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

2016 ◽  
Vol 4 ◽  
Author(s):  
TERENCE TAO
Keyword(s):  
Arbitrary Function ◽  
Circle Method ◽  
Concentration Of Measure ◽  
Subsequent Paper ◽  
Multiplicative Functions ◽  
Restriction Theorem ◽  
Liouville Function ◽  
Short Intervals ◽  
Point Correlation ◽  
Image Position

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 $$\begin{eqnarray}\mathop{\sum }_{x/\unicode[STIX]{x1D714}(x)<n\leqslant x}\frac{\unicode[STIX]{x1D706}(a_{1}n+b_{1})\unicode[STIX]{x1D706}(a_{2}n+b_{2})}{n}=o(\log \unicode[STIX]{x1D714}(x))\end{eqnarray}$$ 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.

scholarly journals Correlations of multiplicative functions and applications

2017 ◽  
Vol 153 (8) ◽  
pp. 1622-1657 ◽  
Author(s):  
Oleksiy Klurman
Keyword(s):  
Asymptotic Formula ◽  
Multiplicative Function ◽  
Divided Difference ◽  
Circle Method ◽  
Partial Sums ◽  
Multiplicative Functions ◽  
Image Position ◽  
Bounded Partial Sums

We give an asymptotic formula for correlations $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}f_{1}(P_{1}(n))f_{2}(P_{2}(n))\cdots f_{m}(P_{m}(n)),\end{eqnarray}$$ where $f,\ldots ,f_{m}$ are bounded ‘pretentious’ multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences. First, we characterize all multiplicative functions $f:\mathbb{N}\rightarrow \{-1,+1\}$ with bounded partial sums. This answers a question of Erdős from $1957$ in the form conjectured by Tao. Second, we show that if the average of the first divided difference of the multiplicative function is zero, then either $f(n)=n^{s}$ for $\operatorname{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an old conjecture of Kátai. Third, we apply our theorem to count the number of representations of $n=a+b$, where $a,b$ belong to some multiplicative subsets of $\mathbb{N}$. This gives a new ‘circle method-free’ proof of a result of Brüdern.

Multiplicative functions at consecutive integers

1986 ◽  
Vol 100 (2) ◽  
pp. 229-236 ◽  
Author(s):  
Adolf Hildebrand
Keyword(s):  
Random Sequence ◽  
Multiplicative Functions ◽  
Liouville Function ◽  
Prime Factors ◽  
Consecutive Integers ◽  
Image Position

Let λ(n) denote the Liouville function, i.e. λ(n) = 1 if n has an even number of prime factors, and λ(n) = − 1 otherwise. It is natural to expect that the sequence λ(n) (n ≥ 1) behaves like a random sequence of ± signs. In particular, it seems highly plausible that for any choice of εi = ± 1 (i = 0,…, k) we have

Multiplicative Functions in Short Intervals

1987 ◽  
Vol 39 (3) ◽  
pp. 646-672 ◽  
Author(s):  
Adolf Hildebrand
Keyword(s):  
Number Theory ◽  
Mean Value ◽  
Considerable Progress ◽  
Central Problem ◽  
Partial Sums ◽  
Multiplicative Functions ◽  
Short Intervals ◽  
Probabilistic Number ◽  
The Mean ◽  
Image Position

A central problem in probabilistic number theory is to evaluate asymptotically the partial sumsof multiplicative functions f and, in particular, to find conditions for the existence of the “mean value”1.1In the last two decades considerable progress has been made on this problem, and the results obtained are very satisfactory.

scholarly journals Sarnak’s conjecture for sequences of almost quadratic word growth

2020 ◽  
pp. 1-56
Author(s):  
REDMOND MCNAMARA
Keyword(s):  
Box Dimension ◽  
Multiplicative Functions ◽  
Logarithmic Density ◽  
Liouville Function ◽  
Sign Patterns

