Functions In Short Intervals Research Articles

On certain unbounded multiplicative functions in short intervals

On certain unbounded multiplicative functions in short intervals

Multiplicative functions in short arithmetic progressions

AbstractWe study for bounded multiplicative functions sums of the form establishing that their variance over residue classes is small as soon as , for almost all moduli , with a nearly power‐saving exceptional set of . This improves and generalizes previous results of Hooley on Barban–Davenport–Halberstam type theorems for such , and moreover our exceptional set is essentially optimal unless one is able to make progress on certain well‐known conjectures. We are nevertheless able to prove stronger bounds for the number of the exceptional moduli in the cases where is restricted to be either smooth or prime, and conditionally on GRH we show that our variance estimate is valid for every . These results are special cases of a “hybrid result” that we establish that works for sums of over almost all short intervals and arithmetic progressions simultaneously, thus generalizing the Matomäki–Radziwiłł theorem on multiplicative functions in short intervals. We also consider the maximal deviation of over all residue classes in the square root range , and show that it is small for “smooth‐supported” , again apart from a nearly power‐saving set of exceptional , thus providing a smaller exceptional set than what follows from Bombieri–Vinogradov type theorems. As an application of our methods, we consider Linnik‐type problems for products of exactly three primes, and in particular prove a ternary approximation to a conjecture of Erdős on representing every element of the multiplicative group as the product of two primes less than .

Higher uniformity of bounded multiplicative functions in short intervals on average

Let $\lambda$ denote the Liouville function. We show that, as $X \rightarrow \infty$, $$ \int^{2X}_X \sup_{\begin{smallmatrix} P(Y)\in\mathbb{R}[Y] \\ \mathrm{deg}{P} \leq k\end{smallmatrix}} \left| \sum_{x\leq n \leq x+H} \lambda(n) e(-P(n))\right| \ dx=o (XH) $$ for all fixed $k$ and $X^{\theta} \leq H \leq X$ with $0 \lt \theta \lt 1$ 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 $\overline{F}(g(n) \Gamma )$. By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result $$ \int_{X}^{2X} \| \lambda \|_{U^{k+1}([x,x+H])}\ dx = o ( X ) $$ 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 $\lambda$ 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 $H\geq \mathrm{exp}((\mathrm{log} X)^{5/8+\varepsilon})$, thus strengthening also previous work on the Fourier uniformity of the Liouville function.

Divisor-bounded multiplicative functions in short intervals

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 ge 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 |lambda _f(n) |^2, where {lambda _f(n)}_n is the sequence of normalized Fourier coefficients of a primitive non-CM holomorphic cusp form, persists in typical short intervals of length hlog X, if h = h(X) rightarrow infty . We also generalize this result to sequences {|lambda _{pi }(n) |^2}_n, where lambda _{pi }(n) is the nth coefficient of the standard L-function of an automorphic representation pi with unitary central character for GL_m, m ge 2, provided pi satisfies the generalized Ramanujan conjecture. Second, using recent developments in the theory of automorphic forms we estimate the variance of averages of all positive real moments {|lambda _f(n) |^{alpha }}_n over intervals of length h(log X)^{c_{alpha }}, with c_{alpha } > 0 explicit, for any alpha > 0, as h = h(X) rightarrow infty . Finally, we show that the (non-multiplicative) Hooley Delta -function has average value gg log log X in typical short intervals of length (log X)^{1/2+eta }, where eta >0 is fixed.

Superscars for arithmetic point scatters II

Abstract We consider momentum push-forwards of measures arising as quantum limits (semiclassical measures) of eigenfunctions of a point scatterer on the standard flat torus ${\mathbb T}^2 = {\mathbb R}^2/{\mathbb Z}^{2}$ . Given any probability measure arising by placing delta masses, with equal weights, on ${\mathbb Z}^2$ -lattice points on circles and projecting to the unit circle, we show that the mass of certain subsequences of eigenfunctions, in momentum space, completely localizes on that measure and are completely delocalized in position (i.e., concentration on Lagrangian states). We also show that the mass, in momentum, can fully localize on more exotic measures, for example, singular continuous ones with support on Cantor sets. Further, we can give examples of quantum limits that are certain convex combinations of such measures, in particular showing that the set of quantum limits is richer than the ones arising only from weak limits of lattice points on circles. The proofs exploit features of the half-dimensional sieve and behavior of multiplicative functions in short intervals, enabling precise control of the location of perturbed eigenvalues.

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

Abstract We study higher uniformity properties of the Möbius function $\mu $ , the von Mangoldt function $\Lambda $ , and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$ for a fixed constant $0 \leq \theta < 1$ and any $\varepsilon>0$ . More precisely, letting $\Lambda ^\sharp $ and $d_k^\sharp $ be suitable approximants of $\Lambda $ and $d_k$ and $\mu ^\sharp = 0$ , we show for instance that, for any nilsequence $F(g(n)\Gamma )$ , we have $$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$ when $\theta = 5/8$ and $f \in \{\Lambda , \mu , d_k\}$ or $\theta = 1/3$ 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,\theta $ . 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 multiparameter nilsequence equidistribution theorems to control type $II$ sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type $I_2$ sums.

