Some Simplest Integral Equalities Equivalent to the Riemann Hypothesis

It is shown that the following integral equalities are equivalent to the Riemann hypothesis for any real a > 0 and any real 0 < ∈ < 1, ∈ ≠ 1:

$$\begin{array}{c}\underset{-\infty }{\overset{\infty }{\int }}\frac{\mathrm{ln}(\zeta (\frac{1}{2}+it))}{a+it}dt=-2\pi{\hspace{0.17em}ln}\frac{a+\frac{1}{2}}{a},\\ \underset{-\infty }{\overset{\infty }{\int }}\frac{\mathrm{ln}(\zeta (\frac{1}{2}+it))}{{(a+it)}^{\in }}dt=-\frac{2\pi}{1-\in }\text{\hspace{0.17em}}({(a+\frac{1}{2})}^{1-\in }-{a}^{1-\in }).\end{array}$$