The Gamma and Zeta Functions with Infinite Polynomial Products

Abstract

A mathematical hypothesis that was initially put forward in 1859 and is still unverified as of 2015. The phrase "the Holy Grail of mathematics" has been used to describe this issue, which is possibly the most well-known of all open problems in mathematics. Although it has connections to other branches of mathematics, it is most frequently associated with the distribution of prime numbers. The infinite product representation of the gamma function and the zeta function are composed of an exponential and a trigonometric component, and this representation is demonstrated by starting with the binomial coefficient and employing its infinite product form.  It is proved, that all these components define imaginary roots on the critical line, if written in the form as they are in the functional equation of the zeta function.