The Zeta Function Derivation of the Ramanujan’s Summation

Please note that the famous Ramanujan Summation has three derivations: Heuristic (Infinity Manipulation), Zeta Function Regularisation and the Cutoff Regularisation. This article presents represents the Zeta Function Regularisation proof of the summation. To read more on the other methods of summation, click on this. Moving on to the article:

In one of my previous articles, I talked about the Ramanujan’s summation and its derivation with countable Heuristics method, infinity manipulation. It was an interesting approach, however, the hypothesis that 1+2+3+4+5+… = -1/12 is a viable hypothesis in many perspectives. For the purpose of this article, I will be talking about the Riemann Zeta Function, a complex analysis method, and how it is used to prove the Ramanujan Summation through a different perspective.

The Riemann Zeta Function is defined as follows:

This function is interesting as it always converges into a fraction including a power of Pi for even integers (Bernoulli Numbers) and uses the natural exponential function for odd integers. For example:

This function is fascinating because of the process of its derivation using the complex plane (To understand this further, look at the Basel problem). The complex plane has the x-axis  Re(z) and the y-axis Im(z), representing with “i”.  For example, if you want to represent (1/2) to the power of 2 +i, you would multiply (1/2) to the power of 2 times (1/2) to the power of i. when this is plotted in the complex plane, a vector multiplication is required, which bends both the vectors smaller than its previous region. When this base is changed and added, like the Zeta Function, it leads to an elliptical path:

Graph 1: Riemann Zeta Function

The following graph plots the Zeta Function from the domain 1 to 7. A rather elegant pattern of circles is formed and tempts us to complete. So, mathematicians followed and tried to replicate it to the other side of the domain: < 1 or =1. However, a problem occurred, the shape doesn’t necessarily need to be symmetrical of the domain 1 to infinity as we have no bases other than geometrically validating it. So, mathematicians sought out to find a trend. Using calculus, they found an odd trend with this graph. Whenever, two lines met in the graph, for example, Zeta(x) and Zeta(x+h), the angle of intersection of their tangents would remain constant for all values of x and h. This trend was called “Analytical” or angle-preserving, ging precedence to repeat a similar pattern on the other domain. In addition, other functions like trigonometric, logarithmic, and natural function were also analytical, forming a trend which mathematicians could follow.

So, if we were to extend the domain of the zeta function using the angle preserving property, an infinite jigsaw puzzle is formed. Therefore, by the process of Analytic Continuation, the following extended Zeta Function Graph is obtained:

Graph 2: Analytical Continuation of Riemann Zeta Function

With this in hand, the Ramanujan Summation can be proved using the Zeta Function:

Take the Function:

This is the Zeta Function’s equivalent to the Ramanujan’s Summation. However, as the domain falls into <1, it undergoes the process of analytical continuation. Using calculus and all the aforementioned properties, we get the value:

Bizarre yet sensible, don’t you think?

Reference:

Graph Images by Grant Sanderson, 3Blue1Brown, Boclips, https://www.nagwa.com/en/videos/307120764964/