Properties of sum appearing in Riemann's explicit formula

Question

Let $R(x)$ be Riemann's function defined as $R(x) = \sum_{k=1}^\infty\frac{\mu(k)\text{li}(x^{1/k})}{k}$ where $\mu$ is the Moebius function and li the logarithmic integral. Let $\pi(x)$ be the prime counting function. Let $\{\rho\}$ be the collection of non-trivial zeros on the critical line of the Riemann zeta function $\zeta$. Then it is known that $\pi(x)\sim R(x)-\sum_{\rho}R(x^\rho)$, or Riemann's explicit formula. It can be shown that the contribution of conjugate pairs of zeta zeros of the form $\rho=\frac{1}{2}\pm i\gamma$ that $R(x^{\rho})+R(x^{\bar{\rho}})$ grows as $\frac{\sqrt{x}}{\log(x)}\frac{2\sin(\gamma\log(x))}{\gamma}$. Thus $\sum_{\rho}R(x^\rho)\sim\frac{\sqrt{x}}{\log(x)}\sum_{\gamma>0}\frac{2\sin(\gamma\log(x))}{\gamma}$. Now let $f(x)=\sum_{\gamma>0}\frac{2\sin(\gamma\log(x))}{\gamma}$ such that $\sum_{\rho}R(x^\rho)\sim\frac{\sqrt{x}}{\log(x)}f(x)$. The term $\sqrt{x}/\log(x)$ is interpreted as the amplitude of the oscillatory sum, whereas $f(x)$ is suggested to be some term with no growth. A related quantity is $V(x)=\frac{R(x)-\pi(x)}{\sqrt{x}/\log(x)}$. My question is: what do we know about $V$ or $f$? I haven't been able to locate any information on these functions. In particular, I am curious if these are bounded, such as by unity, or if not, their asymptotic growth.