Additive functions in short intervals, gaps and a conjecture of Erdős

With the aim of treating the local behaviour of additive functions, we develop analogues of the Matomäki–Radziwiłł theorem that allow us to approximate the average of a general additive function over a typical short interval in terms of a corresponding long average. As part of this treatment, we use a variant of the Matomäki–Radziwiłł theorem for divisor-bounded multiplicative functions recently proven in Mangerel (Divisor-bounded multiplicative functions in short intervals. arXiv: 2108.11401). We consider two sets of applications of these methods. Our first application shows that for an additive function {varvec{g:}} mathbb {N} rightarrow mathbb {C} any non-trivial savings in the size of the average gap |{varvec{g}}{} {textbf {(}}{varvec{n}}{} {textbf {)}}-{varvec{g}}{} {textbf {(}}{varvec{n}}-{textbf {1}}{} {textbf {)}} | implies that {varvec{g}} must have a small first centred moment i.e. the discrepancy of {varvec{g}}{} {textbf {(}}{varvec{n}}{} {textbf {)}} from its mean is small on average. We also obtain a variant of such a result for the second moment of the gaps. This complements results of Elliott and of Hildebrand. As a second application, we make partial progress on an old question of Erdős relating to characterizing constant multiples of {{textbf {log}}} ,{varvec{n}} as the only almost everywhere increasing additive functions. We show that if an additive function is almost everywhere non-decreasing then it is almost everywhere well approximated by a constant times a logarithm. We also show that if the set {{varvec{n}} in mathbb {N} : {varvec{g}}{} {textbf {(}}{varvec{n}}{} {textbf {)}} < {varvec{g}}{} {textbf {(}}{varvec{n}}-{textbf {1}}{} {textbf {)}}} is sufficiently sparse, and if {varvec{g}} is not extremely large too often on the primes (in a precise sense), then {varvec{g}} is identically equal to a constant times a logarithm.

On the Liouville Function in Short Intervals

Abstract Let $\lambda $ denote the Liouville function. Assuming the Riemann Hypothesis, we prove that $\int _X^{2X}\Big |\sum _{x\leq n \leq x+h}\lambda (n) \Big |^2{\textrm{d}} x \ll Xh(\log X)^6,$ 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 Matomäki and Radziwiłł in their work on multiplicative functions in short intervals, as well as some standard results concerning smooth numbers.

Square-root cancellation for sums of factorization functions over short intervals in function fields

We present new estimates for sums of the divisor function and other similar arithmetic functions in short intervals over function fields. (When the intervals are long, one obtains a good estimate from the Riemann hypothesis.) We obtain an estimate that approaches square-root cancellation as long as the characteristic of the finite field is relatively large. This is done by a geometric method, inspired by work of Hast and Matei, where we calculate the singular locus of a variety whose Fq-points control this sum. This has applications to highly unbalanced moments of L-functions.

Multiplicative Functions in Short Intervals, with Applications

Multiplicative Functions in Short Intervals, with Applications

Correction to: Fourier uniformity of bounded multiplicative functions in short intervals on average

Correction to: Fourier uniformity of bounded multiplicative functions in short intervals on average

Fourier uniformity of bounded multiplicative functions in short intervals on average

Let $\lambda$ denote the Liouville function. We show that as $X \rightarrow \infty$, $$ \int_{X}^{2X} \sup_{\alpha} \left | \sum_{x < n \leq x + H} \lambda(n) e(-\alpha n) \right | dx = o ( X H) $$ for all $H \geq X^{\theta}$ with $\theta > 0$ fixed but arbitrarily small. Previously, this was only known for $\theta > 5/8$. For smaller values of $\theta$ 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 $\lambda(n) \Lambda(n + h) \Lambda(n + 2h)$ over the ranges $h < X^{\theta}$ and $n < X$, and where $\Lambda$ is the von Mangoldt function.

Correlations and distribution of essentially bounded functions in short intervals

We say that an arithmetic function f is essentially bounded if $$f(n)\ll _{\varepsilon } n^{\varepsilon }$$ for all $$\varepsilon >0$$ . A particular case of essentially bounded function is the convolution product of the constantly 1 function and an essentially bounded g such that $$g(q)=0$$ for $$q>Q$$ . In this case, we say that $$f=g*{\mathbf{1}}$$ is a sieve function of range Q. Here we survey some of our recent results concerning the theory of the correlations and the distribution of essentially bounded functions in short intervals, with a special attention to the case of the sieve functions. Further, we take this opportunity to refine some proofs and generalize some of the achievements by giving their weighted version for pairs of arithmetic functions.

