arXiv subjects

Kaisa Matomäki

Publications and source records attributed to Kaisa Matomäki.

At least 19 recordsLinked to original sources

arXiv2026

Linnik's problem for multiplicative functions

We study a multiplicative function analogue of Linnik's problem on the least prime in an arithmetic progression. Let $h\colon \mathbb{N}\to\mathbb{R}\setminus\{0\}$ be a multiplicative function, and let $a \pmod q$ be a reduced residue class. We ask how far one must go before finding square-free integers $n_1,n_2\equiv a \pmod q$ with $h(n_1)<0<h(n_2)$. We show that one can always find such integers with $n_1,n_2\le q^{2+o(1)}$, unless the sign of $h$ strongly pretends to be a real Dirichlet character modulo $q$. Thus, apart from this natural character obstruction, sign changes of a multiplicative function occur in every reduced residue class at a scale corresponding essentially to the square root barrier. In the special case of the Liouville function $λ$ this improves on a recent result of Ford and Radziwiłł and matches, up to $q^{o(1)}$ factors, what was previously known conditionally under the generalized Riemann hypothesis.

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arXiv2026

Higher uniformity of arithmetic functions in short intervals II. Almost all intervals

We study higher uniformity properties of the von Mangoldt function $Λ$, the Möbius function $μ$, and the divisor functions $d_k$ on short intervals $(x,x+H]$ for almost all $x \in [X, 2X]$. Let $Λ^\sharp$ and $d_k^\sharp$ be suitable approximants of $Λ$ and $d_k$, $G/Γ$ a filtered nilmanifold, and $F\colon G/Γ\to \mathbb{C}$ a Lipschitz function. Then our results imply for instance that when $X^{1/3+\varepsilon} \leq H \leq X$ we have, for almost all $x \in [X, 2X]$, \[ \sup_{g \in \text{Poly}(\mathbb{Z} \to G)} \left| \sum_{x < n \leq x+H} (Λ(n)-Λ^\sharp(n)) \overline{F}(g(n)Γ) \right| \ll H\log^{-A} X \] for any fixed $A>0$, and that when $X^{\varepsilon} \leq H \leq X$ we have, for almost all $x \in [X, 2X]$, \[ \sup_{g \in \text{Poly}(\mathbb{Z} \to G)} \left| \sum_{x < n \leq x+H} (d_k(n)-d_k^\sharp(n)) \overline{F}(g(n)Γ) \right| = o(H \log^{k-1} X). \] As a consequence, we show that the short interval Gowers norms $\|Λ-Λ^\sharp\|_{U^s(X,X+H]}$ and $\|d_k-d_k^\sharp\|_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$ in the same ranges of $H$. This in turn allows us to establish the Hardy-Littlewood conjecture and the divisor correlation conjecture with a short average over one variable. Our main new ingredients are type $II$ estimates obtained by developing a "contagion lemma" for nilsequences and then using this to "scale up" an approximate functional equation for the nilsequence to a larger scale. This extends an approach developed by Walsh for Fourier uniformity.

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arXiv2025

On optimality of mollifiers

Mollifiers are used in a variety of contexts, for instance to study the non-vanishing of $L$-functions. In this paper, we study the general question of finding optimal mollifiers and provide criteria to identify them provided the corresponding mollified moments can be computed. As an application, we study the non-vanishing of central values of Dirichlet $L$-functions. In particular we show that the Michel-Vanderkam mollifier is optimal in a wide class of balanced two-piece mollifiers as well as provide a new proof that the Iwaniec-Sarnak mollifier is optimal in a wide class of one-piece mollifiers.

