# A sum involving fractional part function

I was exploring some sum when I came across this sum which I have no idea the value, here is the sum

Let $N$ be an integer with the prime decomposition $N = p_1^{k_1} p_2^{k_2} ... p_m^{k_m}$.

Let $a$ be another integer such that $0 < a < N$. Consider the sum

\begin{align} \sum_{1 \leq i \leq m} \left \{ \frac{a}{p_i} \right \} - \sum_{1 \leq i < j \leq m} \left \{ \frac{a}{p_i p_j} \right \} + ... + (-1)^m \left \{ \frac{a}{p_1 p_2 ... p_m} \right \} \end{align} where $\{ x \} = x - \lfloor x \rfloor$ is the factional part function.

I did some numerical calculations and found that the sum is generally small. For some values of $a$, the sum may get large, but its absolute value seems to be bounded by $m - 1$, where $m$ is the number of distinct prime factors of $N$ as above.

Any idea how I can evaluate the sum?

• Two short comments: What about the $k_i$? They don't appear in your sums at all... Also, you might want to multiply everything by $N$, this might give nice results (e.g. $p \cdot \left\{ \frac{a}{p}\right\} = a \mod{p}$). – Dirk Liebhold Jul 5 '17 at 13:38
• See question 88777. The sum can get nearly as big as 2^{m-1}, and does so for some integers N. Gerhard "Is Interested In Partial Totients" Paseman, 2017.07.05. – Gerhard Paseman Jul 5 '17 at 16:12