Geodesic
On summatory functions for automorphic $L$-functions
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 26, Tome 392 (2011), pp. 202-217

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Let $\lambda_f(n)$ denote the $n$th normalized Fourier coefficient of a primitive holomorphic cusp form $f$ for the full modular group. Let $\Delta(x,f\otimes f)$ be the error term in the asymptotic formula of Rankin and Selberg for $$ \sum_{n\le x}\lambda_f(n)^2. $$ It is proved that $\Delta(x,f\otimes f)=\Omega(x^{3/8})$ and $$ \sum_{n\le x}\lambda_f(n^2)=\Omega(x^{1/3}). $$ Other summatory functions associated with automorphic $L$-functions are also studied. 
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     author = {O. M. Fomenko},
     title = {On summatory functions for automorphic $L$-functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {202--217},
     publisher = {mathdoc},
     volume = {392},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_392_a10/}
}
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O. M. Fomenko. On summatory functions for automorphic $L$-functions. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 26, Tome 392 (2011), pp. 202-217. http://geodesic.mathdoc.fr/item/ZNSL_2011_392_a10/