arXiv subjects

Lasse Grimmelt

Publications and source records attributed to Lasse Grimmelt.

13 recordsLinked to original sources

arXiv2026

The exceptional set of the Goldbach problem

We study the estimates for the number of exceptions to the representation of integers as the sum of at most two prime numbers. Most of this article is a survey that gives an overview of existing results. We begin with the legendary Hardy-Littlewood circle method and show how it paved the way to a power saving by Montgomery-Vaughan in 1975 and Pintz in 2018. We conclude with a new result that is a fully explicit formula for the major arcs. Another new observation is the non-existence of exceptional zeros under a sparse version of the Hardy-Littlewood conjecture. The survey part of this article aims to be accessible to an audience that has not encountered these techniques before.

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arXiv2026

Random linear configurations in dense sets and primes

We prove that every polylogarithmically dense subset of $[N]$ contains a nontrivial configuration $x+b_1m,\ldots,x+b_km$ for almost all choices of the coefficient vector $(b_1,\ldots, b_k)$ in a wide range of scales. We prove the same statement for polylogarithmically relatively dense subsets of the primes, in a shorter range of scales. The main ingredients are a new quantitative generalised von Neumann theorem, degree lowering to the $U^{1+}$ norm, and densification arguments that transfer the result to the primes.

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arXiv2026

The Exceptional Set in Goldbach's Problem with two Chen Primes

We show that all natural numbers $n\equiv 4\pmod 6$ are the sum of two Chen primes (primes $p$ such that $p+2$ has at most two prime factors), apart from a power-saving set of exceptions. This improves on various previous results and is optimal, barring substantial progress on the twin prime or binary Goldbach conjectures. The proof is based on constructing a non-negative model for the Chen primes in a suitable approximate sense. To do this, we develop an efficient sieving strategy that makes use of a power-saving variant of the Bombieri--Vinogradov theorem. Furthermore, we show that the primes are well approximated in additive problems by the Cramér model (rough numbers) with a sifting parameter of power size.

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arXiv2025

The divisor function along arithmetic progressions and binary cubic polynomials

We prove a new equidistribution estimate for the divisor function in arithmetic progression to moduli that have two small factors. We give two applications. First, we show an asymptotic formula for the divisor function over arithmetic progressions to almost all moduli of exponent $2/3$. Second, we show an asymptotic formula for the divisor function along the nonhomogeneous binary cubic polynomial $X Y^2+1$.

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arXiv2025

Weighted averages of $\operatorname{SL}_2(\mathbb{R})$ automorphic kernel, Part I: non-oscillatory functions

We prove a theorem that evaluates weighted averages of sums parametrised by congruence subgroups of $\operatorname{SL}_2(\mathbb{Z})$. In the proof, spectral methods are applied directly to the automorphic kernel instead of going over sums of Kloosterman sums. In number theoretical applications this better preserves the specific symmetries throughout the application of spectral methods. In a separate paper we apply the main theorem to quadratic polynomials and obtain new results about their greatest prime factor and the equidistribution of their roots to prime moduli.

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arXiv2025

On the greatest prime factor and uniform equidistribution of quadratic polynomials

We show that the greatest prime factor of $n^2+h$ is at least $n^{1.312}$ infinitely often. This gives an unconditional proof for the range previously known under the Selberg eigenvalue conjecture. Furthermore, we get uniformity in $h \leq n^{1+o(1)}$ under a natural hypothesis on real characters. The same uniformity is obtained for the equidistribution of the roots of quadratic congruences modulo primes. We also prove a variant of the divisor problem for $ax^2+by^3$, which was used by the second author to give a conditional result about primes of that shape.

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arXiv2024

On an Erdős--Kac-type conjecture of Elliott

Elliott and Halberstam proved that $\sum_{p<n} 2^{ω(n-p)}$ is asymptotic to $ϕ(n)$. In analogy to the Erdős--Kac Theorem, Elliott conjectured that if one restricts the summation to primes $p$ such that $ω(n-p)\le 2 \log \log n+λ(2\log \log n)^{1/2}$ then the sum will be asymptotic to $ϕ(n)\int_{-\infty}^λ e^{-t^2/2}dt/\sqrt{2π}$. We show that this conjecture follows from the Bombieri--Vinogradov Theorem. We further prove a related result involving Poisson--Dirichlet distribution, employing deeper lying level of distribution results of the primes.

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arXiv2024

Twisted correlations of the divisor function via discrete averages of $\operatorname{SL}_2(\mathbb{R})$ Poincaré series

We prove a theorem that allows one to count solutions to determinant equations twisted by a periodic weight with high uniformity in the modulus. It is obtained by using spectral methods of $\operatorname{SL}_2(\mathbb{R})$ automorphic forms to study Poincaré series over congruence subgroups. By keeping track of interactions between multiple orbits we get advantages over the widely used sums of Kloosterman sums techniques. We showcase this with applications to correlations of the divisor functions twisted by periodic functions and the fourth moment of Dirichlet $L$-functions on the critical line.

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arXiv2023

Additive problems with almost prime squares

We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We likewise treat representations of shifted primes p-1 as sums of two almost prime squares. The methods involve a combination of analytic, automorphic and algebraic arguments to handle representations by restricted binary quadratic forms with a high degree of uniformity.

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arXiv2022

The Exceptional Set in Goldbach's Problem with Almost Twin Primes

We consider the exceptional set in the binary Goldbach problem for sums of two almost twin primes. Our main result is a power-saving bound for the exceptional set in the problem of representing $m=p_1+p_2$ where $p_1+2$ has at most $2$ prime divisors and $p_2+2$ has at most $3$ prime divisors. There are three main ingredients in the proof: a new transference principle like approach for sieves, a combination of the level of distribution estimates of Bombieri--Friedlander--Iwaniec and Maynard with ideas of Drappeau to produce power savings, and a generalisation of the circle method arguments of Montgomery and Vaughan that incorporates sieve weights.

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arXiv2022

Goldbach Numbers in Short Intervals -- A Nonnegative Model Approach

We decrease the length of the shortest interval for which almost all even integers in it are the sum of two primes. This is achieved by applying a version of the Circle Method that uses two minorants together with a nonnegative model for one of them. Compared to Harman's previous strongest result of this type, we in this way do not need any vector sieve type inequality and so require neither additional majorants nor strong density requirements for the minorants.

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arXiv2022

Representation of Squares by Nonsingular Cubic Forms

We prove an asymptotic formula for the number of representations of squares by nonsingular cubic forms in six or more variables. The main ingredients of the proof are Heath-Brown's form of the Circle Method and various exponential sum results. The depth of the exponential sum results is comparable to Hooley's work on cubic forms in nine variables, in particular we prove an analogue of Katz' bound.

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arXiv2022

Vinogradov's Theorem with Fouvry-Iwaniec Primes

We show that every sufficiently large $x\equiv 3(4)$ can be written as the sum of three primes, each of which is a sum of a square and a prime square. The main tools are a transference version of the circle method and various sieve related ideas.

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