Irregularities in the distributions of primes in function fields

We adapt the Maier matrix method to the polynomial ring Fq[t], and prove analogues of results of Maier [H. Maier, Primes in short intervals, Michigan Math. J. 32 (1985) 221–225] and Shiu [D.K.L. Shiu, Strings of congruent primes, J. London Math. Soc. 61 (2000) 359–373] concerning the distribution of primes in short intervals.