Anomalous magnetic moment of a bound electron

We study binding corrections to the gyromagnetic factor $g_{\mathrm{e}}$ of an electron in hydrogenlike ions. We argue that the leading order binding effects in radiative corrections $\Delta g_{\mathrm{rad}}$ are universal to all orders in $\alpha /\pi$ and the complete result reads $\Delta g_{\mathrm{rad}}{=(g}_{\mathrm{free}}-2\left)\right[1+(Z\alpha {)}^2/6\mathrm{}]+O\left(\right(\alpha /\pi \left)\right(Z\alpha {)}^4).$ The theoretical uncertainty in the prediction for the experimentally interesting carbon ion is decreased by a factor of about 3.