Let $\Lambda$ be the von Mangoldt function. 
I think the following is known to hold under GRH: Given any $q \geq 1$, $(a,q)=1$, and $X \geq 1$, we have
$$
\sum\_{\substack{1 \leq n \leq X \\ n \equiv a\,(\text{mod }q) }} \Lambda(n) = \frac{X}{\phi(q)} + O(X^{1/2} \log (qX)).
$$
I was wondering if someone could give me a reference for this. I keep finding a slight variant of this without the von Mangoldt, but I just wanted this version so that I can directly reference in a paper. Thank you very much!



Reference for Siegel-Walfisz Theorem under GRH
The usual error term in this problem is not $O(X^{1/2} \log (qX))$ but $O(X^{1/2} \log^2(X))$. This version of the result is part of Corollary 13.8 in Montgomery-Vaughan: Multiplicative number theory I. For a more general result (also featuring the slightly weaker error term above) see Theorem 5.15 in Iwaniec-Kowalski: Analytic number theory.


