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Distribution of square-free palindromes

Aleksandr Tuxanidy

Abstract

An exponent of distribution 1/16 is established for square-free palindromes. The main input is an upper bound for the number of palindromes, in arithmetic progressions to large moduli, divisible by large squares. Our argument combines a simplifying reformulation with exponential-sum estimates, recent work on 6-almost-prime palindromes, and the large sieve with square moduli of Baier-Zhao.

Distribution of square-free palindromes

Abstract

An exponent of distribution 1/16 is established for square-free palindromes. The main input is an upper bound for the number of palindromes, in arithmetic progressions to large moduli, divisible by large squares. Our argument combines a simplifying reformulation with exponential-sum estimates, recent work on 6-almost-prime palindromes, and the large sieve with square moduli of Baier-Zhao.
Paper Structure (19 sections, 35 theorems, 352 equations)

This paper contains 19 sections, 35 theorems, 352 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Structural ingredients
  4. A-type process
  5. Paper structure
  6. Preparatory lemmas
  7. Square divisors and quasi-palindromes
  8. Translation to exponential sums
  9. The sum $E_<$
  10. Upper bound for $\Sigma$
  11. Upper bound for $\mathcal{S}(m)$
  12. Concluding
  13. Large $N$ relative to conductors
  14. Large conductors relative to $N$
  15. Proof of Theorem \ref{['thm: square divisors']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $L,q\geq 1$ be integers with $(q,b)=1$ and let $N\geq 1$. Then for any $\epsilon > 0$. In particular, for any $\epsilon > 0$.

Theorems & Definitions (65)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1: Poisson summation
  • proof
  • Lemma 3.2: Coprime sums
  • proof
  • Lemma 3.3: Bézout's identity
  • proof
  • Lemma 3.4: Sums of GCDs
  • proof
  • ...and 55 more