Geodesic
An additive divisor problem with a~growing number of factors
Matematičeskie zametki, Tome 61 (1997) no. 3, pp. 391-406

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Let $\tau_k(n)$ be the number of representations of $n$ as the product of $k$ positive factors, $\tau_2(n)=\tau(n)$. The asymptotics of $\sum_{n\le x}\tau_k(n)\tau(n+1)$ for $80k^{10}(\ln\ln x)^3\le\ln x$ is shown to be uniform with respect to $k$. 
@article{MZM_1997_61_3_a7,
     author = {N. M. Timofeev},
     title = {An additive divisor problem with a~growing number of factors},
     journal = {Matemati\v{c}eskie zametki},
     pages = {391--406},
     publisher = {mathdoc},
     volume = {61},
     number = {3},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_61_3_a7/}
}
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N. M. Timofeev. An additive divisor problem with a~growing number of factors. Matematičeskie zametki, Tome 61 (1997) no. 3, pp. 391-406. http://geodesic.mathdoc.fr/item/MZM_1997_61_3_a7/