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arXiv2025

Low-lying zeros in families of Maass form L-functions: an extended density theorem

We study the one-level density of low-lying zeros in the family of Maass form $L$-functions of prime level $N$ tending to infinity. Generalizing the influential work of Iwaniec, Luo and Sarnak to this context, Alpoge et al. have proven the Katz-Sarnak prediction for test functions whose Fourier transform is supported in $(-\frac32,\frac32)$. In this paper, we extend the unconditional admissible support to $(-\frac{15}8,\frac{15}8)$. The key tools in our approach are analytic estimates for integrals appearing in the Kutznetsov trace formula, as well as a reduction to bounds on Dirichlet polynomials, which eventually are obtained from the large sieve and the fourth moment bound for Dirichlet $L$-functions. Assuming the Grand Density Conjecture, we extend the admissible support to $(-2,2)$. In addition, we show that the same techniques also allow for an unconditional improvement of the admissible support in the corresponding family of $L$-functions attached to holomorphic forms.

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arXiv2025

Weighted sieves with switching

Weighted sieves are used to detect numbers with at most $S$ prime factors with $S \in \mathbb{N}$ as small as possible. When one studies problems with two variables in somewhat symmetric roles (such as Chen primes, that is primes $p$ such that $p+2$ has at most two prime factors), one can utilize the switching principle. Here we discuss how different sieve weights work in such a situation, concentrating in particular on detecting a prime along with a product of at most three primes. As applications, we improve on the works of Yang and Harman concerning Diophantine approximation with a prime and an almost prime, and prove that, in general, one can find a pair $(p, P_3)$ when both the original and the switched problem have level of distribution at least $0.267$.

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arXiv2024

The sixth moment of Dirichlet L-functions at the central point

In 1970, Huxley obtained a sharp upper bound for the sixth moment of Dirichlet $L$-functions at the central point, averaged over primitive characters $χ$ modulo $q$ and all moduli $q \leq Q$. In 2007, as an application of their ``asymptotic large sieve'', Conrey, Iwaniec and Soundararajan showed that when an additional short $t$-averaging is introduced into the problem, an asymptotic can be obtained. In this paper we show that this extraneous averaging can be removed, and we thus obtain an asymptotic for the original moment problem considered by Huxley. The main new difficulty in our work is the appearance of certain challenging ``unbalanced'' sums that arise as soon as the $t$-aspect averaging is removed.

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arXiv2024

A note on zero density results implying large value estimates for Dirichlet polynomials

In this note we investigate connections between zero density estimates for the Riemann zeta function and large value estimates for Dirichlet polynomials. It is well known that estimates of the latter type imply estimates of the former type. Our goal is to show that there is an implication to the other direction as well, i.e. zero density estimates for the Riemann zeta function imply large value estimates for Dirichlet polynomials.

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arXiv2024

Higher uniformity of arithmetic functions in short intervals I. All intervals

We study higher uniformity properties of the Möbius function $μ$, the von Mangoldt function $Λ$, and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{θ+\varepsilon} \leq H \leq X^{1-\varepsilon}$ for a fixed constant $0 \leq θ< 1$ and any $\varepsilon>0$. More precisely, letting $Λ^\sharp$ and $d_k^\sharp$ be suitable approximants of $Λ$ and $d_k$ and $μ^\sharp = 0$, we show for instance that, for any nilsequence $F(g(n)Γ)$, we have \[ \sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) Γ) \ll H \log^{-A} X \] when $θ= 5/8$ and $f \in \{Λ, μ, d_k\}$ or $θ= 1/3$ and $f = d_2$. As a consequence, we show that the short interval Gowers norms $\|f-f^\sharp\|_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$ for these choices of $f,θ$. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals, and show that multiple ergodic averages along primes in short intervals converge in $L^2$. Our innovations include the use of multi-parameter nilsequence equidistribution theorems to control type $II$ sums, and an elementary decomposition of the neighbourhood of a hyperbola into arithmetic progressions to control type $I_2$ sums.

