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?

  • 1
    $\begingroup$ 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. $p \cdot \left\{ \frac{a}{p}\right\} = a \mod{p}$). $\endgroup$ – Dirk Liebhold Jul 5 '17 at 13:38
  • $\begingroup$ 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. $\endgroup$ – Gerhard Paseman Jul 5 '17 at 16:12

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