On a new approach to the Riemann hypothesis

Abstract

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