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Suddenly the microphone of my phone does not work anymore. I have checked with Xperia Diagnostics, and also with the internal diagnostics and also the second microphone does not work. A factory reset did not help either.
Any ideas how to fix it and whether this is a software issue (the sudden failure of both microphones seems like one to me)?
Hm contacted Sony support 4 days ago but no reaction yet 
Try performing a system repair using Pc Companion as this will often fix issues - This will factory reset your phone
Switch off your phone and unplug from Pc (Hold volume up and power for around 10 seconds)
Start PC Companion and select Support zone then Update my phone/tablet then in Blue Repair my phone/tablet and follow the onscreen instructions - When prompted connect your phone still switched off holding volume down or back button - This should start the repair or reformat process
If you are using Windows 8/8.1 or a 64bit operating system then adjust the settings for PC Companion and run in compatibility mode and chose Windows 7 or XP
If that fails then you need to either return your device to the place of purchase or contact Xperia care http://www.sonymobile.com/global-en/support/contact-us/