On sums of Hecke series in short intervals

• Published in: Journal de theorie des nombres de bordeaux Vol. 13; no. 2; pp. 453 - 468
• Main Author: IVIC, Aleksandar
• Format: Journal Article
• Language: English
• Published: Talence Institut de mathématiques de Bordeaux 01.01.2001; Université de Bordeaux 1, laboratoire de mathématiques pures
• Subjects: Algebra; Eigenvalues; Exact sciences and technology; Fourier coefficients; Integers; Log integral function; Mathematical lattices; Mathematics; Number theory; Sciences and techniques of general use; Series convergence; Spectral theory; Waveforms; Hyperbolic measure; Hecke algebra; Integrability; Hecke series; Eigenvalue; Hecke operator; Hilbert space; Fourier series; Summation; Laplacian; Number theory
• ISSN: 1246-7405; 2118-8572
• DOI: 10.5802/jtnb.333

On a $\sum\nolimits_{_{K - G \leqslant {k_j} \leqslant K + G}} {{\alpha _j}H_j^3\left( {\frac{1}{2}} \right)} { \ll _\varepsilon }G{K^{1 + \varepsilon }}$ pour Kε ≤ G ≤ K, ou αj = |ρj(1)|²(cosh πkj)⁻¹, et ρj(1) est le premier coefficient de Fourier de forme de Maass correspondent à la valeur propre ${\lambda _j} = k_j^2 + \frac{1}{4}$ à laquelle le série de Hecke Hj(s) est attachée. Ce resultat fournit l'estimation nouvelle ${H_j}\left( {\frac{1}{2}} \right){ \ll _\varepsilon }k_j^{\frac{1}{3} + \varepsilon }$. We have $\sum\nolimits_{_{K - G \leqslant {k_j} \leqslant K + G}} {{\alpha _j}H_j^3\left( {\frac{1}{2}} \right)} { \ll _\varepsilon }G{K^{1 + \varepsilon }}$ for Kε ≤ G ≤ K, where αj = |ρj(1)|² (cosh πkj)⁻¹, and ρj(1) is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue ${\lambda _j} = k_j^2 + \frac{1}{4}$ to which the Hecke series Hj(s) is attached. This result yields the new bound ${H_j}\left( {\frac{1}{2}} \right){ \ll _\varepsilon }k_j^{\frac{1}{3} + \varepsilon }$.