Zeros of $\mathcal{R}(s)$ on the fourth quadrant

Abstract

We show that there is a sequence of zeros of $\mathcal{R}(s)$ in the fourth quadrant. We show that the $n$-th zero ${\rho }_{-n}={\beta }_{-n}+i{\gamma }_{-n}$, with ${\beta }_{-n}\sim 4{\pi }^{2}n/{\log }^{2}n$ and ${\gamma }_{-n}\sim -4\pi n/\log n$. We give the first terms of an asymptotic development of ${\rho }_{-n}$ and an algorithm to calculate ${\rho }_{-n}$ from $n$.

arXiv comment14 pages, 3 figures