Sur une \'equation fonctionnelle approch\'ee due \`a J. R. Wilton

TL;DR

This paper presents a new proof of Wilton's approximate functional equation for a divisor sum, improving the error term and providing an explicit formula for an associated function.

Contribution

The authors offer a novel proof of Wilton's functional equation, enhancing the error estimate and explicitly characterizing the involved function.

Findings

Improved error term in Wilton's approximate functional equation

Explicit formula derived for the auxiliary function

Enhanced understanding of the divisor sum functional equation

Abstract

We give a new proof of an approximate functional equation, due to J. R. Wilton, for a trigonometric sum involving the divisor function. This allows us to improve on Wilton's error term and to give an explicit formula for an unspecified function involved in the functional equation.