Question

I noticed that for primes $p \le 109$, the following seems to be true:

$\sum_{i | p\#}^{p\#} \lfloor{\frac{p}{i}\mu(i)}\rfloor = 1$

where $\mu(i)$ is the Mobius function.

For example:

$\frac{2}{1} + \frac{2}{2}(-1) = 1$

$\frac{3}{1} + \lfloor\frac{3}{2}(-1)\rfloor + \frac{3}{3}(-1) + \lfloor\frac{3}{6}\rfloor = 1$

and so on...

I verified this up to $p=109$ using a simple java application.

I might be making a mistake in my code or my thinking. This seems like a very surprising result to me.

Is it correct? If it is, does it stop being true for some prime? Could anyone help me to understand this result.

Thanks very much,

-Larry

Answer 1

Assuming that $p\text{#}$ means the product over primes $\prod_{q\leq p}q$, then it is clear that $$\sum_{i\leq p} \mu(i)[\frac{p}{i}] = \sum_{i | p\text{#}} \mu(i)[\frac{p}{i}].$$

But this formula is very well known: $$\sum_{d\leq n} \mu(d)[\frac{n}{d}]=\sum_{k\leq n}\sum_{d | k} \mu(d)=1+0+0+\ldots$$