Geodesic
Mean value of the Hecke series associated with the cusp forms of weight zero
Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part IV, Tome 109 (1981), pp. 93-130

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In this work an asymptotic formula (with a powerlike residue of degree) is proved for the quadratic mean over the critical line of the Dirichlet series associated with the cusp form of weight zero for modular group. The proof is based on author's different formulas of summation for Kloosterman sums. 
@article{ZNSL_1981_109_a4,
     author = {N. V. Kuznetsov},
     title = {Mean value of the {Hecke} series associated with the cusp forms of weight zero},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {93--130},
     publisher = {mathdoc},
     volume = {109},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1981_109_a4/}
}
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N. V. Kuznetsov. Mean value of the Hecke series associated with the cusp forms of weight zero. Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part IV, Tome 109 (1981), pp. 93-130. http://geodesic.mathdoc.fr/item/ZNSL_1981_109_a4/