The Bombieri-Vinogradov theorem for primes of the form $\mathbf{p}={\mathbf{x}}^{\mathbf{2}}+{\mathbf{y}}^{\mathbf{2}}+\mathbf{1}$

In this paper, we establish a Bombieri-Vinogradov type result for prime numbers of the form $p=x^2+y^2+1$. The proof is based on the enveloping sieve.