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Problem 850

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Fractions of Powers

Any positive real number $x$ can be decomposed into integer and fractional parts $\lfloor x \rfloor + \{x\}$, where $\lfloor x \rfloor$ (the floor function) is an integer, and $0\le \{x\} < 1$.

For positive integers $k$ and $n$, define the function
$$f_k(n) = \sum_{i=1}^{n}\left\{ \frac{i^k}{n} \right\}$$
For example, $f_5(10)=4.5$ and $f_7(1234)=616.5$.

Let
$$S(N) = \sum_{\substack{k=1 \\ k\text{ odd}}}^{N} \sum_{n=1}^{N} f_k(n)$$
You are given that $S(10)=100.5$ and $S(10^3)=123687804$.

Find $\lfloor S(33557799775533) \rfloor$. Give your answer modulo $977676779$.

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幂的小数部分

任意正实数$x$都可以分解成整数部分和小数部分$\lfloor x \rfloor + \{x\}$，其中$\lfloor x \rfloor$（下取整函数）是一个整数，而$0\le \{x\} < 1$。

对于正整数$k$和$n$，定义函数
$$f_k(n) = \sum_{i=1}^{n}\left\{ \frac{i^k}{n} \right\}$$
例如，$f_5(10)=4.5$，$f_7(1234)=616.5$。

记
$$S(N) = \sum_{\substack{k=1 \\ k\text{为奇数}}}^{N} \sum_{n=1}^{N} f_k(n)$$
已知$S(10)=100.5$，$S(10^3)=123687804$。

求$\lfloor S(33557799775533) \rfloor$，并将你的答案对$977676779$取余。

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