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Does there exist an asymptotic estimate for the following sum over primes
$$
\sum\_{p\leq x} \frac{\tau(p-1)}{p}\;,
$$
where $\tau(n)=\sum\_{d|n}1$ is the divisor function?



sum over primes involving divisor function (variation of the Titchmarsh divisor problem)
For this kind of things it's always a good idea to check in the two volumes of the Handbook of Number Theory of J. Sandor, D. S. Mitrinovic and B. Crstici.
Here I found the formula:


$$(\star)\quad \sum\_{p \leq x} \tau(p - 1) = \frac{315 \,\zeta(3)}{2 \pi^4} \cdot x + O\!\left(\frac{x}{(\log x)^\alpha}\right), $$
as $x \to +\infty$, for any $\alpha \in \;]0,1[$. *Yu. V. Linnik. New versions and new uses of the dispersion method in binary additive problems (Russian). Dokl.
Akad. Nauk SSSR 137 (1961), 1299–1302.*


Using ($\star$) and partial summation (Tenenbaum - Introduction to Analytic and Probabilistic Number Theory, Chapter I.0, Theorem 1 with $a\_n := \tau(n-1)$ if $n$ is prime, $a\_n := 0$ is $n$ is composite and $b(t) := 1/t$) you should be able to get an asymptotic formula for $\sum\_{p \leq x} \tau(p-1)/p$.


