Sharp local estimates for smooth numbers
We establish sharp local estimates for smooth numbers. Corollaries are derived, including optimal estimates for square-free smooth numbers.
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arXiv subjects
Ofir Gorodetsky
Publications and source records attributed to Ofir Gorodetsky.
A Kubilius model for sieve-theoretic sequences
We bound the total variation distance in the Kubilius model for sequences with positive level of distribution. We obtain a result that we expect is qualitatively optimal. As a special case, it recovers a recent result of Ford on shifted primes, with a slightly simplified proof. In the classical case considered by Kubilius, our theorem gives a simple proof of the optimal bound discovered by Tenenbaum, up to factors of $x^{o(1)}$ and $u^{o(u)}$.
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Multiplicative chaos measure for multiplicative functions: the $L^1$-regime
Let $α$ be a Steinhaus random multiplicative function. For a wide class of multiplicative functions $f$ we construct a multiplicative chaos measure arising from the Dirichlet series of $αf$, in the whole $L^1$-regime. Our method does not rely on the thick point approach or Gaussian approximation, and uses a modified second moment method with the help of an approximate Girsanov theorem. We also employ the idea of weak convergence in $L^r$ to show that the limiting measure is independent of the choice of the approximation schemes, and this may be seen as a non-Gaussian analogue of Shamov's characterisation of multiplicative chaos. Our class of $f$-s consists of those for which the mean value of $|f(p)|^2$ lies in $(0,1)$. In particular, it includes the indicator of sums of two squares. As an application of our construction, we establish a generalised central limit theorem for the (normalised) sums of $αf$, with random variance determined by the total mass of our measure.
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On the limiting distribution of sums of random multiplicative functions
We establish the limiting distribution of $\frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{n\le x}α(n)$ where $α$ is a Steinhaus random multiplicative function, answering a question of Harper. The distributional convergence is proved by applying the martingale central limit theorem to a suitably truncated sum. This truncation is inspired by work of Najnudel, Paquette, Simm and Vu on subcritical holomorphic multiplicative chaos setting, but analysed with a different conditioning argument generalised from Harper's work on fractional moments to circumvent integrability issues at criticality. A significant part of the proof is devoted to the convergence in probability of the associated partial Euler product to a critical multiplicative chaos measure, independent of the mild shift away from the critical line. Our approach to the universality of critical non-Gaussian multiplicative chaos bypasses the barrier analysis with the help of a modified second moment method, and employs a novel argument based on coupling and homogenisation by change of measure, which could be of independent interest.
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A short proof of Helson's conjecture
Let $α\colon \mathbb{N} \to S^1$ be the Steinhaus multiplicative function: a completely multiplicative function such that $(α(p))_{p\text{ prime}}$ are i.i.d.~random variables uniformly distributed on the complex unit circle $S^1$. Helson conjectured that $\mathbb{E}|\sum_{n\le x}α(n)|=o(\sqrt{x})$ as $x \to \infty$, and this was solved in a strong form by Harper. We give a short proof of the conjecture using a result of Saksman and Webb on a random model for the zeta function.
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New representations for all sporadic Apéry-like sequences, with applications to congruences
We find new representations, in terms of constant terms of powers of Laurent polynomials, for all the 15 sporadic Ap{é}ry-like sequences discovered by Zagier, Almkvist-Zudilin and Cooper. The new representations lead to binomial expressions for the sequences, which, as opposed to previous expressions, do not involve powers of 3 or 8. We use these to establish the supercongruence $B_{np^k} \equiv B_{np^{k-1}} \bmod p^{2k}$ for all primes $p \ge 3$ and integers $n,k \ge 1$, where $B_n$ is a sequence discovered by Zagier, known as Sequence $\mathbf{B}$. Additionally, for 14 of the 15 sequences, the Newton polytopes of the Laurent polynomials contain the origin as their only interior integral point. This property allows us to prove that these sequences satisfy a strong form of the Lucas congruences, extending work of Malik and Straub. Moreover, we obtain lower bounds on the $p$-adic valuation of these sequences via recent work of Delaygue.
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High and odd moments in the Erdős--Kac theorem
Granville and Soundararajan showed that the $k$th moment in the Erdős--Kac theorem is equal to the $k$th moment of the standard Gaussian distribution in the range $k=o((\log \log x)^{1/3})$, up to a negligible error term. We show that their range is sharp: when $k/(\log \log x)^{1/3}$ tends to infinity, a different behavior emerges, and odd moments start exhibiting similar growth to even moments. For odd $k$ we find the asymptotics of the $k$th moment when $k=O((\log \log x)^{1/3})$, where previously only an upper bound was known. Our methods are flexible and apply to other distributions, including the Poisson distribution, whose centered moments turn out to be excellent approximations for the Erdős--Kac moments.
