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.

中文翻译：

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维诺格拉多夫的三个素数定理，素数具有给定的原根

我们论文的第一个目的是展示 Hooley 著名的方法如何导致他对原始根的 Artin 猜想的条件证明可以与 Hardy-Littlewood 圆方法相结合。我们通过将奇数表示为三个素数之和来研究这一点，所有这些素数都具有规定的原根。第二个目的是分析单数级数。特别是，使用 Lenstra、Stevenhagen 和 Moree 的结果，我们提供了部分因式分解作为欧拉积，并证明这不会扩展到完全因式分解。