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On consecutive values of random completely multiplicative functions. (English) Zbl 07206398
One can begin with author’s abstract:
“In this article, we study the behavior of consecutive values of random completely multiplicative functions \((X_n)_{n\ge 1}\) whose values are i.i.d. at primes. We prove that for \(X_2\) uniform on the unit circle, or uniform on the set of roots of unity of a given order, and for fixed \(k \ge 1\), \(X_{n+1}, \dots , X_{n+k}\) are independent if \(n\) is large enough. Moreover, with the same assumption, we prove the almost sure convergence of the empirical measure \(N^{-1}\sum^{N} _{n=1}{\delta_{(X_{n+1}, \dots , X_{n+k})}}\) when \(N\) goes to infinity, with an estimate of the rate of convergence. At the end of the paper, we also show that for any probability distribution on the unit circle followed by \(X_2\), the empirical measure converges almost surely when \(k = 1\).”
In a survey of this paper, the special attention is given to one discussion showing “that it is often much less difficult to prove accurate results for random multiplicative function than for the arithmetic functions which are usually considered”. Also, it is noted that in some informal sense, the arithmetic difficulties are diluted into the randomization, which is much simpler to deal with.
The motivation of the present investigation, used techniques, and obtained results are explained.
The present research consists of the following sections:
– Independence in the uniform case. Here \((X_p)_{p\in \mathcal P}\) are i.i.d. uniform random variables on the unit circle.
– Independence in the case of roots of unity. In this section, \((X_p)_{p\ge 1}\) are i.i.d., uniform on the set of \(q\)-th roots of unity, \(q \ge 1\) being a fixed integer.
– Convergence of the empirical measure in the uniform case. In this section, \((X_p)_{p\in \mathcal P}\) are uniform on the unit circle, and \(k \ge 1\) is a fixed integer.
– Moments of order different from two.
– Convergence of the empirical measure in the case of roots of unity. Here \((X_p)_{p\in \mathcal P}\) are i.i.d. uniform on the set \(\mathbb U_q\) of \(q\)-th roots of unity, \(q \ge 1\) being fixed.
– More general distributions on the unit circle. In this section, \((X_p)_{p\in \mathcal P}\) are i.i.d., with any distribution on the unit circle.
MSC:
11K65 Arithmetic functions in probabilistic number theory
11N37 Asymptotic results on arithmetic functions
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
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References:
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