Abstract We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many words which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the ( $\kappa -1$ )-Fourier uniformity conjecture holds then the Liouville function does not correlate with sequences with $O(n^{t-\varepsilon })$ many words of length n where $t = \kappa (\kappa +1)/2$ . We prove a variant of the $1$ -Fourier uniformity conjecture where the frequencies are restricted to any set of box dimension less than $1$ .

scholarly journals Extremal behaviour of ± 1-valued completely multiplicative functions in function fields

2021 ◽  
Vol 56 (1) ◽  
pp. 79-94
Author(s):  
Nikola Lelas ◽  
Keyword(s):  
Rational Function ◽  
Finite Fields ◽  
Function Fields ◽  
Mean Value ◽  
Multiplicative Functions ◽  
Irreducible Polynomials ◽  
Liouville Function ◽  
The Mean

We investigate the classical Pólya and Turán conjectures in the context of rational function fields over finite fields 𝔽q. Related to these two conjectures we investigate the sign of truncations of Dirichlet L-functions at point s=1 corresponding to quadratic characters over 𝔽q[t], prove a variant of a theorem of Landau for arbitrary sets of monic, irreducible polynomials over 𝔽q[t] and calculate the mean value of certain variants of the Liouville function over 𝔽q[t].

scholarly journals The Computational Strength of Extensions of Weak König’s Lemma

1998 ◽  
Vol 5 (41) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Natural Number ◽  
Arbitrary Function ◽  
The Other ◽  
Binary Trees ◽  
Subsequent Paper ◽  
Bounded Functions ◽  
The One ◽  
Weak König’S Lemma ◽  
König’S Lemma

The weak König's lemma WKL is of crucial significance in the study of fragments of mathematics which on the one hand are mathematically strong but on the other hand have a low proof-theoretic and computational strength. In addition to the restriction to binary trees (or equivalently bounded trees), WKL<br />is also `weak' in that the tree predicate is quantifier-free. Whereas in general the computational and proof-theoretic strength increases when logically more complex trees are allowed, we show that this is not the case for trees which are<br />given by formulas in a class Phi where we allow an arbitrary function quantifier prefix over bounded functions in front of a Pi^0_1-formula. This results in a schema Phi-WKL.<br />Another way of looking at WKL is via its equivalence to the principle<br /> For all x there exists y<=1 for all z A0(x; y; z) -> there exists f <= lambda x.1 for all x, z A0(x, fx, z);<br />where A0 is a quantifier-free formula (x, y, z are natural number variables). <br /> We generalize this to Phi-formulas as well and allow function quantifiers `there exists g <= s'<br />instead of `there exists y <= 1', where g <= s is defined pointwise. The resulting schema is called Phi-b-AC^0,1.<br />In the absence of functional parameters (so in particular in a second order context), the corresponding versions of Phi-WKL and Phi-b-AC^0,1 turn out to<br />be equivalent to WKL. This changes completely in the presence of functional<br />variables of type 2 where we get proper hierarchies of principles Phi_n-WKL and<br />Phi_n-b-AC^0,1. Variables of type 2 however are necessary for a direct representation<br />of analytical objects and - sometimes - for a faithful representation of<br />such objects at all as we will show in a subsequent paper. By a reduction of<br />Phi-WKL and Phi-b-AC^0,1 to a non-standard axiom F (introduced in a previous paper) and a new elimination result for F relative to various fragment of arithmetic in all finite types, we prove that Phi-WKL and Phi-b-AC^0,1 do<br />neither contribute to the provably recursive functionals of these fragments nor to their proof-theoretic strength. In a subsequent paper we will illustrate the greater mathematical strength of these principles (compared to WKL).

scholarly journals Square-full numbers in short intervals

1991 ◽  
Vol 110 (1) ◽  
pp. 1-3 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Positive Integer ◽  
Riemann Hypothesis ◽  
Prime Factor ◽  
Short Interval ◽  
Short Intervals ◽  
Image Position

A positive integer n is called square-full if p2|n for every prime factor p of n. Let Q(x) denote the number of square-full integers up to x. It was shown by Bateman and Grosswald [1] thatBateman and Grosswald also remarked that any improvement in the exponent would imply a ‘quasi-Riemann Hypothesis’ of the type for . Thus (1) is essentially as sharp as one can hope for at present. From (1) it follows that, for the number of square-full integers in a short interval, we havewhen and y = o (x½). (It seems more suggestive) to write the interval as (x, x + x½y]) than (x, x + y], since only intervals of length x½ or more can be of relevance here.)

