On a new approach to the Riemann hypothesis

Suppose that the Riemann hypothesis is false and $\rho_{*} = 1/2 + \eta_{*} + i \gamma_{*}$, $\eta_{*} > 0$, is a nontrivial zero of the Riemann $\zeta$-function off the critical line. Under the negation of the Riemann hypothesis for the Riemann $\zeta$-function, we establish an asymptotic relation (as $\gamma_{*} \to \infty$) which relates the residues of the series $\sum_{n \geq 1} \Lambda(n) e^{- 2\pi i p n } n^{-s}$ at $s =$ corresponding nontrivial zeros of some Dirichlet $L$-functions to some function, valid for any rational number $p = a/b < 1$ with $b \ll \log \gamma_{*}$. This related function is continuous in $p$ and we mention its implication to the Riemann hypothesis.