Assuming the generalized Riemann hypothesis, we give asymptotic bounds on the size of intervals that contain primes from a given arithmetic progression using the approach developed by Carneiro, Milinovich and Soundararajan [Comment. Math. Helv. 94, no. 3 (2019)]. For this we extend the Guinand-Weil explicit formula over all Dirichlet characters modulo $q \geq 3$, and we reduce the associated extremal problems to convex optimization problems that can be solved numerically via semidefinite programming.

翻译：假设通用的里曼假设,我们使用Carneiro、Milinovich和Soundararajan[评论: Math. Helv. 94, No. 3 (2019] 所制定的方法,对包含特定算术进展的质数的间隔间隔的大小给予无药可治的界限。为此,我们将Guinand-Weil明确公式扩大到所有迪里赫特字符modulo $q\geq 3美元,并减少相关的极端问题,以弥补可以通过半不定期编程从数字上解决的最优化问题。