In my Android App, I would like to take in some audio from the mic of the smartphone and play it immediately, live, like a microphone, with no lag. I am currently thinking of us
This is actually really tricky on Android. Google themselves have a very good (but slightly long) video explaining the issues.
They also have a page explaining their latency testing methods and benchmarks for various devices.
Essentially, stock Android can't do zero-latency audio, however there's nothing stopping hardware partners from adding the required hardware and platform extensions to do so.