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

On a ${\sum }_{K-G\le {k}_{j}\le K+G}{\alpha }_{j}{H}_{j}^{3}\left(\frac{1}{2}\right){\ll }_{ϵ}G{K}^{1+ϵ}$ pour ${K}^{ϵ}\le G\le K,\text{ou}\phantom{\rule{4pt}{0ex}}{\alpha }_{j}={\left|{\rho }_{j}\left(1\right)\right|}^{2}{\left(cosh\pi {k}_{j}\right)}^{-1},\phantom{\rule{4pt}{0ex}}\text{et}\phantom{\rule{4pt}{0ex}}{\rho }_{j}\left(1\right)$ est le premier coefficient de Fourier de forme de Maass correspondant à la valeur propre ${\lambda }_{j}={k}_{j}^{2}+\frac{1}{4}$ à laquelle le série de Hecke ${H}_{j}\left(s\right)$ est attachée. Ce résultat fournit l’estimation nouvelle ${H}_{j}\left(\frac{1}{2}{\ll }_{ϵ}{k}_{j}^{\frac{1}{3}+ϵ}.$
We have ${\sum }_{K-G\le {k}_{j}\le K+G}{\alpha }_{j}{H}_{j}^{3}\left(\frac{1}{2}\right){\ll }_{ϵ}G{K}^{1+ϵ}$ for ${K}^{ϵ}\le G\le K,\phantom{\rule{4pt}{0ex}}\text{where}\phantom{\rule{4pt}{0ex}}{\alpha }_{j}={\left|{\rho }_{j}\left(1\right)\right|}^{2}{\left(cosh\pi {k}_{j}\right)}^{-1},\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}{\rho }_{j}\left(1\right)$ 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 ${H}_{j}\left(s\right)$ is attached. This result yields the new bound ${H}_{j}\left(\frac{1}{2}{\ll }_{ϵ}{k}_{j}^{\frac{1}{3}+ϵ}.$

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