Mich. Comp. Laws § 38.1362

Sec. 62. If a member claims credit for service performed under this act or former Act No. 136 of the Public Acts of 1945, during the period the member was entitled to membership in the retirement system, but the member's employment was not reported to the retirement system and contributions were not remitted for that member, the member may purchase the credit upon presenting acceptable proof of the service and contributing to the retirement system an amount equal to the amount the member would have contributed according to the schedule governing contributions in effect at the time of that service, plus regular interest on the contributions.