Missing digits and sums of two prime squares

Abstract

We investigate integers whose base $g$ expansion omits a fixed digit and which can be represented as a sum of two prime squares. In the first part of the paper, we apply the Hardy--Littlewood circle method to obtain asymptotic formulas for weighted count of representations of such integers up to $g^k$ as $k\to\infty$, where we weight by the von Mangoldt function. In this case, we also get an interesting bias depending on the fixed digit we are missing. In the second part, combining the circle method with sieve methods, we study the second moment of the corresponding unweighted counting function. This allows us to get a nontrivial lower bound for the cardinality of the set $$\{ n \leq g^k : n \text{ omits the digit } b \text{ in its base } g \text{ expansion and } n = p^2 + q^2 \text{ for some primes } p,q \}. $$