On the correlation of the truncated Liouville function

Daboussi, H.; Sárközy, Andràs

doi:10.4064/aa108-1-6

Cited by 4 publications

(12 citation statements)

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“…for ε ∈ (0, 1) and x ≥ x 0 (ε). This result may be compared with that of Daboussi and Sárkőzy [3] and Mangerel [21], which states that if we define λ <y (n) as the completely multiplicative function taking the value −1 at the primes p < y and +1 at the primes p ≥ y (so that λ <y (p) has the opposite sign as λ >y (p)), then 1 x n≤x λ <x ε (n)λ <x ε (n + 1) = o ε→0 (1); (1.6) moreover, they proved this in a quantitative form. The proof of (1.6) is based on sieve theory and is very different from the proof of (1.5).…”

Section: Applications Of the Main Theoremmentioning

confidence: 85%

On Binary Correlations of Multiplicative Functions

Teräväinen¹

2018

Forum of Mathematics, Sigma

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We study logarithmically averaged binary correlations of bounded multiplicative functions g1 and g2. A breakthrough on these correlations was made by Tao, who showed that the correlation average is negligibly small whenever g1 or g2 does not pretend to be any twisted Dirichlet character, in the sense of the pretentious distance for multiplicative functions. We consider a wider class of real-valued multiplicative functions gj, namely those that are uniformly distributed in arithmetic progressions to fixed moduli. Under this assumption, we obtain a discorrelation estimate, showing that the correlation of g1 and g2 is asymptotic to the product of their mean values. We derive several applications, first showing that the numbers of large prime factors of n and n + 1 are independent of each other with respect to logarithmic density. Secondly, we prove a logarithmic version of the conjecture of Erdős and Pomerance on two consecutive smooth numbers. Thirdly, we show that if Q is cube-free and belongs to the Burgess regime Q ≤ x 4−ε , the logarithmic average around x of the real character χ (mod Q) over the values of a reducible quadratic polynomial is small.

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Section: Applications Of the Main Theoremmentioning

confidence: 85%

On Binary Correlations of Multiplicative Functions

Teräväinen¹

2018

Forum of Mathematics, Sigma

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“…We are, unfortunately, unable to establish. 2 This nevertheless, the techniques used to prove Corollary 1•3 above are sufficient to establish the following weak versions of the binary Chowla conjecture 3 . THEOREM 1•6.…”

Section: Conjecture 1•5 (Chowlamentioning

confidence: 98%

“…As a particular case, they show that n≤x µ y (n)µ y (n + 1) ≪ x 1 (log y) 9 + e − β Their result is not explicitly stated for shifts a other than 1, though this restriction appears to be merely technical. We emphasize, though, that our probabilistic view of the problem motivates the methods that we employ in this paper, and these are substantially different from those used in [3]. Among other things, in [3] Brun's sieve is used, while in the present paper we appeal to the Rosser-Iwaniec sieve instead (see Lemma 2.4 below).…”

Section: Introductionmentioning

confidence: 99%

“…We emphasize, though, that our probabilistic view of the problem motivates the methods that we employ in this paper, and these are substantially different from those used in [3]. Among other things, in [3] Brun's sieve is used, while in the present paper we appeal to the Rosser-Iwaniec sieve instead (see Lemma 2.4 below). This difference accounts for the appearance of the additional factor of log β in the second term in (4) above.…”

Section: Introductionmentioning

confidence: 99%

“…We mention the two following notable examples. Tao [21] proved a logarithmically averaged version of (2), that is (3) n≤x µ(n)µ(n + a) n = o(log x).…”

Section: Introductionmentioning

confidence: 99%

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On the bivariate Erdős–Kac theorem and correlations of the Möbius function

Mangerel¹

2019

Math. Proc. Camb. Phil. Soc.

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Let 2 ≤ y ≤ x such that β := log x log y → ∞. Let ω y (n) denote the number of distinct prime factors p of n such that p ≤ y, and let µ y (n) := µ 2 (n)(−1) ωy(n) , where µ is the Möbius function. We prove that if β is not too large (in terms of x) then for each fixed a ∈ N,This can be seen as a partial result towards the binary Chowla conjecture. Our main input is a quantitative bivariate analogue of the Erdős-Kac theorem regarding the distribution of the pairs (ω(n), ω(n + a)), where n and n + a both belong to any subset of the positive integers with suitable sieving properties; moreover, we show that the set of squarefree integers is an example of such a set. We end with a further application of this probabilistic result related to a problem of Erdős and Mirsky on the number of integers n ≤ x such that τ (n) = τ (n + 1).1 8 .

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