Divisor Problem and an Analogue of Euler's Summation Formula

TL;DR

This paper presents a simple elementary approach to express the remainder term of the divisor problem, derives an Euler-Maclaurin analogue involving the divisor function, and links the remainder to the Riemann zeta function's functional equations.

Contribution

It introduces a straightforward method to analyze the divisor problem's remainder and connects it to the Riemann zeta function's properties, providing new insights.

Findings

Derived a simple expression for the divisor problem's remainder

Established an Euler-Maclaurin analogue for divisor function summation

Connected the divisor problem's remainder to the Riemann zeta function's functional equations

Abstract

By simple elementary method,we obtain with ease,a highly simple expression for the remainder term of the divisor problem and use it to obtain an Euler-Maclaurin analogue of summation involving divisor function.We also obtain a relation connecting the remainder term of the divisor problem and the remainders of approximate functional equations for Riemann zeta function and its square,for positive values of arguments.