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).
We give an alternative proof of a recent result by T.D. Browning and A. Haynes (arXiv:1204.6374v1) on multiplicative inverses in sequences of intervals and improve this result under additional conditions on the spacing of these intervals.
In this paper we study the mean values of some multiplicative functions connected with the divisor function on the short interval of summation. The asymptocic values for such mean values are proved.
Using recent results from the theory of integer points close to smooth curves, we give an asymptotic formula for the distribution of values of a class of integer-valued prime-independent multiplicative functions.
We investigate the solubility of the congruence xy=1 (mod p), where p is a prime and x,y are restricted to lie in suitable short intervals. Our work relies on a mean value theorem for incomplete Kloosterman sums.
Recently, several bounds have been obtained on the number of solutions to congruences of the type $$ (x_1+s)...(x_ν+s)\equiv (y_1+s)...(y_ν+s)\not\equiv0 \pmod p $$ modulo a prime $p$ with variables from some short intervals. Here, for almost all $p$ and all $s$ and also for a fixed $p$ and almost all $s$, we derive stronger bounds. We also use similar ideas to show that for almost all primes, one can always find an element of a large order in any rather short interval.
Let ${\mathcal H}$ be a multiplicative subgroup of $\mathbb{F}_p^*$ of order $H>p^{1/4}$. We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where ${\mathbf{\,e}}_p(z) = \exp(2 πi z/p)$, which improves a result of Bourgain and Garaev (2009). We also obtain new estimates for double exponential sums with product $nx$ with $x \in {\mathcal H}$ and $n \in {\mathcal N}$ for a short interval ${\mathcal N}$ of consecutive integers.
We consider random multiplicative functions taking the values $\pm 1$. Using Stein's method for normal approximation, we prove a central limit theorem for the sum of such multiplicative functions in appropriate 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 …
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 + …
We extend the Matomäki-Radziwiłł theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a function $f$ in typical intervals of length $h(\log X)^c$, with $h = h(X) \rightarrow \infty$ and where $c = c_f \geq 0$ is determined by the distribution of $\{|f(p)|\}_p$ in an explicit way. We give three applications. First, we show that the classical Rankin-Selberg-type asymptotic formula for partial sums of $|λ_f(n)|^2$, where $\{λ_f(n)\}_n$ is the sequence of no…
We show that Sarnak's conjecture on Möbius disjointness holds in every uniquely ergodic modelof a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial $P\in\R[x]$ with irrational leading coefficient and for each multiplicative function $\bnu:\N\to\C$, $|\bnu|\leq1$, we have\[ \frac{1}{M} \sum\_{M\le m\textless{}2M} \frac{1}{H} \left| \sum\_{m\le n \textless{} m+H} e^{2πiP(n)}\bnu(n) \right|\longrightarrow 0 \] as $M\to\infty$, $H\to\infty$, $H/M\to 0$.
We prove in Theorem $2.2$ that the multiplicatively closed subset generated by at most two elements in the set of natural numbers $\mathbb{N}$ has arbitrarily large gaps by explicitly constructing large integer intervals with known prime factorization for the end points, which do not contain any element from the multiplicatively closed set apart from the end points, which belong to the multiplicatively closed set. An Example $4.6$ is also illustrated. We also give a criterion in Theorems $7.8,7.12$ by using a geometric correspondence between maximal singly generated multiplicatively closed set…
For a prime $p$ and a polynomial $F(X,Y)$ over a finite field $\mathbb{F}_p$ of $p$ elements, we give upper bounds on the number of solutions $$ F(x,y)=0, \quad x\in\mathcal{A}, \ y\in \mathcal{B}, $$ where $\mathcal{A}$ and $\mathcal{B}$ are very small intervals or subgroups. These bounds can be considered as positive characteristic analogues of the result of Bombieri and Pila (1989) on sparsity of integral points on curves. As an application we prove that distinct consecutive elements in sequences generated compositions of several rational functions are not contained in any short intervals o…
We estimate multiplicative character sums taken on the values of a non-homogeneous Beatty sequence $\{[αn + β] : n =1,2,... \}$, where $α,β\in\R$, and $α$ is irrational. Our bounds are nontrivial over the same short intervals for which the classical character sum estimates of Burgess have been established.
In the present paper we obtain new upper bound estimates for the number of solutions of the congruence $$ x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in\cU, $$ for certain ranges of $H$ and $|\cU|$, where $\cU$ is a subset of the field of residue classes modulo $p$ having small multiplicative doubling. We then use this estimate to show that the number of solutions of the congruence $$ x^n\equiv λ\pmod p; \quad x\in \N, \quad L<x<L+p/n, $$ is at most $p^{\frac{1}{3}-c}$ uniformly over positive integers $n, λ$ and $L$, for some absolute constant $c>0$. This implies, in part…
We obtain new results about the representation of almost all residues modulo a prime $p$ by a product of a small integer and also an element of small multiplicative subgroup of $({\mathbb Z}/p{\mathbb Z})^*$. These results are based on some ideas, and their modifications, of a recent work of J. Cilleruelo and M. Z. Garaev (2014),
We obtain an upper bound for the multiplicative energy of the spectrum of an arbitrary set from $\mathbb{F}_p$, which is the best possible up to the results on exponential sums over subgroups.
