Corrections of order<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">α</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>(<i>Z</i>α<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>to the hyperfine splitting and the Lamb shift

Corrections to the hyperfine splitting and the Lamb shift of order ${\mathrm{\ensuremath{\alpha}}}^{2}$(Z\ensuremath{\alpha}${)}^{5}$ induced by diagrams with radiative photon insertions in the electron line are calculated in the Fried-Yennie gauge. These contributions are as large as -7.724(1)${\mathrm{\ensuremath{\alpha}}}^{2}$(Z\ensuremath{\alpha}${)}^{5}$/(\ensuremath{\pi}${\mathit{n}}^{3}$)(${\mathit{m}}_{\mathit{r}}$/m${)}^{3}$m and -0.6726(4)${\mathrm{\ensuremath{\alpha}}}^{2}$(Z\ensuremath{\alpha})/(\ensuremath{\pi}${\mathit{n}}^{3}$)${\mathit{E}}_{\mathit{F}}$ for the Lamb shift and the hyperfine splitting, respectively. Phenomenological implications of these results are discussed with special emphasis on the accuracy of the theoretical predictions for the Lamb shift and the experimental determination of the Rydberg constant. A precise value of the Rydberg constant is obtained on the basis of the improved theory and experimental data.