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Smooth permutations and polynomials revisited
We study the counts of smooth permutations and smooth polynomials over finite fields. For both counts we prove an estimate with an error term that matches the error term found in the integer setting by de Bruijn more than 70 years ago. The main term is the usual Dickman $ρ$ function, but with its argument shifted. We determine the order of magnitude of $\log(p_{n,m}/ρ(n/m))$ where $p_{n,m}$ is the probability that a permutation on $n$ elements, chosen uniformly at random, is $m$-smooth. We uncover a phase transition in the polynomial setting: the probability that a polynomial of degree $n$ in $\mathbb{F}_q$ is $m$-smooth changes its behavior at $m\approx (3/2)\log_q n$.
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The variance of integers without small prime factors in short intervals
The variance of primes in short intervals relates to the Riemann Hypothesis, Montgomery's Pair Correlation Conjecture and the Hardy--Littlewood Conjecture. In regards to its asymptotics, very little is known unconditionally. We study the variance of integers without prime factors below $y$, in short intervals. We use complex analysis and sieve theory to prove an unconditional asymptotic result in a range for which we give evidence is qualitatively best possible. We find that this variance connects with statistics of $y$-smooth numbers, and, as with primes, is asymptotically smaller than the naive probabilistic prediction once the length of the interval is at least a power of $y$.
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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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Martingale central limit theorem for random multiplicative functions
Let $α$ be a Steinhaus or a Rademacher random multiplicative function. For a wide class of multiplicative functions $f$ we show that the sum $\sum_{n \le x}α(n) f(n)$, normalised to have mean square $1$, has a non-Gaussian limiting distribution. More precisely, we establish a generalised central limit theorem with random variance determined by the total mass of a random measure associated with $αf$. Our result applies to $d_z$, the $z$-th divisor function, as long as $z$ is strictly between $0$ and $\tfrac{1}{\sqrt{2}}$. Other examples of admissible $f$-s include any multiplicative indicator function with the property that $f(p)=1$ holds for a set of primes of density strictly between $0$ and $\tfrac{1}{2}$.
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On Erdős sums of almost primes
In 1935, Erdős proved that the sums $f_k=\sum_n 1/(n\log n)$, over integers $n$ with exactly $k$ prime factors, are bounded by an absolute constant, and in 1993 Zhang proved that $f_k$ is maximized by the prime sum $f_1=\sum_p 1/(p\log p)$. According to a 2013 conjecture of Banks and Martin, the sums $f_k$ are predicted to decrease monotonically in $k$. In this article, we show that the sums restricted to odd integers are indeed monotonically decreasing in $k$, sufficiently large. By contrast, contrary to the conjecture we prove that the sums $f_k$ increase monotonically in $k$, sufficiently large. Our main result gives an asymptotic for $f_k$ which identifies the (negative) secondary term, namely $f_k = 1 - (a+o(1))k^2/2^k$ for an explicit constant $a= 0.0656\cdots$. This is proven by a refined method combining real and complex analysis, whereas the classical results of Sathe and Selberg on products of $k$ primes imply the weaker estimate $f_k=1+O_{\varepsilon}(k^{\varepsilon-1/2})$. We also give an alternate, probability-theoretic argument related to the Dickman distribution. Here the proof reduces to showing a sequence of integrals converges exponentially quickly to $e^{-γ}$, which may be of independent interest.
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Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor
Let $g$ be a random matrix distributed according to uniform probability measure on the finite general linear group $\mathrm{GL}_n(\mathbb{F}_q)$. We show that $\mathrm{Tr}(g^k)$ equidistributes on $\mathbb{F}_q$ as $n \to \infty$ as long as $\log k=o(n^2)$ and that this range is sharp. We also show that nontrivial linear combinations of $\mathrm{Tr}(g^1),\ldots, \mathrm{Tr}(g^k)$ equidistribute as long as $\log k =o(n)$ and this range is sharp as well. Previously equidistribution of either a single trace or a linear combination of traces was only known for $k \le c_q n$, where $c_q$ depends on $q$, due to work of the first author and Rodgers. We reduce the problem to exhibiting cancellation in certain short character sums in function fields. For the equidistribution of $\mathrm{Tr}(g^k)$ we end up showing that certain explicit character sums modulo $T^{k+1}$ exhibit cancellation when averaged over monic polynomials of degree $n$ in $\mathbb{F}_q[T]$ as long as $\log k = o(n^2)$. This goes far beyond the classical range $\log k =o(n)$ due to Montgomery and Vaughan. To study these sums we build on the argument of Montgomery and Vaughan but exploit additional symmetry present in the considered sums.
