Integral Representations of Riemann auxiliary function

Abstract

We prove that the auxiliary function $\mathcal{R}(s)$ has the integral representation \[\mathop{\mathcal R}(s)=-\frac{2^s π^{s}e^{πi s/4}}{Γ(s)}\int_0^\infty y^{s}\frac{1-e^{-πy^2+πωy}}{1-e^{2πωy}}\,\frac{dy}{y},\qquad ω=e^{πi/4}, \quad\Re s>0,\] valid for $\sigma >0$. The function in the integrand $\frac{1-e^{-\pi y^2+\pi \omega y}}{1-e^{2\pi \omega y}}$ is entire. Therefore, no residue is added when we move the path of integration.

arXiv comment10 pages 1 figure