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An Equidistribution Result for Differences Associated to Square Pyramidal Numbers II

Anji Dong, Katerina Saettone, Kendra Song, Alexandru Zaharescu

TL;DR

This work investigates the equidistribution of the sequence $a_n=|P_n-y_n^2|$, where $P_n=\sum_{i=1}^n i^2$, extending previous average results to arithmetic progressions and Dirichlet twists. The authors combine discrepancy bounds (Erd\H{o}s--Tur\'an), uniform distribution results (Kuipers--Niederreiter), and Euler--Maclaurin techniques to obtain a precise main term $A(b,q,x)=\frac{1}{5q\sqrt{3}}x^{3/2}$ with a power-saving error $O\left(\frac{x^{17/12}}{q^{2/3}}\right)$ for residue classes, along with twisted sum asymptotics $\sum_{n\le x} a_n\chi(n)$ and the corresponding Dirichlet-series pole structure. The paper further analyzes $b_n=\sum_{d|n}a_d$ and proves $\sum_{n\le x} b_n (1-n/x)=\frac{2\zeta(5/2)}{35\sqrt{3}}x^{7/2}+O_\delta(x^{41/12+\delta})$, using Perron-type methods and residue calculations. These results collectively advance equidistribution questions for square-pyramidal differences and illustrate a robust use of Dirichlet-series techniques in arithmetic-progression contexts.

Abstract

This paper presents some new results concerned with uniform distribution properties associated with the sequence $(a_n)_{n\in\mathbb{N}}$, which is defined as the distance from the $n$-th square pyramidal number to the closest square. We also extend the results to arithmetic progressions.

An Equidistribution Result for Differences Associated to Square Pyramidal Numbers II

TL;DR

This work investigates the equidistribution of the sequence , where , extending previous average results to arithmetic progressions and Dirichlet twists. The authors combine discrepancy bounds (Erd\H{o}s--Tur\'an), uniform distribution results (Kuipers--Niederreiter), and Euler--Maclaurin techniques to obtain a precise main term with a power-saving error for residue classes, along with twisted sum asymptotics and the corresponding Dirichlet-series pole structure. The paper further analyzes and proves , using Perron-type methods and residue calculations. These results collectively advance equidistribution questions for square-pyramidal differences and illustrate a robust use of Dirichlet-series techniques in arithmetic-progression contexts.

Abstract

This paper presents some new results concerned with uniform distribution properties associated with the sequence , which is defined as the distance from the -th square pyramidal number to the closest square. We also extend the results to arithmetic progressions.
Paper Structure (4 sections, 3 theorems, 54 equations)

This paper contains 4 sections, 3 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. General Notations
  3. Proof of Theorem \ref{['thm: A(x) and M(x) in AP']}
  4. Proof of Theorem \ref{['thm: asymptotic for bn']}

Key Result

Theorem 1.1

For any $b,q\in\mathbb{N}$ and any $x\geq 1$, define where $a_n$ is as in a_n first definition and defn: square pyramidal number. Then, $A(b,q,x)$ satisfies

Theorems & Definitions (7)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • proof
  • proof : Proof of Corollary \ref{['thm: asymptotic for dirichlet series of chi(n)a(n)']}
  • proof
  • Remark 4.1