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

Abstract

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 .

On Binary Correlations of Multiplicative Functions

“…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).…”

“…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…”

Result on the Mobius Function over Shifted Primes

“…respectively, where t = 0 is a fixed parameter, and c > 0 is a constant, as x → ∞, confer [22], [29], [30, Chapter 5], et alii. Another attempt to prove the arithmetic average order of (133) was made in [18], and an improved error term O(( √ log log x) −2 ) was just proposed in [9, Corollary 2]. Observe that a stronger error term, approximately O((log x) −2 ), is required to compute the arithmetic average orders (133) directly from (132), see [7, Exercise 2.12] for explanation.…”

Autocorrelation of the Mobius Function

“…respectively, where a = 0 is a fixed parameter, and c > 0 is a constant, as x → ∞, confer [15], [22], [25], [23], [13], et alii. Another attempt to prove the arithmetic average order of (1) was made in [12], and an improved error term O(( √ log log x) −2 ) was just proposed in [8, Corollary 2]. Observe that a stronger error term, approximately O((log x) −2 ), is required to compute the arithmetic average orders directly from (1), and (2), see [7,Exercise 2.12] for explanation.…”