On polynomials in primes and J. Bourgain's circle method approach to ergodic theorems

1991 ◽  
Vol 11 (3) ◽  
pp. 485-499 ◽  
Author(s):  
R. Nair
Keyword(s):  
Positive Integer ◽  
Circle Method ◽  
Ergodic Theorems ◽  
Almost Everywhere ◽  
Image Position ◽  
Method Approach

In this paper we prove the following theorem.Theorem 1. For a measure-preserving system (X, β, μ, T) and a positive integer k, if f ∈ L2(X, β, μ), the averages,converge μ almost everywhere. Here p runs over the rational primes and πN denotes their number in [1, N].

scholarly journals POSITIVE PROPORTION OF SHORT INTERVALS CONTAINING A PRESCRIBED NUMBER OF PRIMES

2019 ◽  
Vol 100 (3) ◽  
pp. 378-387
Author(s):  
DANIELE MASTROSTEFANO
Keyword(s):  
Short Intervals ◽  
Positive Proportion ◽  
Implicit Constant ◽  
Image Position ◽  
Careful Investigation

We prove that for every $m\geq 0$ there exists an $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}(m)>0$ such that if $0<\unicode[STIX]{x1D706}<\unicode[STIX]{x1D700}$ and $x$ is sufficiently large in terms of $m$ and $\unicode[STIX]{x1D706}$, then $$\begin{eqnarray}|\{n\leq x:|[n,n+\unicode[STIX]{x1D706}\log n]\cap \mathbb{P}|=m\}|\gg _{m,\unicode[STIX]{x1D706}}x.\end{eqnarray}$$ The value of $\unicode[STIX]{x1D700}(m)$ and the dependence of the implicit constant on $\unicode[STIX]{x1D706}$ and $m$ may be made explicit. This is an improvement of the author’s previous result. Moreover, we will show that a careful investigation of the proof, apart from some slight changes, can lead to analogous estimates when allowing the parameters $m$ and $\unicode[STIX]{x1D706}$ to vary as functions of $x$ or replacing the set $\mathbb{P}$ of all primes by primes belonging to certain specific subsets.

Graphs in the plane invariant under an area preserving linear map and general continuous solutions of certain quadratic functional equations

1985 ◽  
Vol 97 (2) ◽  
pp. 261-278 ◽  
Author(s):  
P. J. McCarthy ◽  
M. Crampin ◽  
W. Stephenson
Keyword(s):  
Functional Equation ◽  
Arbitrary Function ◽  
Functional Equations ◽  
Quadratic Functional ◽  
Continuous Solutions ◽  
Linear Map ◽  
Area Preserving Maps ◽  
Image Position ◽  
Area Preserving

AbstractThe requirement that the graph of a function be invariant under a linear map is equivalent to a functional equation of f. For area preserving maps M(det (M) = 1), the functional equation is equivalent to an (easily solved) linear one, or to a quadratic one of the formfor all Here 2C = Trace (M). It is shown that (Q) admits continuous solutions ⇔ M has real eigenvalues ⇔ (Q) has linear solutions f(x) = λx ⇔ |C| ≥ 1. For |c| = 1 or C < – 1, (Q) only admits a few simple solutions. For C > 1, (Q) admits a rich supply of continuous solutions. These are parametrised by an arbitrary function, and described in the sense that a construction is given for the graphs of the functions which solve (Q).

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