We develop a lower bound sieve for primes under the (unlikely) assumption of infinitely many exceptional characters. Compared with the illusory sieve due to Friedlander and Iwaniec which produces asymptotic formulas, we show that less arithmetic information is required to prove nontrivial lower bounds. As an application of our method, assuming the existence of infinitely many exceptional characters we show that there are infinitely many primes of the form $a^2+b^8$.

中文翻译：

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稀疏集中的异常字符和素数

我们在无限多个特殊字符的（不太可能）假设下开发了素数下界筛。与 Friedlander 和 Iwaniec 产生渐进公式的虚幻筛相比，我们证明证明非平凡下界需要更少的算术信息。作为我们方法的应用，假设存在无限多个特殊字符，我们表明存在无限多个 $a^2+b^8$ 形式的素数。