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Missing digits and sums of two prime squares

Cihan Sabuncu

TL;DR

This paper studies integers whose base-$g$ representation omits a fixed digit and that can be written as a sum of two prime squares. Employing the Hardy–Littlewood circle method, it proves a weighted asymptotic for the sum of the two-prime-squares representation function over the missing-digit set, featuring a local density factor $\mathfrak{S}(b,g)$ that depends on the omitted digit and the base. A second-moment analysis, combined with sieve techniques, yields a nontrivial lower bound for the count of missing-digit integers representable as a sum of two primes squares, while highlighting the obstacles to obtaining an unconditional asymptotic for the unweighted count due to limits in major-arc thinness and $L^{\infty}$ Fourier bounds. The work integrates circle-method estimates with a beta-sieve framework and the Gaussian-integer factorization viewpoint to control off-diagonal representations, and it discusses generalizations to multiple missing digits and related representation problems. Overall, it connects missing-digit and Waring-type problems with sums-of-two-primes-squares representations under a unified analytic approach and demonstrates both what can be achieved and where current bounds fall short.

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

Missing digits and sums of two prime squares

TL;DR

This paper studies integers whose base- representation omits a fixed digit and that can be written as a sum of two prime squares. Employing the Hardy–Littlewood circle method, it proves a weighted asymptotic for the sum of the two-prime-squares representation function over the missing-digit set, featuring a local density factor that depends on the omitted digit and the base. A second-moment analysis, combined with sieve techniques, yields a nontrivial lower bound for the count of missing-digit integers representable as a sum of two primes squares, while highlighting the obstacles to obtaining an unconditional asymptotic for the unweighted count due to limits in major-arc thinness and Fourier bounds. The work integrates circle-method estimates with a beta-sieve framework and the Gaussian-integer factorization viewpoint to control off-diagonal representations, and it discusses generalizations to multiple missing digits and related representation problems. Overall, it connects missing-digit and Waring-type problems with sums-of-two-primes-squares representations under a unified analytic approach and demonstrates both what can be achieved and where current bounds fall short.

Abstract

We investigate integers whose base 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 as , 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
Paper Structure (13 sections, 15 theorems, 147 equations)

This paper contains 13 sections, 15 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Idea of the Proof
  3. Preliminary Lemmas
  4. Elementary Results
  5. Exponential Sum Estimates
  6. Sieve Result
  7. Average of the Representation Function
  8. Major Arcs
  9. Minor Arcs
  10. Off-Diagonal Solutions
  11. Major Arcs
  12. Minor Arcs
  13. Putting Everything Together

Key Result

Theorem 1.1

Fix $g\geqslant 2$ sufficiently large. Let $A>0$ be given. We have as $k\to\infty$, for $X=g^k$, where and $\rho(a;q)$ is as before.

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 15 more