The pair correlation of zeros of DirichletL-functions and primes in arithmetic progressions

We define a function which correlates the zeros of two DirichletL-functions to the modulusq and we prove an asymptotic estimate for averages of the pair correlation functions over all pairs of characters to (modq). An analogue of Montgomery’s pair correlation conjecture is formulated as to how this estimate can be extended to a greater domain for the parameters that are involved. Based on this conjecture we obtain results about the distribution of primes in an arithmetic progression (to a prime modulusq) and gaps between such primes.