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On sums of Hecke series in short intervals. (English) Zbl 0994.11020
Let $$H_j(s)$$ be the standard set of Hecke series associated with Maass wave forms of the full modular group with associated eigenvalues $$\lambda_j= \kappa_j^2+ 1/4$$. Then, with the usual notation, it is shown that $\sum_{K-G< \kappa_j\leq K+G} \alpha_j H_j(1/2)^3\ll GK^{1+\varepsilon},$ for $$1\leq G\leq K$$ and any fixed $$\varepsilon> 0$$. As a corollary one has the new upper bound $$H_j(1/2)\ll \kappa_j^{1/3+\varepsilon}$$.

##### MSC:
 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11F37 Forms of half-integer weight; nonholomorphic modular forms
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