Last year, I found very helpful lecture notes on analytic number theory and the Riemann Zeta function on Otto Forster‘s website. You might actually know this author by his (at least in Germany) very widely used real analysis text book series.
He has on his website several nicely typeset sets of lecture notes (but be warned that many of them are in German). Those lecture notes that I would like to point out here are:
These lectures notes overlap in some parts. They helped me a lot in understanding the rough outline of the usually taught proof of the prime number theorem and the concepts in analytic number theory in general. They also gave me a first glimpse of how the Riemann Zeta function can be approached and demystified.
Unfortunately, the chapter on Euler-Maclaurin summation is not available online. That’s why I recently bought H. M. Edward’s book “Riemann’s Zeta Function” about which I will write more in future posts.