1

The Perceivable Pattern of Primes

Aditi Chegu

2

The Perceivable Pattern of Primes

“Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry.”

- Richard Feynman

The pattern of prime numbers is at the heart of mathematics because it offers clues to the underlying structure of mathematics. While most patterns are discernible at first glance, mathematicians have spent ages toiling away at the primes, piecing together seemingly unrelated snippets of work trying to recognize a pattern.

The quest to find a pattern starts with Leonhard Euler, the mathematician who, apart from his many contributions to the sciences, also standardised the mathematical notation that we use today.

Euler worked with infinite series, which is an operation that adds an infinite number of terms in a definite pattern. A series is called convergent if it comes to a natural stopping point at some finite number, and if it does not stop, it is called divergent. 

The following formula is of a series known as the zeta function; this function is the sum of all the inverted natural numbers when each of them are raised to some power “s”.

The variable “s” is the input for the function, and Euler proved that as long as “s” was greater than one, a series would converge to a certain point on a graph. For every value of “s” equal to or less than one, the series would diverge into something indefinite. Because of this property, when graphed the function ends rather abruptly at x = 1. 

Euler went on to discover a connection between the zeta function and the primes, he proved that the zeta function was also equal to: 

Where all the numbers raised to the power of “s” are prime.

This discovery revealed a connection between Euler’s zeta function and the primes.

This story came to a temporary standstill until Bernard Riemann picked it up again.

Riemann was a mathematician who made several notable contributions to the field of complex analysis, which is the study of functions that involve complex numbers. A complex

3

number is any number that can be expressed as “a+bi” where both “a” and “b” are real numbers and “i” is an imaginary number, the square root of -1.

Riemann used his knowledge of complex analysis to extend the study of the zeta function into the plane of complex numbers. In the new Riemann zeta function, “s” could take in complex inputs as long as “a” in “a+bi” was greater than 1 and would produce finite outputs. This output would be a little different compared to one where “s” was a real number greater than one because exponentiation with imaginary numbers works differently, but an intuitive way of understanding it is that instead of a number being multiplied by itself and expanding linearly on the number line, it will spiral and converge to one finite value.

There was however a problem, while it would have been immensely helpful for mathematicians to see what would happen to the function if “s” was less than 1, the function was still indefinite for all values of “s” where the real part “a” was less than one and in the graph, the graph still ended at x =1.

So, Riemann extended the function to be defined even when “s” was less than 1. He did this by using something known as analytic continuation. A function, like Riemann’s zeta function, needs to have a derivative everywhere if it is analytic. Analytic, put loosely, means that the function has an angle-preserving property. Therefore, the angle between any two lines on a graph need to be the same (unless the derivative of an input is 0 in which case the angle gets multiplied by some integer) both before and after transforming the function. With a requirement of the zeta function being analytic even after its extension, there was only one possible way to draw the function out. And this process of drawing the function beyond the plane in which it is defined is known as analytic continuation.

4

This extended version of the graph made sense in an implicit way, because there was only one possible way to have an analytic continuation on the left half of the plane for the zeta function but there is no rigorous proof for it.

In this extended graph, Riemann made an observation that certain inputs of “s” were getting mapped to the origin after their transformation. The negative even numbers get mapped to the origin, but mathematicians understand why that happens, so these inputs are called trivial zeros. Some of the other inputs also map to the origin, they are called non-trivial zeros because all of these numbers lie on the “critical strip” of the graph, and even though Riemann hypothesised that every single one of those non-trivial zeros sat on the line x = ½, mathematicians cannot figure out why they lie there. 

5

This hypothesis, about the non-trivial zeta zeros, put forward by Riemann is known as the “Riemann Hypothesis” and these non-trivial zeros encode some remarkable information about the primes. 

To understand why and how a function that maps some numbers to the origin is related to prime numbers, the story needs to shift from the works of Euler and Riemann to Riemann’s tutor, Carl Friedrich Gauss.

