$\rho$ is a Non-Trivial Zero of the Riemann Zeta function if and only if
$$\displaystyle\int_1^{+\infty} \lfloor x\rfloor x^{-2-\rho} dx =\int_1^{+\infty} \lfloor x\rfloor \{ x \}x^{-2-\rho} dx $$
where $\lfloor x \rfloor$ is the floor function and $\{ x\} = x-\lfloor x \rfloor$.
Note that the left member is just $\frac{\zeta(\rho +1)}{\rho+1}$.
The above equation let us explore the Riemann Zeta function in the plane $\Re(s)>1$ for finding the zeros in the critical strip.
If in some manner we can show that $\rho$ is a zero iff $2\Re(\rho)-1+\rho$ is a zero using the above equation, the Riemann Hypothesis will be true.
My ask however is only to prove the above equation.