right|^2 (\cosh \pi k_j)^\{-1\}, \text\{ and \} \rho _j (1)$ is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue $\lambda _j = k^2_j + \frac\{1\}\{4\}$ to which the Hecke series $H_j(s)$ is attached. This result yields the new bound $H_j (\frac\{1\}\{2\} \ll _\{\epsilon \} k^\{\frac\{1\}\{3\} + \epsilon \}_j.$},
author = {Ivić, Aleksandar},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {cubic moment; short interval; Hecke series; Maass wave forms; upper bound},
language = {eng},
number = {2},
pages = {453-468},
publisher = {Université Bordeaux I},
title = {On sums of Hecke series in short intervals},
url = {http://eudml.org/doc/248725},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Ivić, Aleksandar
TI - On sums of Hecke series in short intervals
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 2
SP - 453
EP - 468
AB - We have $\sum _{K-G \le k_{j} \le K + G} \alpha _j H^3_j (\frac{1}{2}) \ll _\epsilon GK^{1 + \epsilon }$ for $K^\epsilon \le G \le K, \text{ where } \alpha _j = \left|\rho _j (1) \right|^2 (\cosh \pi k_j)^{-1}, \text{ and } \rho _j (1)$ is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue $\lambda _j = k^2_j + \frac{1}{4}$ to which the Hecke series $H_j(s)$ is attached. This result yields the new bound $H_j (\frac{1}{2} \ll _{\epsilon } k^{\frac{1}{3} + \epsilon }_j.$
LA - eng
KW - cubic moment; short interval; Hecke series; Maass wave forms; upper bound
UR - http://eudml.org/doc/248725
ER -

References

top
  1. [1] J.B. Conrey, H. Iwaniec, The cubic moment of central values of automorphic L-functions. Ann. of Math. (2) 151 (2000), 1175-1216. Zbl0973.11056MR1779567
  2. [2] S.W. Graham, G. Kolesnik, Van der Corput's Method of Exponential Sums. LMS Lecture Note Series126, Cambridge University Press, Cambridge, 1991. Zbl0713.11001MR1145488
  3. [3] M.N. Huxley, Area, Lattice Points, and Exponential Sums. London Math. Soc. Monographs13, Oxford University Press, Oxford, 1996. Zbl0861.11002MR1420620
  4. [4] A. Ivic, The Riemann zeta-function. John Wiley and Sons, New York, 1985. Zbl0556.10026MR792089
  5. [5] A. Ivic, Y. Motohashi, On some estimates involving the binary additive divisor problem. Quart. J. Math. (Oxford) 46 (1995), 471-483. Zbl0847.11046MR1366618
  6. [6] H. Iwaniec, Small eigenvalues of Laplacian for Γ0 (N). Acta Arith.56 (1990), 65-82. Zbl0702.11034
  7. [7] H. Iwaniec, The spectral growth of automorphic L-functions. J. Reine Angew. Math.428 (1992), 139-159. Zbl0746.11024MR1166510
  8. [8] S. Katok, P. Sarnak, Heegner points, cycles and Maass forms. Israel J. Math.84 (1993), 193-227. Zbl0787.11016MR1244668
  9. [9] N.N. Lebedev, Special functions and their applications. Dover Publications, Inc., New York, 1972. Zbl0271.33001MR350075
  10. [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. Zbl0866.11034MR1409382
  11. [11] Y. Motohashi, Spectral mean values of Maass wave forms. J. Number Theory42 (1992), 258-284. Zbl0759.11026MR1189505
  12. [12] Y. Motohashi, The binary additive divisor problem. Ann. Sci. l'École Norm. Sup.4e série 27 (1994), 529-572. Zbl0819.11038MR1296556
  13. [13] Y. Motohashi, Spectral theory of the Riemann zeta-function. Cambridge University Press, Cambridge, 1997. Zbl0878.11001MR1489236

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.