A generalization of the Riemann-Siegel formula

TL;DR

This paper presents a novel generalization of the Riemann-Siegel formula that extends its applicability to any vertical strip, incorporating Mordell integrals and new polynomial families.

Contribution

It provides the first known generalization of the Riemann-Siegel formula to include the Hardy-Littlewood approximate functional equation across all vertical strips.

Findings

Inclusion of Mordell integrals in the asymptotics

Introduction of a new family of polynomials

Extension of the formula's validity to any vertical strip

Abstract

The celebrated Riemann-Siegel formula compares the Riemann zeta function on the critical line with its partial sums, expressing the difference between them as an expansion in terms of decreasing powers of the imaginary variable $t$. Siegel anticipated that this formula could be generalized to include the Hardy-Littlewood approximate functional equation, valid in any vertical strip. We give this generalization for the first time. The asymptotics contain Mordell integrals and an interesting new family of polynomials.