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We consider the $1s$ Lamb shift in hydrogen and helium ions, a quantity,required for an accurate determination of the Rydberg constant and the protoncharge radius by means of hydrogen spectroscopy, as well as for precision testsof the bound-state QED. The dominant QED contribution to the uncertaintyoriginates from $\alpha^8m$ external-field contributions (i.e., thecontributions at the non-recoil limit). We discuss the two- and three-loopcases and in particular, we revisit calculations of the coefficients $B_{61},B_{60}, C_{50}$ in standard notation. We have found a missing logarithmiccontribution of order $\alpha^2(Z\alpha)^6m$. We have also obtained leadingpure self-energy logarithmic contributions of order $\alpha^2(Z\alpha)^8m$ and$\alpha^2(Z\alpha)^9m$ and estimated the subleading terms of order$\alpha^2(Z\alpha)^7m$, $\alpha^2(Z\alpha)^8m$, and $\alpha^2(Z\alpha)^9m$. Thedetermination of those higher-order contributions enabled us to improve theoverall accuracy of the evaluation of the two-loop self-energy of the electron.We investigated the asymptotic behavior of the integrand related to thenext-to-leading three-loop term (order $\alpha^3(Z\alpha)^5m$, coefficient$C_{50}$ in standard notation) and applied it to approximate integration overthe loop momentum. Our result for contributions to the $1s$ Lamb shift for thetotal three loop next-to-leading term is$(-3.3\pm10.5)(\alpha^3/\pi^3)(Z\alpha)^5m$. Altogether, we have completed theevaluation of the logarithmic contributions to the $1s$ Lamb shift of order$\alpha^8m$ and reduced the overall $\alpha^8m$ uncertainty by approximately afactor of three for H, D, and He$^+$ as compared with the most recent CODATAcompilation.

The Lamb shift of the $1s$ state in hydrogen: two-loop and three-loop contributions