Olli Järviniemi’s research while affiliated with University of Turku and other places

We introduce FrontierMath, a benchmark of hundreds of original, exceptionally challenging mathematics problems crafted and vetted by expert mathematicians. The questions cover most major branches of modern mathematics -- from computationally intensive problems in number theory and real analysis to abstract questions in algebraic geometry and category theory. Solving a typical problem requires multiple hours of effort from a researcher in the relevant branch of mathematics, and for the upper end questions, multiple days. FrontierMath uses new, unpublished problems and automated verification to reliably evaluate models while minimizing risk of data contamination. Current state-of-the-art AI models solve under 2% of problems, revealing a vast gap between AI capabilities and the prowess of the mathematical community. As AI systems advance toward expert-level mathematical abilities, FrontierMath offers a rigorous testbed that quantifies their progress.

For an algebraic number α we consider the orders of the reductions of α in finite fields. In the case where α is an integer, it is known by the work on Artin’s primitive root conjecture that the order is ‘almost always almost maximal’ assuming the Generalized Riemann Hypothesis (GRH), but unconditional results remain modest. We consider higher-degree variants under GRH. First, we modify an argument of Roskam to settle the case where α and the reduction have degree two. Second, we give a positive lower density result when α is of degree three and the reduction is of degree two. Third, we give higher-rank results in situations where the reductions are of degree two, three, four or six. As an application we give an almost equidistribution result for linear recurrences modulo primes. Finally, we present a general result conditional to GRH and a hypothesis on smooth values of polynomials at prime arguments.

We show that almost all sectors of the disc \{z \in \mathbb{C}: |z|^2\leq X\} of area (\log X)^{15.1} contain products of exactly two Gaussian primes, and that almost all sectors of area (\log X)^{1 + \varepsilon} contain products of exactly three Gaussian primes. The argument is based on mean value theorems, large value estimates and pointwise bounds for Hecke character sums.

Let $d \ge 3$ be an integer and let $P \in \mathbb{Z}[x]$ be a polynomial of degree d whose Galois group is $S_d$ . Let $(a_n)$ be a non-degenerate linearly recursive sequence of integers which has P as its characteristic polynomial. We prove, under the generalised Riemann hypothesis, that the lower density of the set of primes which divide at least one non-zero element of the sequence $(a_n)$ is positive.

We show that almost all sectors of the disc $\{z \in \mathbb{C}: |z|^2\leq X\}$ of area $(\log X)^{15.1}$ contain products of exactly two Gaussian primes, and that almost all sectors of area $(\log X)^{1 + \varepsilon}$ contain products of exactly three Gaussian primes. The argument is based on mean value theorems, large value estimates and pointwise bounds for Hecke character sums.

We show that $\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{1/2}}} (p_{n+1} - p_n) \ll x^{0.57+\epsilon}$ and $\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{0.45}}} (p_{n+1} - p_n) \ll x^{0.63+\epsilon},$ where $p_n$ is the $n$th prime number. The proof combines Heath-Brown's recent work with Harman's sieve, improving and extending his results. We give applications of the results to prime-representing functions, binary digits of primes and approximation of reals by multiplicative functions.

Since Hooley’s seminal 1967 resolution of Artin’s primitive root conjecture under the Generalized Riemann Hypothesis, numerous variations of the conjecture have been considered. We present a framework generalizing and unifying many previously considered variants, and prove results in this full generality (under GRH).

This work concerns Artin's Conjecture on primitive roots and related problems for number fields. Let $K$ be a number field and let $W_1$ to $W_n$ be finitely generated subgroups of $K^\times$ of positive rank. We consider the index map, which maps a prime $\mathfrak p$ of $K$ to the $n$-tuple of the indices of $(W_i \bmod \mathfrak p)$. Conditionally under GRH, any preimage under the index map admits a density, and the aim of this work is describing it. For example, we express the density as a limit in various ways. We study in particular the preimages of sets of $n$-tuples that are defined by prescribing valuations for their entries. Under some mild assumptions we can express the density as a multiple of a (suitably defined) Artin-type constant.

Let F be a system of polynomial equations in one or more variables with integer coefficients. We show that there exists a univariate polynomial $D \in \mathbb {Z}[x]$ such that F is solvable modulo p if and only if the equation $D(x) \equiv 0 \pmod {p}$ has a solution.