Geodesic
Problems similar to the additive divisor problem
Matematičeskie zametki, Tome 64 (1998) no. 3, pp. 443-456.

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For multiplicative functions $f(n)$, let the following conditions be satisfied: $f(n)\ge0$, $f(p^r)\le A^r$, $A>0$, and for any $\varepsilon>0$ there exist constants $A_\varepsilon$, $\alpha>0$ such that $f(n)\le A_\varepsilon n^\varepsilon$ and $\sum_{p\le x}f(p)\ln p\ge\alpha x$. For such functions, the following relation is proved: $$ \sum_{n\le x}f(n)\tau(n-1)=C(f)\sum_{n\le x}f(n)\ln x\bigl(1+o(1)\bigr). $$ Here $\tau(n)$ is the number of divisors of $n$ and $C(f)$ is a constant. 
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N. M. Timofeev; S. T. Tulyaganov. Problems similar to the additive divisor problem. Matematičeskie zametki, Tome 64 (1998) no. 3, pp. 443-456. http://geodesic.mathdoc.fr/item/MZM_1998_64_3_a13/

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