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Abstract

The first purpose of our paper is to show how Hooley's celebrated method
leading to his conditional proof of the Artin conjecture on primitive roots can
be combined with the Hardy-Littlewood circle method. We do so by studying the
number of representations of an odd integer as a sum of three primes, all of
which have prescribed primitive roots.
The second purpose is to analyse the singular series. In particular, using
results of Lenstra, Stevenhagen and Moree, we provide a partial factorisation
as an Euler product and prove that this does not extend to a complete
factorisation.