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Series expansions by generalized Bessel functions for functions related to the lattice point problems for the p-circle

Masaya Kitajima

TL;DR

The paper addresses lattice-point counting for the $p$-circle and the difficulty of obtaining sharp error-term bounds when $0<p<2$. It introduces a $p$-radial Bessel framework, defining $J_0^{[p]}$ and $J_ω^{[p]}$, and derives a series representation for the difference between the discrete and continuous counts in terms of these functions. The main theorem generalizes the Kuratsubo–Nakai display to the $p$-radial context and provides a convergent lattice sum that could yield nontrivial bounds for $0<p≤1$. The work lays groundwork for uniform asymptotics of $J_ω^{[p]}$ and outlines steps to apply oscillatory-integral methods, potentially advancing lattice-point estimates for the unsolved $p$-circle problems.

Abstract

The lattice point problems of the $p$-circle (for example, the astroid), which a generalized circle for positive real numbers $p$, have been solved for approximately $p$ more than 3, based on the series representation of the error term using the generalized Bessel functions by E. Krätzel and the results of G. Kuba. On the other hand, for the cases $0<p<2$, the method via this series representation cannot make progress. Therefore, in such cases, it is necessary to consider another method. In this paper, we prove that certain functions closely related to the problems can be displayed as series by newly generalized Bessel functions based on the property $p$-radial, generalization of spherical symmetry, and highlight the possibility that attempts to solve the problems via this display are suitable especially for the cases $0<p\leq1$. This study is based on the harmonic-analytic method by S. Kuratsubo and E. Nakai, using certain functions generalizing the error term of the circle problem by variables and series representation of the functions by the Bessel functions.

Series expansions by generalized Bessel functions for functions related to the lattice point problems for the p-circle

TL;DR

The paper addresses lattice-point counting for the -circle and the difficulty of obtaining sharp error-term bounds when . It introduces a -radial Bessel framework, defining and , and derives a series representation for the difference between the discrete and continuous counts in terms of these functions. The main theorem generalizes the Kuratsubo–Nakai display to the -radial context and provides a convergent lattice sum that could yield nontrivial bounds for . The work lays groundwork for uniform asymptotics of and outlines steps to apply oscillatory-integral methods, potentially advancing lattice-point estimates for the unsolved -circle problems.

Abstract

The lattice point problems of the -circle (for example, the astroid), which a generalized circle for positive real numbers , have been solved for approximately more than 3, based on the series representation of the error term using the generalized Bessel functions by E. Krätzel and the results of G. Kuba. On the other hand, for the cases , the method via this series representation cannot make progress. Therefore, in such cases, it is necessary to consider another method. In this paper, we prove that certain functions closely related to the problems can be displayed as series by newly generalized Bessel functions based on the property -radial, generalization of spherical symmetry, and highlight the possibility that attempts to solve the problems via this display are suitable especially for the cases . This study is based on the harmonic-analytic method by S. Kuratsubo and E. Nakai, using certain functions generalizing the error term of the circle problem by variables and series representation of the functions by the Bessel functions.
Paper Structure (4 sections, 7 theorems, 51 equations, 1 figure)

This paper contains 4 sections, 7 theorems, 51 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Definitions and properties of generalized Bessel functions
  3. Proof of Theorem \ref{['thm']}
  4. Concluding remarks

Key Result

Theorem 1.1

Let $p>2$. If $\alpha_{p}<1-\frac{1}{p}$ such that $\Delta(r;p)=\mathcal{O}(r^{\alpha_{p}})$ exists, then $P_{p}(r)=\mathcal{O}(r^{1-\frac{1}{p}}),\Omega(r ^{1-\frac{1}{p}})$ holds.

Figures (1)

  • Figure 1: Examples of the $p$-circle and the approximation by unit squares.

Theorems & Definitions (16)

  • Theorem 1.1: Kratzel, Theorem 3.17 A
  • Proposition 1.2: Kuratsubo-2022; Lemma 5.1(Special cases)
  • Theorem 1.3
  • Definition 2.1: $p$-radial
  • Remark 2.2
  • Proposition 2.3: The series expansion of $J_{0}^{[p]}$
  • proof
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6: The series expansions of $J_{\omega}^{[p]}$: Generalizations of Proposition \ref{['series']} to order
  • ...and 6 more