We prove Bombieri–Vinogradov type theorems for primes with a missing digit in their -adic expansion for some large positive integer . The proof is based on the circle method, which relies on the Fourier structure of the integers with a missing digit and the exponential sums over primes in arithmetic progressions. Combining our results with the semi-linear sieve, we obtain an upper bound and a lower bound of the correct order of magnitude for the number of primes of the form with a missing digit in a large odd base .

中文翻译：

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缺失数字的素数：算术级数的分布和筛理论的应用

我们证明了素数中缺少数字的 Bombieri-Vinogradov 型定理-某些大正整数的进数展开。证明基于圆法，该方法依赖于缺少数字的整数的傅立叶结构以及算术级数中素数的指数和。将我们的结果与半线性筛相结合，我们获得了以下形式的素数数量的正确数量级的上限和下界大奇数基数中缺少数字。