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An arithmetic function is one whose domain is the positive integers and whose range is a subset of the complex numbers. There are a number of important number-theoretic examples.
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sum over primes involving divisor function (variation of the Titchmarsh divisor problem)
This question was also asked on MSE.
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 fu …