We present a proof of the prime number theorem for arithmetic progressions based on tools from functional analysis. At its core, the proof
makes use of the spectral theory of Banach algebras.
The general strategy of the proof was first laid out by Tao in a
blog post from 2014 [Tao14]. In 2017, Einsiedler and Ward provided a
detailed proof of the prime number theorem based on these ideas and
proposed a framework for extending the proof to primes in arithmetic
progressions [EW17]. We fill in the missing pieces to obtain a comprehensive proof of the prime number theorem for arithmetic progressions