Prove that \(\displaystyle{\sum_{{{n}={1}}}^{{\infty}}}{\frac{{\mu{\left({n}\right)}}}{{{10}^{{n}}}}}\) is irrational

Averie Ferguson

Averie Ferguson

Answered question

2022-03-31

Prove that n=1μ(n)10n is irrational

Answer & Explanation

Avery Maxwell

Avery Maxwell

Beginner2022-04-01Added 13 answers

Step 1
The central concern about this problem is to determine whether 1+μ(n) is periodic. If it is periodic, we can soon conclude that this number is rational. We will prove the irrationality by contradicting this statement.
If 1+μ(n) is periodic, then within each period T, there should be only a fixed number of occurences of 1+μ(n)=1. This indicates that there should be a fixed number of non square-free integers within the period. However, if we were to let pk be k'th ', we can show that for any M>o the following system of congruences
{n0 (mod pk2)n+10 (mod pk+12)n+20 (mod pk+22)n+M0 (mod pk+M2) 
always has solution by the Chinese remainder theorem, which means that it is possible to have arbitrarily large consecutive non square-free sequence of integers, contradicting our assumption. Therefore, the series
n=1μ(n)10n
converges to an irrational number.

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