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arXiv2024

Products of primes in arithmetic progressions

A conjecture of Erdős states that, for any large prime $q$, every reduced residue class $\pmod q$ can be represented as a product $p_1p_2$ of two primes $p_1,p_2\leq q$. We establish a ternary version of this conjecture, showing that, for any sufficiently large cube-free integer $q$, every reduced residue class $\pmod q$ can be written as $p_1p_2p_3$ with $p_1,p_2,p_3\leq q$ primes. We also show that, for any $\varepsilon > 0$ and any sufficiently large integer $q$, at least $(2/3-\varepsilon)φ(q)$ reduced residue classes $\pmod q$ can be represented as a product $p_1 p_2$ of two primes $p_1, p_2 \leq q$. The problems naturally reduce to studying character sums. The main innovation in the paper is the establishment of a multiplicative dense model theorem for character sums over primes in the spirit of the transference principle. In order to deal with possible local obstructions we use bounds for the logarithmic density of primes in certain unions of cosets of subgroups of $\mathbb{Z}_q^\times$ of small index and study in detail the exceptional case that there exists a quadratic character $ψ\pmod{q}$ such that $ψ(p) = -1$ for almost all primes $p \leq q$.

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arXiv2024

Primes in arithmetic progressions and short intervals without $L$-functions

We develop a sieve that can detect primes in multiplicatively structured sets under certain conditions. We apply it to obtain a new $L$-function free proof of Linnik's problem of bounding the least prime $p$ such that $p\equiv a\pmod q$ (with the bound $p \ll q^{350}$) as well as a new $L$-function free proof that the interval $(x-x^{39/40}, x]$ contains primes for every large $x$. In a future work we will develop the sieve further and provide more applications.

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arXiv2023

A note on exceptional characters and non-vanishing of Dirichlet $L$-functions

We study non-vanishing of Dirichlet $L$-functions at the central point under the unlikely assumption that there exists an exceptional Dirichlet character. In particular we prove that if $ψ$ is a real primitive character modulo $D \in \mathbb{N}$ with $L(1, ψ) \ll (\log D)^{-25-\varepsilon}$, then, for any prime $q \in [D^{300}, D^{O(1)}]$, one has $L(1/2, χ) \neq 0$ for almost all Dirichlet characters $χ\pmod{q}$.

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arXiv2023

The eighth moment of Dirichlet L-functions II

We prove an asymptotic formula for the eighth moment of Dirichlet $L$-functions averaged over primitive characters $χ$ modulo $q$, over all moduli $q\leq Q$ and with a short average on the critical line. Previously the same result was shown conditionally on the Generalized Riemann Hypothesis by the first two authors.

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arXiv2023

Almost primes in almost all short intervals II

We show that, for almost all $x$, the interval $(x, x+(\log x)^{2.1}]$ contains products of exactly two primes. This improves on a work of the second author that had $3.51$ in place of $2.1$. To obtain this improvement, we prove a new type II estimate. One of the new innovations is to use Heath-Brown's mean value theorem for sparse Dirichlet polynomials.

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arXiv2022

Higher uniformity of bounded multiplicative functions in short intervals on average

Let $λ$ denote the Liouville function. We show that, as $X \rightarrow \infty$, $$\int_{X}^{2X} \sup_{\substack{P(Y)\in \mathbb{R}[Y]\\ deg(P)\leq k}} \Big | \sum_{x \leq n \leq x + H} λ(n) e(-P(n)) \Big |\ dx = o ( X H)$$ for all fixed $k$ and $X^θ \leq H \leq X$ with $0 < θ< 1$ fixed but arbitrarily small. Previously this was only established for $k \leq 1$. We obtain this result as a special case of the corresponding statement for (non-pretentious) $1$-bounded multiplicative functions that we prove. In fact, we are able to replace the polynomial phases $e(-P(n))$ by degree $k$ nilsequences $\overline{F}(g(n) Γ)$. By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result $$\int_{X}^{2X} \| λ\|_{U^{k+1}([x,x+H])}\ dx = o ( X )$$ in the same range of $H$. We present applications of this result to patterns of various types in the Liouville sequence. Firstly, we show that the number of sign patterns of the Liouville function is superpolynomial, making progress on a conjecture of Sarnak about the Liouville sequence having positive entropy. Secondly, we obtain cancellation in averages of $λ$ over short polynomial progressions $(n+P_1(m),\ldots, n+P_k(m))$, which in the case of linear polynomials yields a new averaged version of Chowla's conjecture. We are in fact able to prove our results on polynomial phases in the wider range $H\geq \exp((\log X)^{5/8+\varepsilon})$, thus strengthening also previous work on the Fourier uniformity of the Liouville function.

