On Binary Correlations of Multiplicative Functions

Teräväinen, Joni

doi:10.1017/fms.2018.10

Cited by 10 publications

(12 citation statements)

References 34 publications

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“…It was conjectured in the correspondence of Erdős and Turán [35] (and repeated by Erdős in [10]) that the set of n with P + (n) < P + (n + 1) has asymptotic density equal to 1/2, as one would naturally expect. In [43], it was shown that the logarithmic density of this set indeed equals 1/2. For orderings of longer strings of consecutive values of P + (n), little is known, but Wang [44] showed that either of P + (n + i) < min j≤J j =i P + (n + j) and P + (n + i) > max j≤J j =i P + (n + j) happens with positive lower density for any J ≥ 3.…”

Section: Comparison Of Largest Prime Factors Of Consecutive Integersmentioning

confidence: 96%

Value Patterns of Multiplicative Functions and Related Sequences

Tao

Teräväinen

2019

Forum of Mathematics, Sigma

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We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set A whose indicator function is "approximately multiplicative" and uniformly distributed on short intervals in a suitable sense, we show that the density of the pattern n + 1 ∈ A, n + 2 ∈ A, n + 3 ∈ A is positive, as long as A has density greater than 1 3 . Using an inverse theorem for sumsets and some tools from ergodic theory, we also provide a theorem that deals with the critical case of A having density exactly 1 3 , below which one would need nontrivial information on the local distribution of A in Bohr sets to proceed. We apply our results firstly to answer in a stronger form a question of Erdős and Pomerance on the relative orderings of the largest prime factors P + (n), P + (n + 1), P + (n + 2) of three consecutive integers. Secondly, we show that the tuple (ω(n + 1), ω(n + 2), ω(n + 3)) (mod 3) takes all the 27 possible patterns in (Z/3Z) 3 with positive lower density, with ω(n) being the number of distinct prime divisors. We also prove a theorem concerning longer patterns n + i ∈ Ai, i = 1, . . . k in approximately multiplicative sets Ai having large enough densities, generalising some results of Hildebrand on his "stable sets conjecture". Lastly, we consider the sign patterns of the Liouville function λ and show that there are at least 24 patterns of length 5 that occur with positive upper density. In all of the proofs we make extensive use of recent ideas concerning correlations of multiplicative functions.1 One could also include the n = 0 case here if one wished, although it will not affect our main results.2 For the precise definitions of the various densities used in this paper, as well as the standard arithmetic functions and asymptotic notation, see Subsection 1.7. 1 arXiv:1904.05096v2 [math.NT] 3 Sep 2019 1 d 2 , one can easily show that the Chowla conjectures for the Liouville and Möbius functions are equivalent if one generalises the distinct linear forms n + h1, .. . , n + h k to non-parallel affine forms a1n + h1, . . . , a k n + h k ; we omit the details. 5 We say that f is completely multiplicative if f (mn) = f (m)f (n) for all m, n ∈ N and f (1) = 1. 6 In [42], the proof was written only for f = λ, but the exact same argument works for any completely multiplicative bounded f that is not weakly pretentious.

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Section: Comparison Of Largest Prime Factors Of Consecutive Integersmentioning

confidence: 96%

Value Patterns of Multiplicative Functions and Related Sequences

Tao

Teräväinen

2019

Forum of Mathematics, Sigma

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“…We introduce the cocycle ρ : (0, +∞) × (0, +∞) → C by defining ρ(x 1 , x 2 ) for x 1 , x 2 > 0 to be the unique complex number of size O(ε) such that α(x 1 x 2 ) = α(x 1 )α(x 2 ) exp(ρ(x 1 , x 2 )); (34) this is well-defined and measurable for ε small enough. Arguing exactly as in the proof of Lemma 2.8, we obtain the cocycle equation…”

Section: Proof Of Main Theoremmentioning

confidence: 99%

The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures

Tao

Teräväinen

2019

Alg. Number Th.

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We study the asymptotic behaviour of higher order correlations E n≤X/d g 1 (n + ah 1 ) · · · g k (n + ah k ) as a function of the parameters a and d, where g 1 , . . . , g k are bounded multiplicative functions, h 1 , . . . , h k are integer shifts, and X is large. Our main structural result asserts, roughly speaking, that such correlations asymptotically vanish for almost all X if g 1 · · · g k does not (weakly) pretend to be a twisted Dirichlet character n → χ(n)n it , and behave asymptotically like a multiple of d −it χ(a) otherwise. This extends our earlier work on the structure of logarithmically averaged correlations, in which the d parameter is averaged out and one can set t = 0. Among other things, the result enables us to establish special cases of the Chowla and Elliott conjectures for (unweighted) averages at almost all scales; for instance, we establish the k-point Chowla conjecture E n≤X λ(n + h 1 ) · · · λ(n + h k ) = o(1) for k odd or equal to 2 for all scales X outside of a set of zero logarithmic density.

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“…See [20], [22,Theorem A.1], [21] for some papers dealing with (1) and [22], [27], [16], [29] for some papers dealing with the latter problem. These results have also led to a number of applications, including a solution by Tao [26] to the famous Erdős discrepancy problem.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Correlations of multiplicative functions in function fields

Klurman,

Mangerel,

Teräväinen

2020

Preprint

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We develop an approach to study character sums, weighted by a multiplicative function f : Fq[t] → S 1 , of the formwhere χ is a Dirichlet character and ξ is a short interval character over Fq [t]. We then deduce versions of the Matomäki-Radziwi l l theorem and Tao's two-point logarithmic Elliott conjecture over function fields Fq[t], where q is fixed. The former of these improves on work of Gorodetsky, and the latter extends the work of Sawin-Shusterman on correlations of the Möbius function for various values of q.Compared with the integer setting, we encounter some different phenomena, specifically a low characteristic issue in the case that q is a power of 2, as well as the need for a wider class of "pretentious" functions called Hayes characters.As an application of our results, we give a short proof of the function field version of a conjecture of Kátai on classifying multiplicative functions with small increments, with the classification obtained and the proof being different from the integer case.In a companion paper, we will use these results to characterize the limiting behavior of partial sums of multiplicative functions in function fields and in particular to solve the "corrected" form of the Erdős discrepancy problem over Fq[t]. 1 A function f (G) is called a factorization function if it only depends on the values of deg(P ) and vP (G), where P runs through the irreducible divisors of G, and vP (G) denotes the largest integer k with P k | G.

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On Binary Correlations of Multiplicative Functions

Cited by 10 publications

References 34 publications

Value Patterns of Multiplicative Functions and Related Sequences

Value Patterns of Multiplicative Functions and Related Sequences

The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures

Correlations of multiplicative functions in function fields

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