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Kaisa Matomäki

Publications and source records attributed to Kaisa Matomäki.

36 records · Page 2Linked to original sources
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.

math.NT↗
arXiv2020

On the variance of squarefree integers in short intervals and arithmetic progressions

We evaluate asymptotically the variance of the number of squarefree integers up to $x$ in short intervals of length $H < x^{6/11 - \varepsilon}$ and the variance of the number of squarefree integers up to $x$ in arithmetic progressions modulo $q$ with $q > x^{5/11 + \varepsilon}$. On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges can be improved to respectively $H < x^{2/3 - \varepsilon}$ and $q > x^{1/3 + \varepsilon}$. Furthermore we show that obtaining a bound sharp up to factors of $H^{\varepsilon}$ in the full range $H < x^{1 - \varepsilon}$ is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.

math.NT↗
arXiv2020

On the Möbius function in all short intervals

We show that, for the Möbius function $μ(n)$, we have $$ \sum_{x < n\leq x+x^θ}μ(n)=o(x^θ) $$ for any $θ>0.55$. This improves on a result of Ramachandra from 1976, which is valid for $θ>7/12$. Ramachandra's result corresponded to Huxley's $7/12$ exponent for the prime number theorem in short intervals. The main new idea leading to the improvement is using Ramaré's identity to extract a small prime factor from the $n$-sum. The proof method also allows us to improve on an estimate of Zhan for the exponential sum of the Möbius function as well as some results on multiplicative functions and almost primes in short intervals.

math.NT↗
arXiv2020

Multiplicative functions in short intervals II

We determine the behavior of multiplicative functions vanishing at a positive proportion of prime numbers in almost all short intervals. Furthermore we quantify "almost all" with uniform power-saving upper bounds, that is, we save a power of the suitably normalized length of the interval regardless of how long or short the interval is. Such power-saving bounds are new even in the special case of the Möbius function. These general results are motivated by several applications. First, we strengthen work of Hooley on sums of two squares by establishing an asymptotic for the number of integers that are sums of two squares in almost all short intervals. Previously only the order of magnitude was known. Secondly, we extend this result to general norm forms of an arbitrary number field $K$ (sums of two squares are norm-forms of $\mathbb{Q}(i)$). Thirdly, Hooley determined the order of magnitude of the sum of $(s_{n + 1} - s_{n})^γ$ with $γ\in (1, 5/3)$ where $s_{1} < s_2 < \ldots$ denote integers representable as sums of two squares. We establish a similar results with $γ\in (1, 3/2)$ and $s_n$ the sequence of integers representable as norm-forms of an arbitrary number field $K$. This is the first such result for a number field of degree greater than two. Assuming the Riemann Hypothesis for all Hecke $L$-functions we also show that $γ\in (1,2)$ is admissible. Fourthly, we improve on a recent result of Heath-Brown about gaps between $x^{\varepsilon}$-smooth numbers. More generally, we obtain results about gaps between multiplicative sequences. Finally our result is useful in other contexts aswell, for instance in our forthcoming work on Fourier uniformity (joint with Terence Tao, Joni Teraväinen and Tamar Ziegler).

math.NT↗
arXiv2020

Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges

We study the problem of obtaining asymptotic formulas for the sums $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$ and $\sum_{X < n \leq 2X} Λ(n) d_k(n+h)$, where $Λ$ is the von Mangoldt function, $d_k$ is the $k^{\operatorname{th}}$ divisor function, $X$ is large and $k \geq l \geq 2$ are real numbers. We show that for almost all $h \in [-H, H]$ with $H = (\log X)^{10000 k \log k}$, the expected asymptotic estimate holds. In our previous paper we were able to deal also with the case of $Λ(n) Λ(n + h)$ and we obtained better estimates for the error terms at the price of having to take $H = X^{8/33 + \varepsilon}$.

math.NT↗
arXiv2019

Discorrelation between primes in short intervals and polynomial phases

Let $H = N^θ, θ> 2/3$ and $k \geq 1$. We obtain estimates for the following exponential sum over primes in short intervals: \[ \sum_{N < n \leq N+H} Λ(n) e(g(n)), \] where $g$ is a polynomial of degree $k$. As a consequence of this in the special case $g(n) = αn^k$, we deduce a short interval version of the Waring-Goldbach problem.

