A question concerning multiplicative function

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Mehrab4226
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A question concerning multiplicative function

Unread post by Mehrab4226 » Tue May 18, 2021 8:40 pm

The is a Mobius function in number theory which is stated as,

\[ \mu (n)= \begin{cases} 1 & \text{if n=1} \\ 0 & \text{if } p^2 | n \text{ for some prime } p>1 \\ (-1)^k & \text{if } n=p_1 \cdots p_k \text{ where } p_1,\cdots p_k \text{ are distinct primes} \\ \end{cases} \]
And it is also said that the mobius function is multiplicative meaning $\mu (mn)=\mu (m) \mu (n)$. But if we put $m=3$ and $n=2\times 3$ we get,
$\mu (m)=-1$
$\mu (n)=1$
$\mu (m) \mu (n) =-1$
But,
$\mu (mn)= \mu(18) = 0 \neq \mu (m) \mu (n)$.
I am not understanding why???
Proof that the Mobius function is multiplicative,
Screenshot 2021-05-18 20.36.54.png
104 Number Theory problems by Titu Andreescu
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The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

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Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm
Location:Dhaka, Bangladesh

Re: A question concerning multiplicative function

Unread post by Mehrab4226 » Tue May 18, 2021 9:21 pm

Mehrab4226 wrote:
Tue May 18, 2021 8:40 pm
The is a Mobius function in number theory which is stated as,

\[ \mu (n)= \begin{cases} 1 & \text{if n=1} \\ 0 & \text{if } p^2 | n \text{ for some prime } p>1 \\ (-1)^k & \text{if } n=p_1 \cdots p_k \text{ where } p_1,\cdots p_k \text{ are distinct primes} \\ \end{cases} \]
And it is also said that the mobius function is multiplicative meaning $\mu (mn)=\mu (m) \mu (n)$. But if we put $m=3$ and $n=2\times 3$ we get,
$\mu (m)=-1$
$\mu (n)=1$
$\mu (m) \mu (n) =-1$
But,
$\mu (mn)= \mu(18) = 0 \neq \mu (m) \mu (n)$.
I am not understanding why???
Proof that the Mobius function is multiplicative,
Screenshot 2021-05-18 20.36.54.png
Ok got it. Multiplicative has meaning when m and n are coprime. But in my example, they are not. Meaning $\mu (mn)= \mu (m) \mu (n)$ where $G.C.D(m,n)=1$. Ok thanks anyway.
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

otis
Posts:17
Joined:Wed Aug 24, 2022 10:03 am

Re: A question concerning multiplicative function

Unread post by otis » Wed Jul 26, 2023 7:39 am

This problem is really difficult for me and maybe I need your help to solve this problem. The Password Game

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