Towards the Deep Riemann Hypothesis for ${\mathrm{G}\mathrm{L}}_n$

We explicate the deep Riemann hypothesis for the general linear group ${\mathrm{G}\mathrm{L}}_n$ on the convergence of normalised Euler products of standard $L$-functions on the critical line. It conditionally improves upon the error term in the prime number theorem beyond what the grand Riemann hypothesis predicts. Furthermore, we discuss the Chebyshev bias for Satake parameters on ${\mathrm{G}\mathrm{L}}_n$ from the perspective of the deep Riemann hypothesis.