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The Pattern of Primes and the Riemann Hypothesis

Bridgett Burg

Mr. Reith

Calculus 1

20 December 2013

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Abstract

The Riemann Hypothesis is all about one question: What, if any, is the pattern of the

prime numbers among all counting numbers? Riemann’s conjecture states the zeta function

holds the key to this question and that the nontrivial (nonzero solutions that are not easy to

prove) zeros (roots) of the Riemann Zeta function all have a real part (complex number “x+bi”)

and lie on the critical line of x=1/2 or (1/2)+bi, b being any real number and 0<x<1. According

to Riemann, a proof of this conjecture would reveal the pattern of the primes. A proof for this

conjecture has not yet been found and the problem has become one of the seven Millennium

Problems, chosen by the Clay Institution of Mathematics. The many patterns (incomplete

patterns) that primes create through many different means are discussed and analyzed by using

different techniques found by different mathematicians.

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Problem Statement

If any, what is the pattern of the primes? There are already many vague patterns found

and seen when studying primes but no one knows the specific pattern of prime numbers. Several

mathematicians have studied the pattern of the primes and have found the following patterns,

although incomplete:

The Riemann Hypothesis

Ulam Spirals

Sacks Spirals

Quantum Physics

The Riemann Hypothesis currently holds the most clues as to how to uncover the pattern of the

primes. The proof of The Riemann Hypothesis would have unparalleled affects in not only mathematics

but also in technology and sciences.

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Related Research

Georg Friedrich Bernhard Riemann

Georg Friedrich Bernhard Riemann was born September 17, 1826 in Jameln,

Germany and died July 20, 1866 in Verbania, Italy. Riemann’s father wanted him to

become theologian but Riemann switched studies during college to study mathematics.

He attended Humboldt University of Berlin where he studied with Dirichlet and Jacobi

and also attended Georg-August University of Göttingen under the instruction of Gauss.

Riemann married Elise Koch and had one daughter named Ida. He had many

achievements such as building a bridge between analysis and geometry, clarifying the

nature of multiple-valued functions using Riemann surfaces, and the Riemann integral.

One of the biggest unproven problems he contributed to was the pattern of the primes

with the Riemann Hypothesis.

The Riemann Hypothesis

The central question which Riemann based his hypothesis around was “What, if

any, is the pattern of the prime numbers among all counting numbers?” Riemann’s

conjecture or conclusion formed on incomplete information is that the nontrivial zeros

or the roots of the function that are not easy to prove of the Riemann Zeta function all

have a real part or “complex number” and lie on the critical line of x=1/2 or x=(1/2)+bi, b

being any real number while 0<x<1. The reason why a proof of Riemann’s conjecture

would lead to a known pattern of the primes is due to how the original Zeta function

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created by Euler was manipulated in order to relate to prime numbers while Riemann

continued to analyze it adding in the complex number part. The reason it is called the

Riemann Zeta function is due to how Riemann discovered the important relationship

between the zeros and the distribution of the prime numbers.

The Zeta Function

The Riemann Zeta function has proven the conjecture for more than one billion

zeros of the function but cannot be considered true unless a proof is found. In the

function, when s>1 then there is a finite answer while if s<1 there will be an infinite

answer. Since the original function Euler had created was one dimensional, it did not

have enough geometric structure or graphing capability to hold enough information to

find the pattern of the primes. The key step Riemann made to be able to find more

information about primes was to replace the variable “s” or ‘any real number’ with the

variable z which stood for a complex number. When replacing ‘s’ with ‘z’ there was

some gaps created which were filled by analytical continuation. Analytical continuation

provided a way of calculating values (works for most all complex #s) which the original

formula doesn’t answer by extending the domain of the function. When graphed, there is

an obvious pattern seen by visual presentations of the Riemann Zeta function.

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3-D Representation

In the three dimensional representation, although it seems very random, the spikes

downward are all the zeros of the function on the critical line x=1/2.

Complex Plane:

In the complex plane graph which is a two dimensional graph where the closer a

color is to black, the closer the value at that point is to zero. In the graph it can be

observed that there is a pattern due to the ‘fingers’ that continually appear in the graph.

The tips of the ‘fingers’ in the graph all have a black tip at x=1/2 indicating they are zero

on the critical line Riemann suggested.

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Polar Graph

In the polar graph of the Riemann zeta function, the graph generally oscillates in a

carotid form. The unexpected change is when the graph does not meet back at zero and

ends up meandering from the positive half of the graph to the negative half. The one

general similarity is how for the most part each carotid meets up at zero. It can be

predicted that the pattern would continue if the graph where to be continued.

Riemann’s Influences

Riemann lived during an age of new math where there was a focus on the bigger

picture of functions. The ‘bigger picture’ of functions is the formulation and the idea of

understanding the abstract concepts and relationships functions give. During this age,

mathematicians begin to not limit themselves to specific numbers in functions but begin

to look at certain patterns with functions and variables.