We estimate double sums $$ S_χ(a, I, G) = \sum_{x \in I} \sum_{λ\in G} χ(x + aλ), \qquad 1\le a < p-1, $$ with a multiplicative character $χ$ modulo $p$ where $I= \{1,\ldots, H\}$ and $G$ is a subgroup of order $T$ of the multiplicative group of the finite field of $p$ elements. A nontrivial upper bound on $S_χ(a, I, G)$ can be derived from the Burgess bound if $H \ge p^{1/4+\varepsilon}$ and from some standard elementary arguments if $T \ge p^{1/2+\varepsilon}$, where $\varepsilon>0$ is arbitrary. We obtain a nontrivial estimate in a wider range of parameters $H$ and $T$. We also estimate dou…
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.
Recently Tao, Croot and Helfgott invented an algorithm to determine the parity of the number of primes in a given interval in O(x^{1/2-c+\eps}) steps for some absolute constant c. We propose a slightly different approach, which leads to the implicit value of c. To achieve this aim we discuss the summation of multiplicative functions, developing sublinear algorithms and proving several general theorems.
We show that the Erdős-Kac theorem is valid in almost all intervals $\left[x,x+h\right]$ as soon as $h$ tends to infinity with $x$. We also show that for all $k$ near $\log\log x$, almost all interval $\left[x,x+\exp\left(\left(\log\log x\right)^{1/2+\varepsilon}\right)\right]$ contains the expected number of integers $n$ such that $ω(n)=k$. These results are a consequence of the methods introduced by Matomäki and Radziwiłł to estimate sums of multiplicative functions over short intervals.
Let $λ$ denote the Liouville function. Assuming the Riemann Hypothesis, we prove that $$\int_X^{2X}\Big|\sum_{x\leq n \leq x+h}λ(n) \Big|^2 dx \ll Xh(\log X)^6,$$ as $X\rightarrow \infty$, provided $h=h(X)\leq \exp\left(\sqrt{\left(\frac{1}{2}-o(1)\right)\log X \log\log X}\right).$ The proof uses a simple variation of the methods developed by Matom{ä}ki and Radziwiłł in their work on multiplicative functions in short intervals, as well as some standard results concerning smooth numbers.
We obtain new bounds of multivariate exponential sums with monomials, when the variables run over rather short intervals. Furthermore, we use the same method to derive estimates on similar sums with multiplicative characters to which previously known methods do not apply. In particular, in the multiplicative characters modulo a prime $p$ we break the barrier of $p^{1/4}$ for ranges of individual variables.
We cosider the number of r-tuples of squarefree numbers in a short interval. We prove that it cannot be much bigger than the expected value and we also estabish an asymptotic formula if the interval is not very short.
This is a copy of my Bourbaki Seminar on the work of Matomaki and Radziwill showing that the Liouville function exhibits cancelation in almost all short intervals as soon as the length of the interval tends to infinity.
Using estimates on Hooley's $Δ$-function and a short interval version of the celebrated Dirichlet hyperbola principle, we derive an asymptotic formula for a class of arithmetic functions over short segments. Numerous examples are also given.
Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: The primes in an short interval contains many arithmetic progressions of any given length.
In this article, we prove an "equivalence" between two higher even moments of primes in short intervals under Riemann Hypothesis. We also provide numerical evidence in support of these asymptotic formulas.
Let $p$ be a large prime number and $g$ be any integer of multiplicative order $T$ modulo $p$. We obtain a new estimate of the double exponential sum $$ S=\sum_{n\in \mathcal{N}}\left|\sum_{m\in \mathcal{M} }e_p(an g^{m})\right|, \quad \gcd (a,p)=1, $$ where $\mathcal{N}$ and $\mathcal{M}$ are intervals of consecutive integers with $|\mathcal{N}|=N$ and $|\mathcal{M}|=M<T$ elements. One representative example is the following consequence of the main result: if $N=M\approx p^{1/3}$, then $|S|< N^{2-1/8 + o(1)}$. We then apply our estimate to obtain new results on additive congruences involving …
We prove that a suitable explicit formula for the Cesaro-averaged number of representations of an integer as a sum of two primes holds in short intervals.
A numerical study on the distributions of primes in short intervals of length $h$ over the natural numbers $N$ is presented. Based on Cramér's model in Number Theory, we obtain a heuristic expression applicable when $h \gg \log{N}$ but $h \ll N$, providing support to the Montgomery and Soundararajan conjecture on the variance of the prime distribution at this scale.
We show estimates for the distribution of $k$-free numbers in short intervals and arithmetic progressions. We argue that, at least in certain ranges, these estimates agree with a conjecture by H. L. Montgomery.
In this paper, we shall establish a rather general asymptotic formula in short intervals for a classe of arithmetic functions and announce two applications about the distribution of divisors of square-full numbers and integers representable as sums of two squares.
Given good knowledge on the even moments, we derive asymptotic formulas for $λ$-th moments of primes in short intervals and prove "equivalence" result on odd moments. We also provide numerical evidence in support of these results.
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.
We show that there are short intervals $[x,x+y]$ containing $\gg y^{1/10}$ numbers expressible as the sum of two squares, which is many more than the average when $y=o( (\log{x})^{5/9})$. We obtain similar results for sums of two squares in short arithmetic progressions.
We obtain a new bound on certain double sums of multiplicative characters improving the range of several previous results. This improvement comes from new bounds on the number of collinear triples in finite fields, which is a classical object of study of additive combinatorics.
