Let us define, $$r(b)=\sum_{k=1}^{\lfloor \frac{b-1}{2} \rfloor} (b \bmod{k})$$ After playing around with the $r(b)$ function for sometime I noticed that $r(b)$ appreared to be more even than odd. So to see the difference between the number of even and odd terms of $r(b)$, I defined a function, $$z(x)=\sum_{n=1}^x(-1)^{r(n)}$$ When user Peter ran a program for computing values of $z(x)$ in PARI, I observed that for $x\le 10^{10}$, $z(x)\gt 0$. This suggests that there are always more even terms of $r(n)$ than odd terms for any $x$.
This leads to my two questions:
- Is $z(x)$ always positive? If so, then how do we prove this?
- Is $|z(x)|$ bounded by some maximum value? If so, then what is this maximum value? Till now the maximum value of $|z(x)|$ found was $49$ for $x = 5424027859$. I find it odd that $|z(x)|$ goes to these large values and then returns back to small values as small as $1$.
from Hot Weekly Questions - Mathematics Stack Exchange
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