Geodesic
Zeta function of the additive divisor problem and the spectral expansion of the automorphic Laplacian
Zapiski Nauchnykh Seminarov POMI, Automorphic functions and number theory. Part II, Tome 134 (1984), pp. 84-116 
A. I. Vinogradov; L. A. Takhtadzhyan. Zeta function of the additive divisor problem and the spectral expansion of the automorphic Laplacian. Zapiski Nauchnykh Seminarov POMI, Automorphic functions and number theory. Part II, Tome 134 (1984), pp. 84-116. http://geodesic.mathdoc.fr/item/ZNSL_1984_134_a4/
@article{ZNSL_1984_134_a4,
     author = {A. I. Vinogradov and L. A. Takhtadzhyan},
     title = {Zeta function of the additive divisor problem and the spectral expansion of the automorphic {Laplacian}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {84--116},
     year = {1984},
     volume = {134},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1984_134_a4/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The representation for zeta function of the additive divisor problem $\zeta_k(s)=\sum_{n=1}^\infty\frac{\tau(n)\tau(n+k)}{n^s}$, $\operatorname{Re}s>1$, in terms of spectral data of the automorphic Laplacian is presented. With its help the meromorphic continuation of $\zeta_k(s)$ into the whole complex plane is proved and an estimate of the order of $\zeta_k(s)$ in the critical strip $0<\operatorname{Re}s\leqslant1$ is obtained. Using the method of complex integration the asymptotic formula $$ \sum_{n\leqslant x}\tau(n)\tau(n+k)=xP_k(\log x)+O(x^{\frac23+\varepsilon}),\quad\varepsilon>0, $$ is derived where $P_k(x)$ is a quadratic polynomial.