Let $C_{c}^{r}(\mathbb{R}^{\ast})$ denote the set of all functions $f \colon \mathbb{R} \setminus \{0\} \to \mathbb{C}$ that are $r$ times differentiable and have compact support. For any $y \in \mathbb{R} \setminus\{0\}$ and any $f \in C_{c}^{2}(\mathbb{R}^{\ast})$, let $m_{y}(f) := \sum_{n \in \mathbb{N}} y\phi(n) f(y^{\frac{1}{2}}n)$ where $\phi$ is the Euler totient function; in addition, let $m_{0}(y) := \frac{6}{\pi^{2}}\int_{0}^{\infty} u f(u)\, du$.

In the paper Discrete measures and the Riemann hypothesis (Kodai Math. J. 17 (1994), no. 3, 596–608.), Prof. Alberto Verjovsky proved the folllowing:

The Riemann Hypothesis is true if and only if $m_{y}(f) = m_{0}(f) + o(y^{\frac{3}{4}-\epsilon})$ as $y \to 0$ for every $f \in C_{c}^{2}(\mathbb{R}^{\ast})$ and every $\epsilon > 0$.

Let $C_{c}^{r}(\mathbb{R}^{\ast})$ denote the set of all functions $f \colon (0,\infty) \to \mathbb{C}$ that are $r$ times differentiable and have compact support. For any $y \in (0,\infty)$ and any $f \in C_{c}^{2}(\mathbb{R}^{\ast})$, let $m_{y}(f) := \sum_{n \in \mathbb{N}} y\phi(n) f(y^{\frac{1}{2}}n)$ where $\phi$ is the Euler totient function; in addition, let $m_{0}(y) := \int_{0}^{\infty} \left(\frac{6}{\pi^{2}}\right)u f(u)\, du$.

In the paper Discrete measures and the Riemann hypothesis (Kodai Math. J. 17 (1994), no. 3, 596–608.), Prof. Alberto Verjovsky proved the folllowing:

The Riemann Hypothesis is true if and only if $m_{y}(f) = m_{0}(f) + o(y^{\frac{3}{4}-\epsilon})$ as $y \to 0$ for every $f \in C_{c}^{r}(\mathbb{R}^{\ast})$ with $r \in [2,\infty]$ and every $\epsilon > 0$.

You can find the aforementioned paper here: http://projecteuclid.org/euclid.kmj/1138040054