Variance of primes in short residue classes for function fields (Q6562932)

\textit{J. P. Keating} and \textit{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:\N\[\N\int_0^N \Big|\sum_{x<n\leq x+\Delta} \Lambda(n)-\Delta\Big|^2 dx,\N\]\Nand the mean value of primes in short intervals\N\[\N\sum_{\substack{a=1\\ (a,q)=1}}^q \Big| \sum_{\substack{n\leq N\\ n\equiv a\bmod q}} \Lambda(n)-\frac{N}{\varphi(q)}\Big|^2,\N\]\Nin the function field setting.\N\NIn 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.\N\NThe main result is Theorem 3.4 and it computes the variance\N\[\NV(n,h;Q):=\frac {1}{q^n}\sum_{C\in{\mathcal M}_n}\sum_{\substack{ A\bmod Q\\ (A,Q)=1}}\Big| \Psi(C,h;Q,A)-\frac{q^{h+1}}{\varphi(Q)} \Big|^2,\N\]\Nfor 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.