Puzzle in the Primes: Atiyah’s Attempt at the Riemann Hypothesis

Mathematics is constantly evolving under the eager minds of a vigorous and thriving research community. At the cornerstone of mathematics, particularly in number theory, are the prime numbers–natural numbers divisible only by one and themselves. Recently, the mathematics community discovered a pattern in such numbers, attempting at a proof of the Riemann Hypothesis. One of the most famous unsolved mathematical problems in history, the hypothesis seeks to prove a special property of the Riemann function that can approximate the distribution of primes, given by a relationship between two functions: the prime-counting function π(x) and the Riemann zeta function. In 1859, mathematician Bernhard Riemann, presented a paper titled “On the Number of Prime Numbers Less Than a Given Quantity.” At the heart of the paper was an explicit formula for the number of primes up to any predetermined limit, an improvement on the approximation of π(x). For this formula to hold true, the values at which the Riemann zeta function equals 0 must be known. The Riemann zeta function is defined as the analytic continuation of the series (s)=1+12s+13s+14s…, or (s)=n=11ns, for complex-valued s, which converges when the real part of s is greater than 1. Since the series does not converge and is thus undefined when the real part of s is less than 1, the zeta function is “constructed” through a process called analytic continuation on the entire complex plane. This process extends the series from its defined values so that all of its higher derivatives exist, are continuous, and hence “analytic.” For all negative even integers (-2, -4, -6, etc.), (s)=0 ; these values are called trivial zeros. The primary focus of the Riemann Hypothesis deals with the infinite number of non-trivial zeros and claims the following: all non-trivial zeros have a real part equal to 12 (i.e. all non-trivial zeros can be expressed in the form 12+yi, where i =-1. )While values of s within a magnitude of 1013 have computationally been tested to agree with the assertion of the Riemann zeta function, the Riemann Hypothesis has yet to be proven in its general form. For years, many attempted to prove or disprove the assertion, but none has succeeded. This past month, at the Heidelberg Laureate Forum (HLF), Sir Michael Atiyah presented his attempt at proving the original Riemann Hypothesis. In his 45-minute lecture, he described how a seemingly unrelated concept in physics–the fine structure constant, which describes the strength and nature of electromagnetic interactions between charged particles–was the key to proving the Riemann Hypothesis. The two concepts are supposedly related by the Todd Function, introduced in Atiyah’s paper “The Fine Structure Constant.” There, Atiyah introduced certain properties of the Todd function including analyticity on all compact sets of the complex plane. However interesting this connection seems, though, the claim was met by much skepticism from the mathematical community. To his critics, Atiyah responded, “Nobody believes any proof of the Riemann Hypothesis, let alone by someone who’s 90,” remaining adamant about the veracity of his proof. Still, the verdict seems unlikely to come out in his favor. A large part of his theoretical work lies in his proof to the Proceedings of the Royal Society A that has yet to be published. As of the writing of this article, no definitive response to Atiyah’s proof has been published, but the general consensus appears to be that his proof is, at the very least, flawed. While this “proof” may not be the ultimate solution to the Riemann Hypothesis, Atiyah’s shortcomings still provide the opportunity of an open problem for the mathematical community. For the next generation of mathematicians, this problem may entail advances in the discovery of the distribution of the primes and better cryptography techniques. With advances in mathematical machineries, we may see a solid proof of the hypothesis in the next few decades.

References

Heidelberg Laureate Forum. (2018, September 24). 6th HLF - Lecture: Sir Michael Francis Atiyah [Video file]. Retrieved from http://www-history.mcs.st-andrews.ac.uk/Biographies/Atiyah.html

O’Connor, J. J., & Robertson, E. F. (n.d.). Michael Francis Atiyah. Retrieved October 12, 2018, from MacTutor History of Mathematics Archive website: http://www-history.mcs.st-andrews.ac.uk/Biographies/Atiyah.html

Riemann Hypothesis. (n.d.). Retrieved October 12, 2018, from Clay Mathematics Institute website: http://www.claymath.org/millennium-problems/riemann-hypothesis

Riemann Hypothesis. (n.d.). Retrieved October 12, 2018, from Encyclopaedia Britannica website: https://www.britannica.com/science/Riemann-hypothesis

Schembri, F. (2018, September 24). Skepticism Surrounds Renowned Mathematician‘s Attempted Proof of 160-Year-Old Hypothesis. Science. Retrieved from https://www.sciencemag.org/news/2018/09/skepticism-surrounds-renowned-mathematician-s-attempted-proof-160-year-old-hypothesis