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

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

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

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    $\begingroup$ In your example, you seem to be summing mu(i)*floor(p/i) rather than floor(mu*p/i). $\endgroup$ Jun 20, 2012 at 22:59

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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: $$\sum_{d\leq n} \mu(d)[\frac{n}{d}]=\sum_{k\leq n}\sum_{d | k} \mu(d)=1+0+0+\ldots$$

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  • $\begingroup$ But Larry is using the floor function. Are you? Gerhard "Ask Me About System Design" Paseman, 2012.06.20 $\endgroup$ Jun 20, 2012 at 23:13
  • $\begingroup$ Thanks very much! I'll check out the proof behind the very well known equation! Cheers, -Larry $\endgroup$ Jun 20, 2012 at 23:15
  • $\begingroup$ Hi Gerhard, Yes, I am using the floor function. Isn't the equation cited by David also using a floor function? -Larry $\endgroup$ Jun 20, 2012 at 23:16
  • $\begingroup$ I am using [] to denote floor. The first equality in the second equation comes from reversing the order of summation on the right hand side, since d divides exactly [n/d] elements of {1,...,n}. $\endgroup$ Jun 21, 2012 at 2:49

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