The Mu3e experiment searches for the lepton flavor violating decay $mu^+~rightarrow~e^+~e^+~e^-$ with an ultimate aimed sensitivity of 1 event in $10^{16}$ decays. This goal can only be achieved by reducing the material budget per tracking layer to $X/X_0 approx 0.1 %$. High-Voltage Monolithic Active Pixel Sensors (HV-MAPS) which are thinned to $50 mu m$ serve as sensors. Gaseous helium is chosen as coolant. Results of recent studies related to the sensor prototypes, the helium cooling, and module prototyping are presented. The recent chip submission MuPix10 has proven its functionality regarding efficiency and time resolution. The helium cooling system for the inner tracker could be verified using a full-scale prototype. A complete prototype equipped with MuPix10 chips will be tested inside the Mu3e magnet in summer 2021.
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