Digital functions along Squares of Prime Numbers

In the last 20 years the Gelfond conjectures concerning the well distribution of the sum-of-digits function along prime numbers and along squares have been solved and these results, which are strongly connected with the Sarnak conjecture, were generalized to $q$-multiplicative functions and automatic sequences. In this paper we study a combination of both challenges and prove a Prime Number Theorem for $q$-multiplicative functions along squares which can be rewritten into a well distribution result along squares of primes.