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

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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

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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

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