Tuples of polynomials over finite fields with pairwise coprimality conditions

Let q be a prime power. We estimate the number of tuples of degree bounded monic polynomials (Q1, . . . , Qv) ∈ (Fq[z])v that satisfy given pairwise coprimality conditions. We show how this generalises from monic polynomials in finite fields to Dedekind domains with a finite norm

Bounded gaps between primes with a given primitive root

Fix an integer $g \neq -1$ that is not a perfect square. In 1927, Artin conjectured that there are infinitely many primes for which $g$ is a primitive root. Forty years later, Hooley showed that Artin's conjecture follows from the Generalized Riemann Hypothesis (GRH). We inject Hooley's analysis into the Maynard--Tao work on bounded gaps between primes. This leads to the following GRH-conditional result: Fix an integer $m \geq 2$. If $q_1 < q_2 < q_3 < \dots$ is the sequence of primes possessing $g$ as a primitive root, then $\liminf_{n\to\infty} (q_{n+(m-1)}-q_n) \leq C_m$, where $C_m$ is a finite constant that depends on $m$ but not on $g$. We also show that the primes $q_n, q_{n+1}, \dots, q_{n+m-1}$ in this result may be taken to be consecutive.Comment: small corrections to the treatment of \sum_1 on pp. 11--1

Joint distribution in residue classes of families of polynomially-defined multiplicative functions

We study the distribution of families of multiplicative functions among the coprime residue classes to moduli varying uniformly in a wide range, obtaining analogues of the Siegel--Walfisz Theorem for large classes of multiplicative functions. We extend a criterion of Narkiewicz for families of multiplicative functions that can be controlled by values of polynomials at the first few prime powers, and establish results that are completely uniform in the modulus as well as optimal in most parameters and hypotheses. This also significantly generalizes and improves upon previous work done for a single such function in specialized settings. Our results have applications for most interesting multiplicative functions, such as the Euler totient function $\phi(n)$, the sum-of-divisors function $\sigma(n)$, the coefficients of the Eisenstein series, etc., and families of these functions. For instance, an application of our results shows that for any fixed $\epsilon>0$, the functions $\phi(n)$ and $\sigma(n)$ are jointly asymptotically equidistributed among the reduced residue classes to moduli $q$ coprime to $6$ varying uniformly up to $(\log x)^{(1-\epsilon)\alpha(q)}$, where $\alpha(q) = \prod_{\ell \mid q} (\ell-3)/(\ell-1)$; furthermore, the coprimality restriction is necessary and the range of $q$ is essentially optimal. One of the primary themes behind our arguments is the quantitative detection of a certain mixing (or ergodicity) phenomenon in multiplicative groups via methods belonging to the `anatomy of integers', but we also rely heavily on more pure analytic arguments (such as a suitable modification of the Landau-Selberg-Delange method), -- whilst using several tools from arithmetic and algebraic geometry, and from linear algebra over rings as well.Comment: 66 page