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arXiv2022

An averaged form of Chowla's conjecture

Let $λ$ denote the Liouville function. A well known conjecture of Chowla asserts that for any distinct natural numbers $h_1,\dots,h_k$, one has $\sum_{1 \leq n \leq X} λ(n+h_1) \dotsm λ(n+h_k) = o(X)$ as $X \to \infty$. This conjecture remains unproven for any $h_1,\dots,h_k$ with $k \geq 2$. In this paper, using the recent results of the first two authors on mean values of multiplicative functions in short intervals, combined with an argument of Katai and Bourgain-Sarnak-Ziegler, we establish an averaged version of this conjecture, namely $$\sum_{h_1,\dots,h_k \leq H} \left|\sum_{1 \leq n \leq X} λ(n+h_1) \dotsm λ(n+h_k)\right| = o(H^kX)$$ as $X \to \infty$ whenever $H = H(X) \leq X$ goes to infinity as $X \to \infty$, and $k$ is fixed. Related to this, we give the exponential sum estimate $$ \int_0^X \left|\sum_{x \leq n \leq x+H} λ(n) e(αn)\right| dx = o( HX )$$ as $X \to \infty$ uniformly for all $α\in \mathbb{R}$, with $H$ as before. Our arguments in fact give quantitative bounds on the decay rate (roughly on the order of $\frac{\log\log H}{\log H}$), and extend to more general bounded multiplicative functions than the Liouville function, yielding an averaged form of a (corrected) conjecture of Elliott.

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arXiv2022

Siegel zeros, twin primes, Goldbach's conjecture, and primes in short intervals

We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular we prove for \[ \sum_{n \leq X} Λ(n) Λ(\pm n+h) \] an asymptotic formula which holds uniformly for $h = O(X)$. Such an asymptotic formula has been previously obtained only for fixed $h$ in which case our result quantitatively improves those of Heath-Brown (1983) and Tao and Teräväinen (2021). Since our main theorems work also for large $h$ we can derive new results concerning connections between Siegel zeros and the Goldbach conjecture and between Siegel zeros and primes in almost all very short intervals.

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arXiv2022

Almost primes in almost all very short intervals

We show that as soon as $h\to \infty$ with $X \to \infty$, almost all intervals $(x-h\log X, x]$ with $x \in (X/2, X]$ contain a product of at most two primes. In the proof we use Richert's weighted sieve, with the arithmetic information eventually coming from results of Deshouillers and Iwaniec on averages of Kloosterman sums.

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arXiv2021

Singmaster's conjecture in the interior of Pascal's triangle

Singmaster's conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal's triangle; that is, for any natural number $t \geq 2$, the number of solutions to the equation $\binom{n}{m} = t$ for natural numbers $1 \leq m < n$ is bounded. In this paper we establish this result in the interior region $\exp(\log^{2/3+\varepsilon} n) \leq m \leq n-\exp(\log^{2/3 + \varepsilon} n)$ for any fixed $\varepsilon > 0$. Indeed, when $t$ is sufficiently large depending on $\varepsilon$, we show that there are at most four solutions (or at most two in either half of Pascal's triangle) in this region. We also establish analogous results for the equation $(n)_m = t$, where $(n)_m := n(n-1)\ldots(n-m+1)$ denotes the falling factorial.

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