非阿贝尔大筛法与阿廷原根猜想
A non-abelian large sieve and Artin's primitive root conjecture
AI总结：
本文研究经典阿贝尔大筛法不等式的非阿贝尔推广，假设该推广成立可证明阿廷原根猜想，还通过对偶技术得到针对该推广的无条件结果。
AI中文摘要：
阿廷有一个著名猜想：若整数a不为0、±1或完全平方数，则存在无穷多素数p，使得a是模p的原根。本文研究经典（阿贝尔）大筛法不等式在非阿贝尔语境下的推广。假设非阿贝尔大筛法不等式成立，本文给出阿廷原根猜想的一个证明。此外，利用对偶技术，本文推导出针对猜想中非阿贝尔大筛法不等式的无条件结果。
英文摘要：
A well-known conjecture of Artin states that if $a$ is an integer not equal to $0, \pm 1$ or a perfect square, then there exist infinitely many primes $p$ such that $a$ is a primitive root $(\text{ mod } p)$. In this article, we study a generalization of the classical (abelian) large sieve inequality in non-abelian settings. Assuming the non-abelian large sieve inequality, we provide a proof of Artin's primitive root conjecture. Further, using duality techniques, we derive unconditional results towards the conjectured non-abelian large sieve inequality.
