We study the existence of products of primes in arithmetic progressions, building on the work of Ramaré and Walker. One of our main results is that if is a large modulus, then any invertible residue class mod contains a product of three primes where each prime is at most . Our arguments use results from a wide range of areas, such as sieve theory or additive combinatorics, and one of our key ingredients, which has not been used in this setting before, is a result by Heath-Brown on character sums over primes from his paper on Linnik's theorem.

中文翻译：

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论算术级数中素数乘积的存在性

我们以 Ramaré 和 Walker 的工作为基础，研究算术级数中素数乘积的存在性。我们的主要结果之一是，如果是一个大模数，则任意可逆残差类模包含三个素数的乘积，其中每个素数最多。我们的论点使用了来自广泛领域的结果，例如筛子理论或加性组合学，而我们的关键成分之一是希思·布朗（Heath-Brown）对素数的字符和的结果，它以前没有在这种情况下使用过。关于林尼克定理的论文。