I learned that dependent on the Riemann Hypothesis $$d(x)=-\frac{1}{\pi}\sum_{p^n}\frac{\ln(p)}{p^{\frac{n}{2}}}\cos(x\ln(p^n))$$ has peaks converging at the real points $t$ where $\zeta(\frac{1}{2} + it)=0$.
${p}$ are the roots of the prime powers,$\{2,3,2,5,7,2,3,11,\dots\}$ and $p^n$ are the prime powers, $\{2,3,4,5,7,8,9,11,\dots\}.$
(I've seen it sometimes where the $\frac{1}{\pi}$ is left out).
I was just wondering where the proof of this is. I could take a guess that the connection with the Riemann Hypothesis is in the square root in the denominator of the summation of $d(x)$. If so could someone elaborate how we got there? On the converse, if we prove that the root in the denominator $p^\frac{n}{2}$ must be a square root, does that mean RH is true?