Discrete measures and the Riemann hypothesis (Q1345390)

The author's main theorem may be restated as follows. Let \({1\over 2}\leq \gamma\leq 1\); the Riemann zeta-function has no zeros in the half-plane \(\text{Re } s>\gamma\) if and only if \(\sum_{n=1}^ \infty \varphi (n) f({n\over x})= m_ 0 (f,x)+ O(x^{\gamma+ \varepsilon})\), \(\varepsilon>0\), for any smooth function \(f: \mathbb{R}_ +\to \mathbb{R}\) with compact support; here \(\varphi\) denotes Euler's function, \(\mathbb{R}_ +\) is the set of positive real numbers, and \(m_ 0 (f,x):= 6\pi^{-2} x^ 2 \int_ 0^ \infty uf(u) du\). The proof makes use of the standard properties of the Mellin transform and the Riemann zeta-function.