Riemann's auxiliary Function. Basic Results

TL;DR

This paper explores the properties and integral representations of Riemann's auxiliary function, providing known results, specific value calculations, and relations to the Riemann zeta function, serving as a reference for further research.

Contribution

It offers a comprehensive overview of Riemann's auxiliary function, including new integral expressions and explicit value calculations at integers, enhancing understanding of its properties.

Findings

Integral expression for Riemann's auxiliary function

Values at integers except odd natural numbers

Relation to the Riemann zeta function at half-integers

Abstract

We give the definition, main properties and integral expressions of the auxiliary function of Riemann $\mathop{\mathcal R }(s)$. For example we prove $$\pi^{-s/2}\Gamma(s/2)\mathop{\mathcal R }(s)=-\frac{e^{-\pi i s/4}}{ s}\int_{-1}^{-1+i\infty} \tau^{s/2}\vartheta_3'(\tau)\,d\tau.$$ Many of these results are known, but they serve as a reference. We give the values of $\mathop{\mathcal R }(s)$ at integers except at odd natural numbers. We have $$\zeta(\tfrac12+it)=e^{-i\vartheta(t)}Z(t),\quad \mathop{\mathcal R }(\tfrac12+it)=\tfrac12e^{-i\vartheta(t)}(Z(t)+iY(t)),$$ with $\vartheta(t)$, $Z(t)$ and $Y(t)$ real functions.