Levinson Functions

Abstract

Starting from some of Norman Levinson's results, we construct interesting examples of functions $f(s)$ such that for $s=\frac{1}{2}+it$, we have $Z(t)=2\Re \{\ddot{I}{€}^{-\frac{s}{2}}\hat{I}“(s/2)f(s)\}$. For example one such function is \[\begin{aligned}{\mathcal R }_{-3}(s)=\frac12&\int_{0\swarrow1}\frac{x^{-s}e^{3πix^2}}{e^{πi x}-e^{-πi x}}\,dx\\&+\frac{1}{2\sqrt{3}}\int_{0\swarrow1}\frac{x^{-s}e^{\frac{πi}{3}x^2}}{e^{πi x}-e^{-πi x}}\Bigl(e^{\frac{πi}{2}}+2e^{-\frac{πi}{6}}\cos(\tfrac{2πx}{3})\Bigr)\,dx.\end{aligned}\]

14 pages 6 figures