Missing digits and sums of two prime squares

TL;DR

This paper studies integers missing a fixed digit in a given base that can be expressed as a sum of two prime squares, using advanced analytic and sieve methods to derive asymptotic formulas and bounds.

Contribution

It applies the Hardy--Littlewood circle method combined with sieve techniques to analyze the distribution of such integers and establish lower bounds on their count.

Findings

Asymptotic formulas for weighted counts of integers as sums of two prime squares

Identification of a bias depending on the missing digit

Nontrivial lower bounds for the count of such integers

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 $$\{ n \leq g^k : n \text{ omits the digit } b \text{ in its base } g \text{ expansion and } n = p^2 + q^2 \text{ for some primes } p,q \}. $$