An Explicit Formula For The Divisor Function

TL;DR

This paper presents an explicit formula for the divisor function d(n), enabling a unified approach to analyze error terms in the divisor and circle problems.

Contribution

It formalizes an explicit formula for d(n), advancing the study of error terms in classical number theory problems.

Findings

Provides a new explicit formula for d(n)

Enables unified analysis of divisor and circle problems

Facilitates improved error term estimates

Abstract

The details for the construction of an explicit formula for the divisors function d(n) = #{d | n} are formalized in this article. This formula facilitates a unified approach to the investigation of the error terms of the divisor problem and circle problem.