Robin showed that if $a\in(1/2, 1]$ is the supremum of the real parts of the zeros of the Riemann zeta function $\zeta(s)$, then $f(x)=\Omega_{\pm} (x^{-b})$, where $b$ is some number on $(a-1/2, 1/2],$ $$f(x)=\log \Big(e^{\gamma}\log \theta(x)\prod_{p\leq x} (1-p^{-1})\Big),$$ $\theta(x)=\sum_{p\leq x} \log p$, the Chebyshev sum over the primes $p\leq x$ and $\gamma=0.577\cdots$ the Euler constant.

**But is it true that if $\zeta(s)\neq 0$ for $\Re(s)\in(1/2 , 1]$, then $f(x)\neq \Omega_{\pm} (x^{-c})$ for any $c\in (0, 1/2]$?**

**In other words, is it true that $a\in(1/2, 1]$ is the supremum of the real parts of the zeros of the Riemann zeta function if and only if $f(x)=\Omega_{\pm} (x^{-b})$, where $b$ is some number on $(a-1/2, 1/2]$ ?**