The Riemann Zeta Function is an infinite sum that can be proved to add up to a finite value. Apéry’s Constant is a form of the Zeta Function that expresses the sum of the reciprocals of all cubed positive integers (hence the title of this article). In other words, 1 cubed is 1, 2 cubed is 8, 3 cubed is 27, and so on. Now we take the reciprocals of all of these numbers (that is, 1+(1/8)+(1/27)…) and add them. We know that there really is an answer: In mathematical terms, the Zeta Function is said to “converge.” With a finite set of numbers, determining the sum is methodologically easy but possibly tedious. Infinite series like the Zeta Function are different: Just because we know they converge does not mean we know to which value they converge in the limit. The ability to state a simple-sounding problem in the proper mathematical context often facilitates a proof – or, at the least, an educated guess about what a proof might look like. However, finding a precise value for the convergence of the Zeta Function has eluded the greatest mathematicians for over 150 years. No end to this confusion is in sight. Mathematicians did not even know until the 1970s whether Apéry’s Constant could be written as a fraction, let alone what that fraction would be! As it turns out, one cannot write it as a fraction (of whole numbers, anyway).
Why in the world is this problem so difficult? And why would we even care? These questions seem to be related to each other. In short, we should care because the problem is both difficult and fundamental to a variety of seemingly unrelated fields. Like pi, Apéry’s Constant shows up in the strangest of places. In physics, the constant appears in supercooling (Bose-Einstein Gases), superconduction, and the strengths of Casimir Forces (forces between charged plates held at a distance from each other). It appears in Quantum Field Theory as a regularization and as a form of partitioning energy in accordance with temperature. Roughly, regularization is a method of making infinite quantities converge predictably to finite ones. Hawking Regularization, S Regularization, and Heat-Kernel Regularization are all examples of methods that use mathematical sleight-of-hand to re-order sums so that they add to something fixed, known, and calculable. Apéry’s Constant appears in the equations of state for Photon Gases that arise in nuclear fusion and the swirls of light that form around black holes. The Zeta Function appears in Cosmology in diverse contexts. The first context is that of the Cosmic Microwave Background (CMB), which is thought to be radiation left over from the Big Bang. This background radiation can be visualized as a Photon Gas that is spread out roughly uniformly. The uniformity is not there at every scale, which is a problem called CMB Anisotropy. Because Anisotropy consists in determining how and why a gas is distributed in a certain way, the questions it poses may themselves be related to the value of Apéry’s Constant.
Paradoxically, the Zeta Function arises in the analysis of Power Laws, which are determinations of who gets what from whom – and thus of inequality and Anisotropy. In economics, Power Laws are used to model why the rich get richer and the poor stay poor. In physics, these laws are related to the distribution of the velocities of subatomic particles at different energy levels, like electrons around an atom. There is inherent inequality even in randomness: Although electrons may cluster around an average speed, there are outliers each way. Even if we cool them to near absolute zero, there are still inequalities. Apéry’s Constant appears in these inequalities as well. The ubiquity of Apéry’s Constant is inexplicable. However, its appearance in physical calculations may provide some vantage by which a closed form can be found. A predictive paradigm for physics like Quantum Gravity purports to describe the relative strengths of fundamental forces and the relative ratios of the constants of nature. Once sufficiently developed, physics may provide a way of saying what Apéry’s Constant must be. Very likely, such an explicit calculation would not appear like a widget out of some magic box. Rather, it would be embedded in a series of arguments that said, in a clever way, that if the Apéry is something other than a predicted value, we would incur a contradiction in an underlying physical theory of reality that could be tested by numerical or physical experiments. This approach would not constitute a proof in the mathematical sense, but would at least provide a roadmap toward a proof.
Further Reading (Technical).
[1]. Aoki, Takashi, et al., eds. Zeta functions, topology and quantum physics. Vol. 14. Springer Science & Business Media, 2008.
[2]. Ivic, Aleksandar. The Riemann zeta-function: theory and applications. Courier Corporation, 2012.
[3]. Kirsten, Klaus. “Basic zeta functions and some applications in physics.” A window into zeta and modular physics 57 (2010): 101-143.
[4]. Osler, Thomas J., and Brian Seaman. “A computer hunt for Apery’s constant.” Mathematical Spectrum 35.1 (2002): 5-8.
