For certain classes of functions \(f\) the author considers the sums \(\sum_{p \leq x} f(i_p)\), where \(i_p\) denotes the index of \(2 \bmod p\). When \(f(1)=1\) and \(f(x)=0\) otherwise, this sum counts the number of primes \(p \leq x\) for which \(2\) is a primitive root, in which case an asymptotic formula was deduced by C. Hooley [J. Reine Angew. Math. 225, 209-220 (1967; Zbl 0221.10048)] from a very general form of the Generalised Riemann Hypothesis.
The author examines the consequences of a similar approach to the more general question, using also the Brun-Titchmarsh theorem, and a result of L. Murata [Arch.Math. 57, 555-565 (1991; Zbl 0755.11029)] on the number of primes for which 2 has index \(m\). This leads to a putative asymptotic formula in the case when \(|f(m)|\leq \log^C x\) for \(m \leq x\), and weaker results for functions \(f\) which do not satisfy this condition. A number of specific applications are also considered.