Examples of Bessel Functions of Order Zero Related to the Functional Equation of the Riemann Zeta Function

Source abstract

In 2010, M. A. Chaudhry et al. introduced a class of functions Sigma satisfying an L^1-condition. They proved that every function phi in Sigma gives rise to the functional equation for the Riemann zeta function. They also provided an example of a function outside Sigma for which the functional equation still holds. In this paper, we present several examples of Bessel functions of order zero that do not belong to Sigma but nevertheless give rise to the functional equation for the Riemann zeta function. In particular, we show that J_0(a sqrt(x)), J_0(ax), and Y_0(x) exhibit this phenomenon.