Applications of the large sieve inequality for \(\mathbb{F}_q[T]\) (Q1273729)

This paper gives two applications of the author's large sieve inequality for a polynomial ring \({\mathbb F}_q(T)\) in a single variable over a finite field, from which a corresponding theorem of Brun-Titchmarsh type was inferred. Both of these results were given in [\textit{C.-N. Hsu}, J. Number Theory 58, 267-287 (1996; Zbl 0852.11049)]. As an analogue of the upper bound for the number of primes in an interval, it is now shown that if \(f \in {\mathbb F}_q(T)\) and \(N<\deg f\) then the number of irreducible polynomials \(p\) in \({\mathbb F}_q(T)\) with \(\deg(p-f)\leq N\) does not exceed \(2q^{N+1}/(N+2)\). The second result is an analogue of the Titchmarsh divisor problem, an asymptotic formula for \(\sum \tau(p-g)\), where \(g\) is a fixed polynomial in \({\mathbb F}_q(T)\), the summation is over irreducibles \(p\) of fixed degree \(N\), and \(\tau(f)\) denotes the number of polynomial divisors of \(f\). The treatment follows the classical lines: an available asymptotic formula for the number \(\pi(N;d,g)\) of irreducibles \(p\) of degree \(N\) with \(p \equiv g \bmod d\) is taken from the author's paper [J. Number Theory 61, 85-96 (1996)]. This result is not sufficient on its own, but the proof is completed by use of the ``Brun-Titchmarsh'' upper bound for \(\pi(N;d,g)\).