Arithmetic correlations over large finite fields (Q2796766)

This paper investigates the analogs of some correlation problems for primes in the setting of polynomials over finite fields. Let \(q\) be an odd prime power, \(P_n\) and \(M_n\) the set of polynomials and of monic polynomials of degree \(n\) over \(F_q\), resp. For a polynomial \(f\) put \(\Lambda(f) = \text{deg}\, p\) if \(f\) is a power of an irreducible polynomial \(p\), and 0 otherwise. It is known that NEWLINE\[CARRIAGE_RETURNNEWLINE q^{-n} \sum_{f\in M_n} \Lambda(f) \Lambda(f+K) = 1 + E(K, n, q),CARRIAGE_RETURNNEWLINE\]NEWLINE where \(E(K, n, q) = O(q^{-1/2})\) for a nonvanishing \(K\). The main results of the paper give estimates of the average of this error term \(E\) in the form NEWLINE\[CARRIAGE_RETURNNEWLINE \sum_{K\in M_k} E(K, n, q) = \frac{1}{1-q} + O(q^{-3/2}) CARRIAGE_RETURNNEWLINE\]NEWLINE for \(k<n-3\), and of a twisted average NEWLINE\[CARRIAGE_RETURNNEWLINE \sum_{j \leq n-\text{deg}\, Q -1} \sum_{K\in M_j} E(KQ, n, q) = \frac{ n-\text{deg}\, Q}{ 1-q} + O(q^{-3/2}). CARRIAGE_RETURNNEWLINE\]NEWLINE The second has the corollary that for \( \text{deg}\, Q = n-1\) we have \( E(Q, n, q) = \frac{1}{1-q} + O(q^{-3/2}) \).