A Riemann Function Sum

Problem

The Riemann function is defined on $[0, 1]$ by

R(x) = \begin{cases} \dfrac{1}{p}, & x = \dfrac{q}{p},\ \gcd(p, q) = 1,\ p > q,\\[4pt] 0, & x \text{ irrational in } [0,1] \text{ or } x \in \{0, 1\}. \end{cases}

A function $f(x)$ on $\mathbb{R}$ satisfies $f(3x) + f(2 - 3x) = 0$ , the function $f(x + 2)$ is even, and $f(x) = R(x)$ for $x \in [0, 1)$ . Find

\sum_{k=1}^{2025} f\left(\frac{k}{4}\right).

Answer

Solution

Difficulty	8/10
Topics	functions, Symmetry

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