Like origami art, where we can fold a piece of paper to create a boat and then refold the same piece of paper differently to build a totally different thing (for example, a bird), the simple steps below show that we can unfold an infinite series (commonly known as Riemann's zeta function) and then refold it to get a finite series. It is fascinating to see unfolding a divergence function in 6 steps and then refolding the same function in 6 steps gives us a convergent function. This is the most elegant method to analytically continue Riemann's zeta function to the critical strip because it shows mathematics overlaps with art.
Note: Consider the infinite series transition below for step 6. (s=1 has been shown for simplicity) :
The connection between mathematics and art has been a topic of fascination for centuries. In particular, the process of origami - folding and unfolding paper to create intricate designs - has been likened to mathematical problem-solving. In a similar way, the process of unfolding and refolding the Riemann zeta function can be seen as a kind of mathematical origami. By unfolding and refolding this series in a certain way, we can arrive at a finite series, which is an elegant and fascinating result. This process, known as analytic continuation, is essential to understanding the properties of the zeta function, and it is an important tool in number theory.
Moreover, this process of unfolding and refolding the Riemann zeta function is not just a dry mathematical exercise, but it also shows how mathematics can overlap with art. The elegance of the steps involved, and the beauty of the final result, are reminiscent of the intricate designs that can be created through origami. By exploring this connection between mathematics and art, we can gain a deeper appreciation for the beauty and power of both fields. In the following video, you can witness the remarkable simplicity of the connection between two seemingly complex mathematical concepts: Analytic continuation and the Riemann hypothesis. The beauty of this connection lies in its elegant logic and accuracy, demonstrating that we do not have to sacrifice one aspect for the other. It is a rare example of a harmonious combination of elegance, beauty, logic, and correctness in the world of mathematics.