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      A definite recursive relation and some statistical properties for M\"obius function

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          Abstract

          An elementary recursive relation for M\(\ddot{\mathrm{o}}\)bius function \(\mu (n)\) is introduced by two simple ways. With this recursive relation, \(\mu (n)\) can be calculated without directly knowing the factorization of the \(n\). \(\mu (1) \sim \mu (2 \times 10^7) \) are calculated recursively one by one. Based on these \(2\times 10^7\) samples, the empirical probabilities of \(\mu (n)\) of taking \(-1\), 0, and 1 in classic statistics are calculated and compared with the theoretical probabilities in number theory. The numerical consistency between these two kinds of probability show that \(\mu (n)\) could be seen as an independent random sequence when \(n\) is large. The expectation and variance of the \(\mu (n)\) are \(0\) and \(6 n/ \pi^2\), respectively. Furthermore, we show that any conjecture of the Mertens type is false in probability sense, and present an upper bound for cumulative sums of \(\mu (n)\) with a certain probability.

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          Author and article information

          Journal
          2016-08-15
          2016-08-21
          Article
          1608.04606
          bb348ab8-c94a-4d0e-b99a-dd604cb00bad

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          28 pages, 6 figues, 4 tables
          math.NT math.CO

          Combinatorics,Number theory
          Combinatorics, Number theory

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