The sum of digits of primes in Z[i]

We study the distribution of the complex sum-of-digits function sq with basis q = −a ± i, a ∈ Z+ for Gaussian primes p. Inspired by a recent result of Mauduit and Rivat [16] for the real sum-of-digits function, we here get uniform distribution modulo 1 of the sequence (αsq(p)) provided α ∈ R \Q and q is prime with a ≥ 28. We also determine the order of magnitude of the number of Gaussian primes whose sum-of-digits evaluation lies in some fixed residue class mod m. 1 Preliminaries and Notation Let q = −a ± i (choose a sign) with a ∈ Z and denote Q = |q|2 = a + 1. Then every z ∈ Z[i] has a unique finite representation