Fourier coefficients of non-holomorphic modular forms and sums of Kloosterman sums (Q1175126)

Using the work of \textit{N. V. Kuznetsov} [Mat. Sb., Nov. Ser. 111(153), 334-383 (1980; Zbl 0427.10016)] the author proves an estimate for the mean value of the squares of the Fourier coefficients of Maass cusp forms averaged over the index of eigenfunctions of the Laplacian in short intervals. As a main result he states \[ \sum_{| k-t_ j|<1}|\rho_ j(n)|^ 2/ch(t_ j)\ll t, \hbox{ valid for } t\gg n^{1+\epsilon}. \] One should notice, that this is better by the factor \(\log t\) than what could be deduced from Kuznetsov's asymptotic formula in [loc.cit]. Combining this with the results of \textit{D. Goldfeld} and \textit{P. Sarnak} [Invent. Math. 71, 243-250 (1983; Zbl 0507.10029)] he obtains an integral mean value formula for Kloosterman sums \[ \int_ Y^{eY}\biggl(\sum_{c\leq x}{S(m,n,c)\over c}\biggr)^ 2{dx \over x}\ll_{m,n} \log Y. \]