Transcendental Numbers and Zeta Functions

The concept of “number” has formed the basis of civilzation since time immemorial. Looking back from our vantage point of the digital age, we can agree with Pythagoras that “all is number”. The study of numbers and their properties is the mathematical equivalent of the study of atoms and their structure. It is in fact more than that. The famous physicist and Nobel Laureate Eugene Wigner spoke of the “unreasonable effectiveness of mathematics in the natural sciences” to refer to the miraculous power of abstract mathematics to describe the physical universe. Numbers can be divided into two groups: algebraic and transcendental. Algebraic numbers are those that satisfy a non-trivial polynomial equation with integer coefficients. Transcendental numbers are those that do not. Numbers such as √ 2, √ −1 are algebraic, whereas, numbers like π and e are transcendental. To prove that a given number is transcendental can be quite difficult. It is fair to say that our knowledge of the universe of transcendental numbers is still in its infancy. A dominant theme that has emerged in the recent past is the theory of special values of zeta and L-functions. In this article, we will touch only the hem of the rich tapestry that weaves transcendental numbers and values of L-functions in an exquisite way. This idea can be traced back to Euler and his work. In 1735, Euler discovered experimentally that