A large sieve inequality for rational function fields
The author derives a large sieve inequality for rational function fields \(\mathbb{F}_q (t)\), where \(\mathbb{F}_q\) denotes a fixed finite field with \(q\) elements. He applies this inequality to obtain an analogue of the Brun-Titchmarsh theorem: Let \(a\), \(b\) be non-zero polynomials in \(\mathbb{F}_q [t]\) with \(a\) monic, \((a, b)= 1\) and \(\deg a>\deg b\geq 0\). For any positive integer \(N\), let \(\pi (N; a, b)\) denote the number of monic irreducibles \(f\in \mathbb{F}_q [t]\) such that \(\deg f= N\) and \(f\equiv b\pmod a\). Then \[ \pi (N; a, b)\leq 2\cdot {{q^N} \over {\Phi (a) (N-\deg a+1)}} \] for \(N> \deg a\), where \(\Phi (a)= \text{card} (\mathbb{F}_q [t]/ (a))^*\). Moreover, the author shows that the set \(\{P\); \(P\), \(P+D\) are monic irreducibles in \(\mathbb{F}_q [t]\}\) has Dirichlet density zero for any non-zero polynomial \(D\in \mathbb{F}_q [t]\).
scientific article; zbMATH DE number 894607
scientific article