I have been playing around with the Möbius Function and primorials and I am finding results that I am not yet able to understand which I suspect are very elementary.


Here's the current result which is I am working through. Any help is greatly appreciated!


Let $p\_k$ be any prime. Let $x$ be any integer.


It seems based on my calculations that $\sum\_{i | p\_{k}\#} \lfloor{\frac{(x \% i) + (p\_k \% i)}{i}}\rfloor\mu(i) \ge -1$ 


where **%** is the remainder so that $5 \% 3 = 2$ and $7 \% 3 = 1$


But if we let $x,y$ be any integer, we can find that there exists $x,y$ such that:


$\sum\_{i | p\_{k}\#} \lfloor{\frac{(x \% i) + (y \% i)}{i}}\rfloor\mu(i) < -1$


For example: 


If $x=13$, $y=23$, $p\_k = 5$ , then the sum is $-2$


Is there a well known explanation for this? Thanks.



A question about the Mobius Function
Note that


$$ \lfloor \frac{a \% p + b \% p}{p} \rfloor = \lfloor \frac{a+b}{p} \rfloor - \lfloor \frac{a}{p} \rfloor - \lfloor \frac{b}{p} \rfloor$$


so your expression can be written as $F\_{p\_k\#}(x+p\_k) - F\_{p\_k\#}(x) - F\_{p\_k\#}(p\_k)$, where


$$ F\_n(x) := \sum\_{i|n} \lfloor \frac{x}{i} \rfloor \mu(i).$$


Now, the expression $F\_n$ can be simplified by elementary number theoretic computations. Indeed, observe that


$$ \lfloor \frac{x}{i} \rfloor = \sum\_{m \leq x: i|m} 1$$


and thus after interchanging summations,


$$ F\_n(x) = \sum\_{m \leq x} \sum\_{i|n, i|m} \mu(i).$$


But by Mobius inversion, $\sum\_{i|n, i|m} \mu(i)$ equals 1 when n,m are coprime and 0 otherwise. Thus $F\_n(x)$ is nothing more than the number of natural numbers $m$ less than $x$ that are coprime to $n$. In particular, this gives


$$ F\_{p\_k\#}(x+p\_k) \geq F\_{p\_k\#}(x)$$


and


$$ F\_{p\_k\#}(p\_k) = 1$$


which explains your numerically observed phenomenon.


The quantities here are somewhat reminiscent of the quantity $\pi(x+y)-\pi(x)-\pi(y)$ that occurs in the second [Hardy-Littlewood conjecture](http://en.wikipedia.org/wiki/Second_Hardy%25E2%2580%2593Littlewood_conjecture) (which, by the way, is widely believed to be false). Indeed, the quantity $F\_{p\_k\#}(x)$ (that is, the number of natural numbers up to x that have no prime factor less than or equal to $p\_k$) is more commonly denoted $\pi(x,p\_k)$ in the analytic number theory literature.


EDIT: I also encountered some related expressions and inequalities in my paper <http://arxiv.org/abs/0908.4323> (and with more advanced versions of these inequalities also in this paper of Granville and Soundararajan: <http://arxiv.org/abs/math/0508361> ).


