Values of Riemann’s auxiliary function at positive odd integers

Determine the values of Riemann’s auxiliary function (s), defined by (s) = ∫_{0↘1} x^{-s} e^{π i x^2} / (e^{π i x} − e^{−π i x}) dx (Equation (E:defRzeta), Section 3), at the positive odd integers s = 2n + 1 with n ≥ 0. The Hankel-contour representation in Equation (E:Rintegral3) implies the integral term is zero at these s, but the prefactor has a pole, preventing evaluation by that method.

Background

The paper defines Riemann’s auxiliary function (s) via a Siegel-style contour integral and develops several integral and series representations linking it to the Riemann zeta function. In Section 4, the author derives integral formulas and evaluates (s) at even integers, establishing explicit expressions and the trivial zeros at negative even integers.

The author also obtains explicit values for odd negative integers. However, when attempting to evaluate (s) at positive odd integers via the Hankel-contour representation (Equation (E:Rintegral3)), the integral component vanishes while the multiplicative prefactor has a pole at s = 2n + 1. This obstruction prevents deducing the values of (s) at positive odd integers using that approach, leaving their explicit determination unresolved within this work.