Yıldırım's pair correlation conjecture for Dirichlet $L$-functions

Let $q=1$ or let $q$ be prime, let $0<\eta\le 1$ be fixed, and define

$$G^+_{\chi_1,\chi_2}(x,T)=\sum_{0<\gamma_j\le T}x^{i(\gamma_1-\gamma_2)}W(\gamma_1-\gamma_2),$$

where the ordinates are zeros of the relevant Dirichlet $L$-functions and $W(u)=4/(4+u^2)$. Set

$$F_q(x,T)=\sum_{\chi_1,\chi_2\pmod* q}\overline{\chi_1}(a)\chi_2(a)G_{\chi_1,\chi_2}(x,T).$$

Pair correlation for Dirichlet $L$-functions. Under GRH, as $x\to\infty$,

$$\sum_{\chi_1,\chi_2\pmod* q}\overline{\chi_1}(a)\chi_2(a)G^+_{\chi_1,\chi_2}(x,T)\sim\frac{\varphi(q)}{2\pi}T\log(qT),$$

uniformly for

$$q\le\min(\sqrt{x}\log^{-3}x,x^{1-\eta}\log x)$$

and

$$x^\eta\le T<\frac{x}{q}\log x.$$

This is the Dirichlet-$L$ analogue of Montgomery's pair-correlation conjecture and is used in the paper as an input toward bounds for primes in arithmetic progressions.