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

TL;DR

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Contribution

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Abstract

We show that there is a sequence of zeros of $\mathop{\mathcal R}(s)$ in the fourth quadrant. We show that the $n$-th zero $\rho_{-n}=\beta_{-n}+i\gamma_{-n}$, with $\beta_{-n}\sim 4\pi^2 n/\log^2n$ 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 $n$.