  As a teenager, Gauss drew up large tables of primes, all the way to three million in attempts to find patterns. He also graphed the prime numbers in a way that with every prime he found on the x-axis his function would step up by one unit on the y-axis. With a small set of numbers, the function looked jagged, but as he started graphing it with larger sets of numbers the sharp edges were no longer significant, the function began to smoothen out. This was a revelation to Gauss because the graph of prime numbers that he drew seemed similar to the graph of the logarithmic integral function. A logarithm is a function that undoes exponentiation, much like division undoes multiplication. And the logarithmic integral function is a function that has a slope of 1/log(x). When Gauss noticed this connection, he proceeded to conjecture that for any given number “x” that was inputted in the function, the number of primes would be roughly the same as 1/log(x). This conjecture went through some modification since it was first proposed and a simple way to explain it is that instead of having the graph of primes step up by one unit every time it encountered a prime “y” or the exponent of that prime, the graph stepped up by log(y).

This conjecture came to be known as the “Prime Counting Function” and as the numbers grow larger and approach infinity, the percentage error of the function reduces.

This changed with Riemann’s work with the zeta function and non-trivial zeta zeros, because he discovered a wave that corresponds with the zeta function at s = 1 which closely

6

approximated the jagged graph of the Prime Counting Function. The percentage error of this wave is controlled by the zeta zeros. Meaning that this wave when encoded with more and more non-trivial zeta zeros began to match the Prime Counting Function more accurately, and when an infinite amount of non-trivial zeta zeros were added to it, the wave that Riemann discovered looked exactly like the jagged prime counting function.

So, when Riemann rigorously proved that the non-trivial zeros are needed to create an accurate prime counting function and predict the distribution of primes, it showed that there was definitely a pattern to the primes and a proof of why the non-trivial zeros all lie on the x=½ line would give mathematicians a tight grasp on this pattern.

While I, as a maths enthusiast, would love to say that the story comes to a close here and the non-trivial zeta zeros’ strange affinity to x=½ was proved mathematically (hence having the work of many mathematicians come to fruition by solving the mystery of the pattern of the primes), I cannot, because then I would be lying.

The Riemann Hypothesis remains unproven, and as the hypothesis is somewhat of a holy grail in mathematics because it opens up avenues for a lot more theorems and problems, it becomes more important that a rigorous proof be found.

7

References

1. https://www.theguardian.com/science/2005/mar/04/ research.highereducation#:~:text=A%20prime%20number%20is%20simply,number%20except%201%20and%20itself.&text=He%20noticed%20that%20primes%20become,is%20only%20one%20(113).

2. https://kids.frontiersin.org/article/10.3389/frym.2018.00040 3. https://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/devlin.pdf 4. https://en.wikipedia.org/wiki/Bernhard_Riemann 5. https://www.brainyquote.com/quotes/richard_p_feynman_160463#:~:text=Feynman

%20Quotes&text=Nature%20uses%20only%20the%20longest%20threads%20to%20weave%20her%20patterns,organization%20of%20the%20entire%20tapestry.

6. https://en.wikipedia.org/wiki/ Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function

7. https://kellivogstad.com/2015/10/04/why-is-my-kid-building-patterns-again-the- importance-of-patterning/#:~:text=Patterns%20are%20at%20the%20heart,an%20important%20skill%20in%20math.&text=We%20use%20patterns%20to%20represent,see%20relationships%20and%20develop%20generalizations.

8. https://www.researchgate.net/figure/The-critical-strip-of-s-Image-from- fieldofsciencecom-9_fig2_327043089

9. https://youtu.be/d6c6uIyieoo 10. https://youtu.be/VTveQ1ndH1c 11. https://youtu.be/zlm1aajH6gY 12. https://youtu.be/sD0NjbwqlYw 13. https://en.wikipedia.org/wiki/Prime-counting_function#:~:text=In%20mathematics

%2C%20the%20prime%2Dcounting,unrelated%20to%20the%20number%20%CF%80).

14. https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss 15. https://en.wikipedia.org/wiki/Prime_number_theorem#Prime-

counting_function_in_terms_of_the_logarithmic_integral16. https://en.wikipedia.org/wiki/Leonhard_Euler 17. https://thatsmaths.com/2014/02/27/the-prime-number-theorem/ 18. https://youtu.be/rGo2hsoJSbo