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

TL;DR

This paper introduces new series expansions using generalized Bessel functions for functions related to lattice point problems of the p-circle, especially effective for cases where 0<p≤1, expanding the analytical tools for these geometric problems.

Contribution

The paper develops a novel series representation using generalized Bessel functions for p-circle related functions, particularly addressing the challenging cases where 0<p<2.

Findings

Series representations are established for functions related to p-circle lattice problems.

The new method is especially suitable for cases 0<p≤1.

The approach extends harmonic-analytic techniques to broader p-values.

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. Kr\"{a}tzel 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.