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What is the hardest math ever?

The Riemann Hypothesis, famously called the holy grail of mathematics, is considered to be one of the toughest problems in all of mathematics.

cantorsparadise.com - Understanding the Hardest Problem in Mathematics | by Kasper Müller

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Understanding the Hardest Problem in Mathematics

A simple formulation of the Riemann hypothesis

I love mathematical problems… I can’t help it - I am an addict when it comes to these mind-bending and intriguing questions. I guess the reason for this affection is in part because of the mental challenge that the problems pose and in part because of the inherent beauty of the hunt for truth in this mysterious, alien and beautiful world called mathematics. Some problems are harder to solve than others though and many of the very hard problems are in fact so difficult to wrap our heads around that the mere formulation of them may require several years of dedicated studies at university level just to understand them in the first place. This is a pity because oftentimes the problems can be reformulated in a much simpler way. These reformulations are called equivalences and the proof of one of those would prove the original statement (and vice versa). The Riemann Hypothesis, famously called the holy grail of mathematics, is considered to be one of the toughest problems in all of mathematics. But more importantly, its truth is essential in order to understand the distribution of the prime numbers which are the fundamental multiplicative building blocks of the natural numbers.

To be fair though, we don’t know if it really is true.

Natural numbers are of course the central topic of study in the field of number theory. This problem is therefore central to the whole field. Carl Friedrich Gauss called mathematics “the queen of the sciences" and he referred to number theory as “the queen of mathematics”. G.H. Hardy also talked about the intrinsic beauty of pure mathematics and number theory which has played an important role in the development and direction of the entire field seen from a historical point of view. In this article, I will show you an elementary version of the Riemann hypothesis that was discovered by Jeffrey Lagarias in 2001. An equivalence that only requires basic mathematical knowledge. This makes mere mortals able to play along and take up the battle with this giant of a problem!

Prerequisites

Before formulating the actual question, I want to make sure that I don’t lose anybody and so we will go through the simple prerequisites one by one until I am sure that we are all on the same page. The first ingredient we need in order to state the Riemann hypothesis in this simple way is that of the harmonic numbers.

The harmonic numbers

The nth harmonic number is defined by the following expression:

They grow approximately like the natural logarithm and in fact, the difference between the harmonic numbers and the logarithm is in the limit a constant known as the Euler-Mascheroni constant or sometimes just Euler’s constant. We know a lot about the harmonic numbers. We have several formulas and generating functions at our disposal and they are worth an entire article by themselves.

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e

Other than being the weirdest subheading of all time and a commonly used letter in the English alphabet, this little symbol represents one of the most remarkable and important numbers in all of mathematics. Without e there would be no solutions to differential equations and calculus would fall apart faster than Euler could do arithmetic! The number itself is about 2.71828…though we will never be able to write all the decimals down because e is irrational. This means that the decimals continue in a non-cyclic way forever. It is in fact transcendental meaning that it is not a root of a polynomial with whole-number coefficients! We have this beautiful formula first discovered by Euler himself in the 1730s:

But what does this number represent and why is it special?

It turns out that if you do continuous compounding of interests, the factor you end up with is e, but more importantly, the exponential function with e as its base f(x) = e^x, has the interesting property f`(x) = f(x), that is, the function e^x is an identity and more generally e^(ax) is an eigen function for the derivative operator. Moreover, this function is a homomorphism from the group of real numbers with respect to addition to the positive real numbers with respect to multiplication i.e. f(x + y) = f(x)f(y).

e is a special number!

The natural logarithm

This function, denoted ln, is the inverse of e^x. That is, ln(e^x) = e^ln(x) = x. This alone makes the natural logarithm one of the most important functions out there but again there is more. ln(x) can be defined as the area under the graph (or integral) of the function f(t)=1/t from 1 to x and as stated above, it is a continuous version of the harmonic numbers. It is also a homomorphism since it has the fantastic property ln(xy) = ln(x) + ln(y) and it has a plethora of other properties that are never too late to study.

The sum of divisors function

This function denoted σ is a very important function in number theory.

In order to define it, recall that a divisor of a number n is a number k such that n/k is a whole number. For example, the positive divisors of 6 are 1, 2, 3 and 6. σ(6) is thus the sum of the divisors of 6 i.e. σ(6) = 1 + 2 + 3 + 6 = 12. σ is multiplicative which means if n and m have greatest common divisor 1, then σ(nm) = σ(n)σ(m). As an example of this we have σ(36) = 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 + 36 = 91 but on the other hand since σ(4) = 1 + 2 + 4 = 7 and σ(9) = 1 + 3 + 9 = 13, we can calculate the same using σ(36) = σ(4⋅ 9) = σ(4) ⋅ σ(9) = 7 ⋅ 13 = 91.

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It is surprisingly hard to find good formulas for the sum of divisors function. This has been a pursuit for many famous mathematicians including Ramanujan and Euler.

The Riemann Hypothesis

The problem description usually revolves around the so-called non-trivial zeros of an analytic continuation of a certain complex holomorphic function called the Riemann zeta function, typically defined by an infinite series and corresponding Euler product. This was what I meant by “hard to understand the problem description in the first place” by the way. However, as promised, we won’t take that route here. We shall see an elementary problem that is equivalent to the Riemann hypothesis. A formulation that only uses the above ingredients.

Lagarias showed that the following assertion is equivalent to the Riemann hypothesis.

Conjecture (Lagarias)

For each n ≥ 1,

That is it! This ladies and gents is the Riemann hypothesis stated a little differently than usual. If you prove this, you will get a million-dollar prize and your name will be on a short list of geniuses that changed scientific history.

The original paper by Lagarias can be found below.

Note that it is convention in many number theoretic texts to use the notation log for natural logarithm. This is because other logarithms are rarely used in number theory but in the end, it is just notation of course.

Does this formulation of the problem then make it easier to solve the Riemann hypothesis?

To be honest, I am not sure. The problem should in principle be just as hard (it is an equivalence after all), but it might open some doors into unexplored terrain. The sum of divisors function is tricky to work with because it requires some knowledge about a number’s prime factorization or at least some way of detecting it. That being said, I think it is intriguing that this problem has such a simple equivalence. Hopefully, this version of it will make more people interested in number theory and mathematics in general.

At least I hope so.

cantorsparadise.com - Understanding the Hardest Problem in Mathematics | by Kasper Müller

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