By Gyulim Jessica Kang
Introduction
The great unsolved problems of pure mathematics have stood as an unbroken mystery for centuries. The Riemann hypothesis, with a million-dollar prize on the line for the final solutions, is the holy grail of these baffling enigmas. The Riemann hypothesis is related to the precise distribution of prime numbers and was made by renowned German mathematician Georg Riemann in 1859 (Clay Mathematics Institute, n.d.). If proven, the hypothesis will have many implications on a range of fields including cryptography, e-commerce, and banking (Simons Foundation, 2020).
For now, though, it is just a conjecture.
In order for a conjecture to be fully addressed, it must either be proven to be true, proven to be false, or proven to be unprovable (the continuum hypothesis and Euler’s parallel postulate are some examples). Before discussing the Riemann hypothesis, we must first delve into the Riemann zeta function that forms its basis. What exactly is a zeta function, and what kinds of values does it yield?
Zeta functions are analytic functions of complex variables, and they form the fundamental basis of modern analytic number theory (3Blue1Brown, 2017). Analytic continuation is a crucial technique utilized in a branch of mathematics called complex analysis. Zeta functions have multiple applications in a wide variety of interdisciplinary fields: these include the Casimir effect and Bose-Einstein condensation in physics, the prime number theorem in number theory, and more.
The Riemann zeta function is a function of a complex variable s, which is equivalent to σ (sigma) + it. When Re(s), or σ, is greater than 1, the Riemann zeta function can be written as a converging integral or summation (Simons Foundation, 2020).
The red graph on the bottom right is a partial representation of the zeroes of the zeta function, a connection to the Riemann Hypothesis.
Important Terminology
Complex analysis is a branch of mathematics that involves the study of functions that have complex numbers as inputs and outputs.
Analytic continuation is a technique within complex analysis that extends the domain over which a complex function is defined (3Blue1Brown, 2017). Since the Riemann zeta function is defined for a complex variable s with a real part greater than 1, an analytical continuation would involve examining the function’s behavior when defined for a complex variable s with a real part less than or equal to 1. This is because the series will not converge and instead diverge for these values.
The Basel Problem
The Basel Problem is a problem that involves finding the sum of the reciprocal squares of natural numbers until infinity. This is the same as finding the value of the Riemann zeta function when n = 2. What’s interesting is that although Euler, who proved it first, relied on certain manipulations that weren’t justified yet, he was later proven correct and he produced a fully rigorous proof in 1741 (“Basel Problem”, n.d.).
Here, we can write sin x as equal to the polynomial of infinite order on the right: we call this the Taylor series of sin x. We can observe from the graph of sin x that all the zeros of sin x are in the form kπ, where k is an integer.
By dividing sin x by x and then using the Weierstrass factorization theorem, we can factor sinx/x into a product of linear factors given by its roots, just as we do for finite polynomials.This was the manipulation Euler employed that wasn’t proven at the time; the possibility of factorizing polynomials of infinite order was proven later by Weierstrass.
The linear factors are written as is because the function's roots are x=k*pi for any integer k. These roots are the zeros of sinx/x because they are also the zeros of sin x.
If we expand the product in the second line of the image above, all of the terms will be in the form of a coefficient times x to the power of an even integer ≥ 0. This is because each of the factors contains an x^2 term. Next, the third line represents the coefficient of the x^2 term after expansion. The only way to get an x^2 term when expanding would be to multiply an x^2 term in any one of the factors with the 1 in every other factor.
Going further, what’s really interesting is that certain infinite series like the sum of all natural numbers is a divergent series, meaning that it increases without bound to infinity; therefore, the series does not converge to a finite limit. However, it is possible for us to utilize advanced methods such as the zeta function regularization or Ramanujan summation to assign finite numerical values even to those divergent series (T. Editors of Encyclopaedia, 2008). Using these methods gives us the result that the sum of all natural numbers is -1/12; if you can't believe me, try searching it up! In order to do this, however, it is important to understand the infinite series as a value derived through these summation methods and not as the sum of an infinite series how we usually understand it. Otherwise, it would diverge! These methods have applications in fields besides complex analysis such as quantum field theory and string theory.
Riemann Hypothesis and Its Implications
In summary, the zeta function becomes zero for all negative even integers, which are called trivial zeroes. However, the function also becomes zero at other, more significant values; these are called the nontrivial zeroes. The Riemann hypothesis asserts that the zeta function’s nontrivial zeros all lie on a single line on the complex plane—the critical strip—where the real part of the zero is ½ (Simons Foundation, 2020).
If proven, the Riemann hypothesis will have groundbreaking implications. This is because it would define the seemingly random distribution of primes by pinpointing a pattern. This is significant because the whole of areas such as internet security and cryptography relies on prime numbers in the process of encryption and decryption. It takes exponentially more time to factor a number the larger and larger it gets; involving unfathomably large primes within a code ensures that no computer could crack this code within a human lifetime (T. Editors of Encyclopaedia, 2008).
Decades or centuries may pass before this conjecture is proven if it can even be proven at all. Of course, this uncertainty without closure may be frustrating. However, it is often these very doubts that spur progress in mathematics, paving the way for discussions and discoveries that have resounding effects in countless interconnected fields.
References
Basel Problem. Brilliant.org. Retrieved 11:25, February 24, 2022, from https://brilliant.org/wiki/sum-of-reciprocal-of-squares-basel-theorem/
Britannica, T. Editors of Encyclopaedia (2008, July 23). Riemann zeta function. Encyclopedia Britannica. https://www.britannica.com/science/Riemann-zeta-function
Riemann Hypothesis. Clay Mathematics Institute. (n.d.). Retrieved March 25, 2022, from https://www.claymath.org/millennium-problems/riemann-hypothesis
The Continuing Challenge to Prove the Riemann Hypothesis. Simons Foundation. (2020, May 6). Retrieved from https://www.simonsfoundation.org/2020/05/06/finding-prime-locations-the-continuing-challenge-to-prove-the-riemann-hypothesis/
3Blue1Brown. (2017). Visualizing the Riemann Zeta Function and Analytic Continuation. Youtube. Retrieved from https://www.youtube.com/watch?v=sD0NjbwqlYw
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