Arithmetic functions in short intervals and the symmetric group

We consider the variance of sums of arithmetic functions over random short intervals in the function field setting. Based on the analogy between factorizations of random elements of $\mathbb{F}_q[T]$ into primes and the factorizations of random permutations into cycles, we give a simple but general formula for these variances in the large $q$ limit for arithmetic functions that depend only upon factorization structure. From this we derive new estimates, quickly recover some that are already known, and make new conjectures in the setting of the integers. In particular we make the combinatorial observation that any function of this sort can be decomposed into a sum of functions $u$ and $v$, depending on the size of the short interval, with $u$ making a negligible contribution to the variance, and $v$ asymptotically contributing diagonal terms only. This variance evaluation is closely related to the appearance of random matrix statistics in the zeros of families of L-functions and sheds light on the arithmetic meaning of this phenomenon.

Higher Moments of Arithmetic Functions in Short Intervals: A Geometric Perspective

Abstract We study the geometry associated to the distribution of certain arithmetic functions, including the von Mangoldt function and the Möbius function, in short intervals of polynomials over a finite field $\mathbb{F}_{q}$. Using the Grothendieck–Lefschetz trace formula, we reinterpret each moment of these distributions as a point-counting problem on a highly singular complete intersection variety. We compute part of the ℓ-adic cohomology of these varieties, corresponding to an asymptotic bound on each moment for fixed degree n in the limit as $q \to \infty $. The results of this paper can be viewed as a geometric explanation for asymptotic results that can be proved using analytic number theory over function fields.

Short interval results for certain prime-independent multiplicative functions

Using bounds for certain sums of fractional parts over r-full numbers coming 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 in short intervals.

On the variance of sums of divisor functions in short intervals

Given a positive integer n the k-fold divisor function dk(n) equals the number of ordered k-tuples of positive integers whose product equals n. In this article we study the variance of sums of d(k) ...

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 M\obius function we show that there are cancellations in the sum of $\mu(n)$ in almost all intervals of the form $[x, x + \psi(x)]$ with $\psi(x) \rightarrow \infty$ 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 $x^{\epsilon}$-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 $\lambda(n)\lambda(n+1)$, with $\lambda(n)$ Liouville's function, is non-trivially bounded in absolute value by $1 - \delta$ for some $\delta > 0$. 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 M\obius 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.

THE LOGARITHMICALLY AVERAGED CHOWLA AND ELLIOTT CONJECTURES FOR TWO-POINT CORRELATIONS

Let $\unicode[STIX]{x1D706}$ denote the Liouville function. The Chowla conjecture, in the two-point correlation case, asserts that $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D706}(a_{1}n+b_{1})\unicode[STIX]{x1D706}(a_{2}n+b_{2})=o(x)\end{eqnarray}$$ as $x\rightarrow \infty$, for any fixed natural numbers $a_{1},a_{2}$ and nonnegative integer $b_{1},b_{2}$ with $a_{1}b_{2}-a_{2}b_{1}\neq 0$. In this paper we establish the logarithmically averaged version $$\begin{eqnarray}\mathop{\sum }_{x/\unicode[STIX]{x1D714}(x)&lt;n\leqslant x}\frac{\unicode[STIX]{x1D706}(a_{1}n+b_{1})\unicode[STIX]{x1D706}(a_{2}n+b_{2})}{n}=o(\log \unicode[STIX]{x1D714}(x))\end{eqnarray}$$ of the Chowla conjecture as $x\rightarrow \infty$, where $1\leqslant \unicode[STIX]{x1D714}(x)\leqslant x$ is an arbitrary function of $x$ that goes to infinity as $x\rightarrow \infty$, thus breaking the ‘parity barrier’ for this problem. Our main tools are the multiplicativity of the Liouville function at small primes, a recent result of Matomäki, Radziwiłł, and the author on the averages of modulated multiplicative functions in short intervals, concentration of measure inequalities, the Hardy–Littlewood circle method combined with a restriction theorem for the primes, and a novel ‘entropy decrement argument’. Most of these ingredients are also available (in principle, at least) for the higher order correlations, with the main missing ingredient being the need to control short sums of multiplicative functions modulated by local nilsequences. Our arguments also extend to more general bounded multiplicative functions than the Liouville function $\unicode[STIX]{x1D706}$, leading to a logarithmically averaged version of the Elliott conjecture in the two-point case. In a subsequent paper we will use this version of the Elliott conjecture to affirmatively settle the Erdős discrepancy problem.

SIGN PATTERNS OF THE LIOUVILLE AND MÖBIUS FUNCTIONS

Let ${\it\lambda}$ and ${\it\mu}$ denote the Liouville and Möbius functions, respectively. Hildebrand showed that all eight possible sign patterns for $({\it\lambda}(n),{\it\lambda}(n+1),{\it\lambda}(n+2))$ 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 $({\it\mu}(n),{\it\mu}(n+1))$. A new feature in the latter argument is the need to demonstrate that a certain random graph is almost surely connected.