J. P. Keating and Z. Rudnick studied in [Int. Math. Res. Not. 2014, No. 1, 259-288 (2014); Zbl 1319.11084] the variances of primes in arithmetic progressions to a fixed large modulus: \[ \int_0^N \Big|\sum_{x<n\leq x+\Delta} \Lambda(n)-\Delta\Big|^2 dx, \] and the mean value of primes in short intervals \[ \sum_{\substack{a=1\\ (a,q)=1}}^q \Big| \sum_{\substack{n\leq N\\ n\equiv a\bmod q}} \Lambda(n)-\frac{N}{\varphi(q)}\Big|^2, \] in the function field setting.
In this paper, the authors consider the hybrid problem of calculating the variance of primes in intersections of arithmetic progressions and short intervals. Keating and Rudnick used an involution to translate short intervals into arithmetic progressions. This paper follows this approach. In addition, this involution is applied to arithmetic progressions.
The main result is Theorem 3.4 and it computes the variance \[ V(n,h;Q):=\frac {1}{q^n}\sum_{C\in{\mathcal M}_n}\sum_{\substack{ A\bmod Q\\ (A,Q)=1}}\Big| \Psi(C,h;Q,A)-\frac{q^{h+1}}{\varphi(Q)} \Big|^2, \] for three different cases. The notations are given in Section 2 and the proof of Theorem 3.4 is given in Sections 5, 6 and 7.