On a new approach to the Riemann hypothesis

Abstract

Suppose that the Riemann hypothesis is false and $\ddot{I}{}_{\ast }=1/2+\hat{I}{\cdot }_{\ast }+i{\hat{I}}_{\ast }^3$, $\hat{I}{\cdot }_{\ast }>0$, is a nontrivial zero of the Riemann $\hat{I}\P$-function off the critical line. Under the negation of the Riemann hypothesis for the Riemann $\hat{I}\P$-function, we establish an asymptotic relation (as ${\hat{I}}_{\ast }^3\to \infty$) which relates the residues of the series ${\sum }_{n\ge 1}\hat{I}›(n)e^{-2\ddot{I}€ipn}{n}^{-s}$ 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 $b\ll \log {\hat{I}}_{\ast }^3$. This related function is continuous in $p$ and we mention its implication to the Riemann hypothesis.

11 pages. To fix an error in Lemma 1.1