Riemann’s Paper

In 1859, Riemann published his only work on the theory of numbers which was

less than ten pages long and devoted to the prime number theorem. The first success in

proving one of Riemann’s statements in his paper was in 1893 by J. Hadamard. All of

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Riemann’s statements have been proven since except one, “The Riemann Hypothesis” or

that all zeros of the zeta function in the strip 0<x<1 lie on the central line x=1/2. Riemann

wrote that, “…it is very probable that all roots are real. One would, however, wish for a

strict proof of this; I have, though, after some fleeting futile attempts, provisionally put

aside the search for such, as it appears unnecessary for the next objective of my

investigation.” which sounds strikingly familiar considering Fermat and his last theorem

as Riemann hints at how he possibly found an un-simplified proof.

Primes

A prime is a number which only has two factors, one and itself. Every non-prime

number is composed of prime numbers such as 30 which can be expressed as the

multiplication of three primes being 2, 3, and 5. The density of prime numbers less than a

large number (n) can also be roughly estimated using Dn(n)≈ 1/ln(n).

Prime Density

Riemann saw a connection between the prime density function [Dn(n)≈ 1/ln(n)]

and the solutions of the zeta function. He found that any complex number which solved

the prime density function is a zero of the zeta function such as -2,-4, etc. or wherever (n)

is negative even whole number. The function showed that there is an infinite number of

zeros beside the real zeros and according to Riemann’s conjecture all other zeros of the

function must have the form n=(1/2)+bi where b is some real number.

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If True…

If all complex zeros of the zeta function really have the real part of ½ in common

then the prime density function n(Dn)≈ 1/ln(n) differs from the actual density of primes

in a way that would prove the overall pattern of primes is regular. Although it can’t

predict with any large accuracy of when the next prime will occur it would be a big step

in helping prove Riemann’s conjecture.

The Distribution of Primes

The distribution of primes can be seen visually through Ulam and Sacks spirals.

Ulam Spirals

Ulam spirals are a square spiral of numbers where if you circle the primes you can

see diagonal lines created by the primes (with gaps). These diagonal lines can be

expressed as equations, some lines having more primes than others. There are two lines

with 12x as many primes as the average line in an Ulam spiral, they are quadratic

equations known as:

- y= x2+x+3399714628553118047

- y=

x2+x+3325181098069687810315008528712950885731284775149819034

9983874538507313

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Sacks Spirals

The Sacks spiral is where the square numbers are in a straight line and then

connected with an archimedean spiral. Once connected, evenly space the other numbers

in between each square number on the spiral which leads to the next square number.

After creating the Sacks spiral, it assumes a volleyball-like shape.

Patterns

Mathematics is known in general as the science of patterns for example, how

differential calculus is the study of patterns of motion and change. The easy way to see

these patterns is too view them topographically (i.e. 3-D) instead of linearly. These

different viewing options are capable due to different advancements in math such as how

complex numbers form a 2-D which allowed applications in geometry and calculus.

Is There an Infinite Number of Primes?

Euclid proved there are an infinite amount of primes through the harmonic series or:

The harmonic series is called harmonic due to its relations to music and the wavelengths

of overtones of a vibrating string.

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Impacts

If a proof were to be found for the Riemann Hypothesis, advancements would be

made in physics, modern communications technology, understanding of numbers,

mathematics, and cryptograms with coding. Prime numbers are used in coding for

encrypting for security purposes; two large primes are multiplied together for an

encryption code and the two large primes used are the decryption code.

Quantum Physics

Hugh Montgomery found a formula to describe spacing between zeros of the zeta

function which also gives the spacing between energy levels of the quantum chaotic system.

Quantum Chaos is a branch of Physics which describes certain systems using quantum

theory, a theory about matter. Alain Connes is trying to use Quantum Physics to prove the

Riemann Hypothesis.

Alain Connes

Alain Connes is trying to solve the conjecture by writing down systems of

equations which specify a hypothetical quantum chaotic system that has the primes built

in. So far, Connes has proved that the system’s energy levels correspond with the zeros of

the zeta function that are on the critical line. If he can prove there are no other zeros

except the ones that are on the critical line or the ‘energy levels’ he will have proved

Riemann’s conjecture. If Connes is successful, it would be the first time quantum physics

was used to solve a pure mathematics problem, the Riemann Conjecture.

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Suggestions for Further Research

Although primes may seem like a short, tiny topic they can only truly be

described as an iceberg. There are many topics which relate to the pattern of the primes

but were not discussed but would be great topics for individual research such as:

Mersenne Primes (2n-1)

Fermat Primes (22^n+1)

Goldbach’s Conjecture

Internet security coding

Twin Primes

Further research about primes could result in more advancements and

understanding of the Riemann Hypothesis and the possible secret pattern of the prime

numbers.

References

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10 Dec. 2013.

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