math.NT↗
arXiv2019

Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges

We show that the expected asymptotic for the sums $\sum_{X < n \leq 2X} Λ(n) Λ(n+h)$, $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$, and $\sum_{X < n \leq 2X} Λ(n) d_k(n+h)$ hold for almost all $h \in [-H,H]$, provided that $X^{8/33+\varepsilon} \leq H \leq X^{1-\varepsilon}$, with an error term saving on average an arbitrary power of the logarithm over the trivial bound. Previous work of Mikawa, Perelli-Pintz and Baier-Browning-Marasingha-Zhao covered the range $H \geq X^{1/3+\varepsilon}$. We also obtain an analogous result for $\sum_n Λ(n) Λ(N-n)$. Our proof uses the circle method and some oscillatory integral estimates (following a paper of Zhan) to reduce matters to establishing some mean-value estimates for certain Dirichlet polynomials associated to "Type $d_3$" and "Type $d_4$" sums (as well as some other sums that are easier to treat). After applying Hölder's inequality to the Type $d_3$ sum, one is left with two expressions, one of which we can control using a short interval mean value theorem of Jutila, and the other we can control using exponential sum estimates of Robert and Sargos. The Type $d_4$ sum is treated similarly using the classical $L^2$ mean value theorem and the classical van der Corput exponential sum estimates.

math.NT↗
arXiv2018

Fourier uniformity of bounded multiplicative functions in short intervals on average

Let $λ$ denote the Liouville function. We show that as $X \rightarrow \infty$, $$ \int_{X}^{2X} \sup_α \left | \sum_{x < n \leq x + H} λ(n) e(-αn) \right | dx = o ( X H) $$ for all $H \geq X^θ$ with $θ> 0$ fixed but arbitrarily small. Previously, this was only known for $θ> 5/8$. For smaller values of $θ$ this is the first `non-trivial' case of local Fourier uniformity on average at this scale. We also obtain the analogous statement for (non-pretentious) $1$-bounded multiplicative functions. We illustrate the strength of the result by obtaining cancellations in the sum of $λ(n) Λ(n + h) Λ(n + 2h)$ over the ranges $h < X^θ$ and $n < X$, and where $Λ$ is the von Mangoldt function.

math.NT↗
arXiv2017

Multiplicative functions in short intervals

We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the Möbius function we show that there are cancellations in the sum of $μ(n)$ in almost all intervals of the form $[x, x + ψ(x)]$ with $ψ(x) \rightarrow \infty$ arbitrarily slowly. This goes beyond what was previously known conditionally on the Density Hypothesis or the stronger Riemann Hypothesis. Second, we settle the long-standing conjecture on the existence of $x^ε$-smooth numbers in intervals of the form $[x, x + c(\varepsilon) \sqrt{x}]$, recovering unconditionally a conditional (on the Riemann Hypothesis) result of Soundararajan. Third, we show that the mean-value of $λ(n)λ(n+1)$, with $λ(n)$ Liouville's function, is non-trivially bounded in absolute value by $1 - δ$ for some $δ> 0$. This settles an old folklore conjecture and constitutes progress towards Chowla's conjecture. Fourth, we show that a (general) real-valued multiplicative function $f$ has a positive proportion of sign changes if and only if $f$ is negative on at least one integer and non-zero on a positive proportion of the integers. This improves on many previous works, and is new already in the case of the Möbius function. We also obtain some additional results on smooth numbers in almost all intervals, and sign changes of multiplicative functions in all intervals of square-root length.

math.NT↗
arXiv2017

Vinogradov's three primes theorem with almost twin primes

In this paper we prove two results concerning Vinogradov's three primes theorem with primes that can be called almost twin primes. First, for any $m$, every sufficiently large odd integer $N$ can be written as a sum of three primes $p_1, p_2$ and $p_3$ such that, for each $i \in \{1,2,3\}$, the interval $[p_i, p_i + H]$ contains at least $m$ primes, for some $H = H(m)$. Second, every sufficiently large integer $N \equiv 3 \pmod{6}$ can be written as a sum of three primes $p_1, p_2$ and $p_3$ such that, for each $i \in \{1,2,3\}$, $p_i + 2$ has at most two prime factors.

math.NT↗
arXiv2015

When the sieve works

We are interested in classifying those sets of primes $\mathcal{P}$ such that when we sieve out the integers up to $x$ by the primes in $\mathcal{P}^c$ we are left with roughly the expected number of unsieved integers. In particular, we obtain the first general results for sieving an interval of length $x$ with primes including some in $(\sqrt{x},x]$, using methods motivated by additive combinatorics.

