Question 175
(1) Given coprime positive integers $a, b$. Prove: There exist real numbers $\lambda, \beta$ such that for any positive integer $m$, it holds that
$$
\lambda m - \beta \le \sum_{k=1}^{m-1} \left\{ \frac{ak}{m} \right\} \cdot \left\{ \frac{bk}{m} \right\} \le \lambda m + \beta.
$$
(2) Prove: there exists a positive integer $N$, such that for any prime $p \gt N$, the following proposition holds:
If the positive integers $a, b, c$ satisfy that $(a+b)(a+c)(b+c)$ is not divisible by $p$, then there exist at least $\lfloor \frac{p}{12} \rfloor$ elements $k$ in the set $\{1, 2, \dots, p-1\}$ such that
$$
\left\{ \frac{ak}{p} \right\} + \left\{ \frac{bk}{p} \right\} + \left\{ \frac{ck}{p} \right\} \le 1.
$$
Here, $\lfloor x \rfloor$ represents the greatest integer not exceeding $x$, and $\{x\} = x - \lfloor x \rfloor$ represents the fractional part of $x$.
$$
\lambda m - \beta \le \sum_{k=1}^{m-1} \left\{ \frac{ak}{m} \right\} \cdot \left\{ \frac{bk}{m} \right\} \le \lambda m + \beta.
$$
(2) Prove: there exists a positive integer $N$, such that for any prime $p \gt N$, the following proposition holds:
If the positive integers $a, b, c$ satisfy that $(a+b)(a+c)(b+c)$ is not divisible by $p$, then there exist at least $\lfloor \frac{p}{12} \rfloor$ elements $k$ in the set $\{1, 2, \dots, p-1\}$ such that
$$
\left\{ \frac{ak}{p} \right\} + \left\{ \frac{bk}{p} \right\} + \left\{ \frac{ck}{p} \right\} \le 1.
$$
Here, $\lfloor x \rfloor$ represents the greatest integer not exceeding $x$, and $\{x\} = x - \lfloor x \rfloor$ represents the fractional part of $x$.