Levinson Functions

TL;DR

This paper constructs new functions related to the Riemann zeta function using Levinson's results, providing explicit integral examples that connect to the critical line and explore properties of these functions.

Contribution

It introduces explicit integral functions inspired by Levinson's work that model the behavior of the zeta function on the critical line.

Findings

Explicit integral functions related to the zeta function are constructed.

These functions exhibit interesting properties on the critical line.

The work provides new examples connecting Levinson's results to function construction.

Abstract

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