math.NT↗
arXiv2015

Sign patterns of the Liouville and Möbius functions

Let $λ$ and $μ$ denote the Liouville and Möbius functions respectively. Hildebrand showed that all eight possible sign patterns for $(λ(n), λ(n+1), λ(n+2))$ occur infinitely often. By using the recent result of the first two authors on mean values of multiplicative functions in short intervals, we strengthen Hildebrand's result by proving that each of these eight sign patterns occur with positive lower natural density. We also obtain an analogous result for the nine possible sign patterns for $(μ(n), μ(n+1))$. A new feature in the latter argument is the need to demonstrate that a certain random graph is almost surely connected.

math.NT↗
arXiv2015

When the sieve works II

For a set of primes $\mathcal{P}$, let $Ψ(x, \mathcal{P})$ be the number of positive integers $n \leq x$ all of whose prime factors lie in $\mathcal{P}$. In this paper we classify the sets of primes $\mathcal{P}$ such that $Ψ(x, \mathcal{P})$ is within a constant factor of its expected value. This task was recently initiated by Granville, Koukoulopoulos and Matomäki and their main conjecture is proved in this paper. In particular our main theorem implies that, if not too many large primes are sieved out in the sense that \[ \sum_{\substack{p \in \mathcal{P} \\ x^{1/v} < p \leq x^{1/u}}} \frac{1}{p} \geq \frac{1 + \varepsilon}{u}, \] for some $\varepsilon > 0$ and $v \geq u \geq 1$, then \[ Ψ(x, \mathcal{P}) \gg_{\varepsilon, v} x \prod_{\substack{p \leq x\\ p \notin\mathcal{P}}} \left(1 - \frac{1}{p}\right). \]

math.NT↗
arXiv2015

Small scale distribution of zeros and mass of modular forms

We study the behavior of zeros and mass of holomorphic Hecke cusp forms on $SL_2(\mathbb Z) \backslash \mathbb H$ at small scales. In particular, we examine the distribution of the zeros within hyperbolic balls whose radii shrink sufficiently slowly as $k \rightarrow \infty$. We show that the zeros equidistribute within such balls as $k \rightarrow \infty$ as long as the radii shrink at a rate at most a small power of $1/\log k$. This relies on a new, effective, proof of Rudnick's theorem on equidistribution of the zeros and on an effective version of Quantum Unique Ergodicity for holomorphic forms, which we obtain in this paper. We also examine the distribution of the zeros near the cusp of $SL_2(\mathbb Z) \backslash \mathbb H$. Ghosh and Sarnak conjectured that almost all the zeros here lie on two vertical geodesics. We show that for almost all forms a positive proportion of zeros high in the cusp do lie on these geodesics. For all forms, we assume the Generalized Lindelöf Hypothesis and establish a lower bound on the number of zeros that lie on these geodesics, which is significantly stronger than the previous unconditional results.

math.NT↗
arXiv2015

Sign changes of Hecke eigenvalues

Let $f$ be a holomorphic or Maass Hecke cusp form for the full modular group and write $λ_f(n)$ for the corresponding Hecke eigenvalues. We are interested in the signs of those eigenvalues. In the holomorphic case, we show that for some positive constant $δ$ and every large enough $x$, the sequence $(λ_f(n))_{n \leq x}$ has at least $δx$ sign changes. Furthermore we show that half of non-zero $λ_f(n)$ are positive and half are negative. In the Maass case, it is not yet known that the coefficients are non-lacunary, but our method is robust enough to show that on the relative set of non-zero coefficients there is a positive proportion of sign changes. In both cases previous lower bounds for the number of sign changes were of the form $x^δ$ for some $δ< 1$.

math.NT↗
arXiv2015

A note on the Liouville function in short intervals

In this note we give a short and self-contained proof that, for any $δ> 0$, $\sum_{x \leq n \leq x+x^δ} λ(n) = o(x^δ)$ for almost all $x \in [X, 2X]$. We also sketch a proof of a generalization of such a result to general real-valued multiplicative functions. Both results are special cases of results in our more involved and lengthy recent pre-print.

math.NT↗
arXiv2012

A new geometric approach to Sturmian words

We introduce a new geometric approach to Sturmian words by means of a mapping that associates certain lines in the n x n -grid and sets of finite Sturmian words of length n. Using this mapping, we give new proofs of the formulas enumerating the finite Sturmian words and the palindromic finite Sturmian words of a given length. We also give a new proof for the well-known result that a factor of a Sturmian word has precisely two return words.

cs.DM↗