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Multiplicative arithmetic functions and the generalized Ewens measure
Random integers, sampled uniformly from $[1,x]$, share similarities with random permutations, sampled uniformly from $S_n$. These similarities include the Erdős--Kac theorem on the distribution of the number of prime factors of a random integer, and Billingsley's theorem on the largest prime factors of a random integer. In this paper we extend this analogy to non-uniform distributions. Given a multiplicative function $α\colon \mathbb{N} \to \mathbb{R}_{\ge 0}$, one may associate with it a measure on the integers in $[1,x]$, where $n$ is sampled with probability proportional to the value $α(n)$. Analogously, given a sequence $\{ θ_i\}_{i \ge 1}$ of non-negative reals, one may associate with it a measure on $S_n$ that assigns to a permutation a probability proportional to a product of weights over the cycles of the permutation. This measure is known as the generalized Ewens measure. We study the case where the mean value of $α$ over primes tends to some positive $θ$, as well as the weights $α(p) \approx (\log p)^γ$. In both cases, we obtain results in the integer setting which are in agreement with those in the permutation setting.
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Magic squares, the symmetric group and Möbius randomness
Diaconis and Gamburd computed moments of secular coefficients in the CUE ensemble. We use the characteristic map to give a new combinatorial proof of their result. We also extend their computation to moments of traces of symmetric powers, where the same result holds but in a wider range. Our combinatorial proof is inspired by gcd matrices, as used by Vaughan and Wooley and by Granville and Soundararajan. We use these CUE computations to suggest a conjecture about moments of characters sums twisted by the Liouville (or by the Möbius) function, and establish a version of it in function fields. The moral of our conjecture (and its verification in function fields) is that the Steinhaus random multiplicative function is a good model for the Liouville (or for the Möbius) function twisted by a random Dirichlet character. We also evaluate moments of secular coefficients and traces of symmetric powers, without any condition on the size of the matrix. As an application we give a new formula for a matrix integral that was considered by Keating, Rodgers, Roditty-Gershon and Rudnick in their study of the $k$-fold divisor function.
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Smooth integers and the Dickman $ρ$ function
We establish an asymptotic formula for $Ψ(x,y)$ whose shape is $x ρ(\log x/\log y)$ times correction factors. These factors take into account the contributions of zeta zeros and prime powers and the formula can be regarded as an (approximate) explicit formula for $Ψ(x,y)$. With this formula at hand we prove oscillation results for $Ψ(x,y)$, which resolve a question of Hildebrand on the range of validity of $Ψ(x,y) \asymp xρ(\log x/\log y)$. We also address a question of Pomerance on the range of validity of $Ψ(x,y) \ge x ρ(\log x/\log y)$. Along the way we improve classical estimates for $Ψ(x,y)$ and, on the Riemann Hypothesis, uncover an unexpected phase transition of $Ψ(x,y)$ at $y=(\log x)^{3/2+o(1)}$.
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Smooth integers and de Bruijn's approximation $Λ$
This paper is concerned with the relationship of $y$-smooth integers and de Bruijn's approximation $Λ(x,y)$. Under the Riemann hypothesis, Saias proved that the count of $y$-smooth integers up to $x$, $Ψ(x,y)$, is asymptotic to $Λ(x,y)$ when $y \ge (\log x)^{2+\varepsilon}$. We extend the range to $y \ge (\log x)^{3/2+\varepsilon}$ by introducing a correction factor that takes into account the contributions of zeta zeros and prime powers. We use this correction term to uncover a lower order term in the asymptotics of $Ψ(x,y)/Λ(x,y)$. The term relates to the error term in the prime number theorem, and implies that large positive (resp. negative) values of $\sum_{n \le y} Λ(n)-y$ lead to large positive (resp. negative) values of $Ψ(x,y)-Λ(x,y)$, and vice versa. Under the Linear Independence hypothesis, we show a Chebyshev's bias in $Ψ(x,y)-Λ(x,y)$.
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Uniform estimates for smooth polynomials over finite fields
We establish new estimates for the number of $m$-smooth polynomials of degree $n$ over a finite field $\mathbb{F}_q$, where the main term involves the number of $m$-smooth permutations on $n$ elements. Our estimates imply that the probability that a random polynomial of degree $n$ is $m$-smooth is asymptotic to the probability that a random permutation on $n$ elements is $m$-smooth, uniformly for $m\ge (2+\varepsilon)\log_q n$ as $q^n \to \infty$. This should be viewed as an unconditional analogue of works of Hildebrand and of Saias in the integer setting, which assume the Riemann Hypothesis. Moreover, we show that the range $m \ge (2+\varepsilon)\log_q n$ is sharp; this should be viewed as a resolution of a (polynomial analogue of a) conjecture of Hildebrand. As an application of our estimates, we determine the rate of decay in the asymptotic formula for the expected degree of the largest prime factor of a random polynomial.
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