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Focus on Mathematics

Collaboration between Number Theory and Computers : Mitsugu Mera : Kyushu University, Graduate School of Mathematics, Wakayama Research Laboratory, JSPS Research Fellowship for Young Scientists-DC2

One of my research themes is “Research on zeros of a q-analogue of the complete Riemann zeta function.” On reading this, I’m afraid that high-school students and non-specialists will not help but be totally lost, responding, no doubt, in the following way:

・“Complete?”
・“Riemann zeta function?” (A complex function defined as )
・“q-analogue?”
・“Zeros?” (This one is easy: Points at which the value of a function is equal to zero.)

To obtain a good understanding of these keywords, one would have to study pure mathematics as taught at the university level, particularly a field called “analytic number theory.” However, there is absolutely no need to tense up right now as in “This is unbelievably difficult!”, so please don’t worry.

Here, I will not go into details, but the following terms, which, in actuality, would be fairly common within the range of high-school mathematics, can act as a departure point for obtaining an understanding of the terms I introduced above:

・Power sum of natural numbers
・Series which converges or diverges
・Exponential function
・Prime numbers (2, 3, 5, 7, 11,...)

The following animations (achieved using MATLAB software) show the results of numerical calculations that I recently obtained through my research.

To think that these results—which belong to the realm of deep mathematics—were obtained as a result of one’s own efforts has to be exciting!

(* Click on the image to run the animation. You will need QuickTime Player playback.)

This is, of course, specialized material, but let me explain in simple terms. Each frame in either animation plots the values (about 10,000 points) of parameter q (0<q<1) related to “q-analogue” on the vertical axis and the absolute values of the series (double series) corresponding to the “q-analogue of the complete Riemann zeta function” (the “q-zeta function”) on the horizontal axis while making high-accuracy estimations such that approximation error is under . In addition, the value (complex number) of the independent variable that attends to this q-zeta function is varied at suitable step-widths in the positive imaginary direction to produce the unique behavior shown here.

By the way, the background to the creation of these animations is the “Riemann hypothesis,” which is one of the most important unsolved problems in mathematics. The Riemann hypothesis is a conjecture about the distribution of non-trivial zeros of the Riemann zeta function . It has yet to be solved, and 2009 marked its 150th year as a major difficult-to-solve conjecture. In these animations, particularly Animation 1, there is a direct link to this conjecture. Actually, a proposition equivalent to the Riemann hypothesis can be constructed from the viewpoint of “q-analogue” based on several considerations including the results shown in Animation 1.

It would give me great satisfaction if the reader should come away from this article with a sensation of “This is really interesting!” even if he or she found it difficult to understand. With such a frame of mind, I believe you can enter university and taste the true fascination of mathematics. You will move from the basics of counting-oriented arithmetic to the intriguing world of number theory, and as shown by the above animations, you will encounter the “unbelievable hidden world of mathematics” that computer simulation is making possible for the first time. Wouldn’t you like to experience this amazing world for yourself?