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 $k$. 
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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/

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