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

zbMATH Open1397.11011arXiv1608.04606MaRDI QIDQ2960928

Publication date: 17 February 2017

Abstract: An elementary recursive relation for M

d d o t m a t h r m o

bius function

m u (n)

is introduced by two simple ways. With this recursive relation,

m u (n)

can be calculated without directly knowing the factorization of the

n

.

m u (1) s i m m u (2 i m e s 1 0^{7})

are calculated recursively one by one. Based on these

2 i m e s 1 0^{7}

samples, the empirical probabilities of

m u (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

m u (n)

could be seen as an independent random sequence when

n

is large. The expectation and variance of the

m u (n)

are

0

and

6 n / p i^{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

m u (n)

with a certain probability.

Full work available at URL: https://arxiv.org/abs/1608.04606

zbMATH Keywords

Möbius function recursive relation Mertens conjecture probability sense

Mathematics Subject Classification ID

Arithmetic functions; related numbers; inversion formulas (11A25) Analytic computations (11Y35) Calculation of integer sequences (11Y55) Values of arithmetic functions; tables (11Y70)