Analytic expressions for some Mellin transforms with their application to prime counting function and interpolation formulas for the zeta function

TL;DR

This paper derives new Mellin transform formulas using Ramanujan's method and applies them to analyze the prime counting function and the Riemann zeta function, providing insights into their properties and potential interpolation formulas.

Contribution

It introduces novel Mellin transform expressions based on Ramanujan's approach and applies these to key number-theoretic functions, advancing analytical techniques in number theory.

Findings

Derived new Mellin transform formulas using Ramanujan's method

Applied results to prime counting and zeta functions

Proposed interpolation formulas for the zeta function

Abstract

The aim of our present work here is to present few results in the theory of Mellin transforms using the method that S. Ramanujan used in proving his Master Theorem. Further applications of our results for some number-theoretic functions such as the prime counting function and the zeta function are established.