New Results Involving Riemann Zeta Function Using Its Distributional Representation

The relation of special functions with fractional integral transforms has a great influence on modern science and research. For example, an old special function, namely, the Mittag-Leffler function, became the queen of fractional calculus because its image under the Laplace transform is known to a large audience only in this century. By taking motivation from these facts, we use distributional representation of the Riemann zeta function to compute its Laplace transform, which has played a fundamental role in applying the operators of generalized fractional calculus to this well-studied function. Hence, similar new images under various other popular fractional transforms can be obtained as special cases. A new fractional kinetic equation involving the Riemann zeta function is formulated and solved. Thereafter, a new relation involving the Laplace transform of the Riemann zeta function and the Fox-Wright function is explored, which proved to significantly simplify the results. Various new distributional properties are also derived.

Automated summary

The relationship between special functions and fractional integral transforms has a significant impact on scientific research, particularly the image of the Mittag-Leffler function under the Laplace transform. By using distributional representation, the Laplace transform of the Riemann zeta function is computed and applied in generalized fractional calculus, resulting in new images under various fractional transforms.