We study the Bateman--Horn conjecture for generalised von Mangoldt functions. As an application, we prove that for $k \in \{2, 3\}$, almost all Bouniakowsky polynomials represent integers that are a product of exactly $k$ primes.

We prove that the average error term when counting square-free values of polynomials is the quartic root of the main term.

We establish the Hasse principle for 100% of conic bundles over ℙ^1_ℚ.

We obtain asymptotic formulas for averages of general multivariate arithmetic functions evaluated at polynomial arguments using recent work of Rydin Myerson and Rome-Yamagishi. We give several applications of our results in analytic number theory and arithmetic geometry. For example, we improve on the number of variables needed to prove the Hasse principle for certain polynomial systems, and we count the number of fibers with a rational point in families of high-dimensional Châtelet varieties, allowing for arbitrarily large subordinate Brauer groups.

For a general family of non-negative functions matching upper and lower bounds are established for their average over the values of any equidistributed sequence. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

We develop a version of the Hardy-Littlewood circle method to obtain asymptotic formulas for averages of general multivariate arithmetic functions evaluated at polynomial arguments in several variables. As an application, we count the number of fibers with a rational point in families of high-dimensional Ch\^atelet varieties, allowing for arbitrarily large subordinate Brauer groups.

We apply the Gärtner–Ellis theorem on large deviations to prove a weak version of the Loughran–Smeets conjecture for general fibrations.

We prove a composite case of the Cohen–Lenstra–Gerth heuristics. Specifically, we establish an asymptotic for the average 6-torsion of the class group of quadratic number fields. We also prove Malle's conjecture for Galois D_6-extensions.

We establish a Galois-theoretic trichotomy governing Diophantine stability for genus $0$ curves. We use it to prove that the curve associated to the Hilbert symbol is Diophantine stable with probability $1$. Our asymptotic formula for the second-order term exhibits strong bias toward instability.

We prove matching upper and lower bounds for the average of the 6-torsion of class groups of quadratic fields. Furthermore, we count the number of integer solutions on an affine quartic threefold.

We determine the order of magnitude for all exponential moments of the rank in a broad class of elliptic fibrations and for the 3 · 2^k-torsion in the class group of quadratic fields.

We estimate the average of any arithmetic function k over the values of any smooth polynomial in many variables provided only that k has a distribution in arithmetic progressions of fixed modulus. We give several applications of this result including the analytic Hasse principle for an intersection of two cubics in 21 variables and asymptotics for the number of integer solutions of a non-algebraic variety.

We prove a stronger form of our previous result that Schinzel's Hypothesis holds for 100% of n-tuples of integer polynomials satisfying the usual necessary conditions, where the primes represented by the polynomials are subject to additional constraints in terms of Legendre symbols, as well as upper and lower bounds. We establish the triviality of the Brauer group of generic diagonal conic bundles over the projective line. Finally, we give an explicit lower bound for the probability that diagonal conic bundles in certain natural families have rational points.

Abstract Erdős considered the second moment of the gap-counting function of prime divisors in 1946 and proved an upper bound that is not of the right order of magnitude. We prove asymptotics for all moments. Furthermore, we prove a generalisation stating that the gaps between primes p for which there is no $\mathbb{Q}_p$ -point on a random variety are Poisson distributed.

Given a variety with coefficients in Z \mathbb {Z} , we study the distribution of the number of primes dividing the coordinates as we vary an integral point. Under suitable assumptions, we show that this has a multivariate normal distribution. We generalise this to more general Weil divisors, where we obtain a geometric interpretation of the covariance matrix. For our results we develop a version of the Erdős–Kac theorem that applies to fairly general integer sequences and does not require a positive exponent of level of distribution.

. We prove asymptotics for Serre’s problem on the number of diagonal planar conics with a rational point and use this to put forward a new conjecture on counting the number of varieties in a family which are everywhere locally soluble.

We prove asymptotics for Serre's problem on the number of diagonal planar conics with a rational point and use this to put forward a new conjecture on counting the number of varieties in a family which are everywhere locally soluble.

We prove that the average of the $k$-th smallest prime quadratic non-residue modulo a prime approximates the $2k$-th smallest prime.

We resolve Schinzel’s Hypothesis (H) for $$100\%$$ of polynomials of arbitrary degrees. We deduce that a positive proportion of diagonal conic bundles over $${\mathbb {Q}}$$ with any given number of degenerate fibres have a rational point, and obtain similar results for generalised Châtelet equations.

With probability 1, we assess the average behaviour of various arithmetic functions at the values of degree d polynomials f that are ordered by height. This allows us to establish averaged versions of the Bateman-Horn conjecture, the polynomial Chowla conjecture and to address a basic question about the integral Hasse principle for norm form equations. Moreover, we are able to quantify the error term in the asymptotics and the size of the exceptional set of f, both with arbitrary logarithmic power savings.