Mathematics and a History of Prime Numbers and theZeta Function

Austin LiuMathematics 333 - Complex Analysis

Duke UniversityDurham, North Carolina 27708

November 26, 2014

Abstract

This paper gives a brief introduction to prime numbers beginning from 500 BCuntil recent times. Important theorems relevant to each time period are given, and therelevant mathematicians are described in detail. In particular, links between complexanalysis and one of the most important theorems in number theory - the Prime NumberTheorem - are provided. Detailed analysis on the Riemann Zeta function is also pro-vided, and relevant links to prime number theory is given. Although this paper givesa wide breadth of numerous topics relating to prime numbers, it is primarily used togive an introduction to the topics presented rather than extremely thorough analysis.This paper is meant to provide both entertainment and information to the reader per-taining to number theory and how prime numbers are essential to our understandingof numbers and our universe as a whole.

1 Introduction to Prime Numbers

1.1 A Brief Overview: What is a Prime Number?

Since the beginning of time, mankind has constantly been obsessed with numbers. It is ourdestiny to try to understand more about numbers and how we can use them to understandthe universe. One of the basic building blocks for understanding numbers is the topic of primenumber theory. However, what is a prime number? To explain simply, a prime number isa natural number that has exactly two distinct natural number divisors (1 and the numberitself). More mathematically:

Definition 1. A natural number a which is greater than 1 is a prime number or is prime ifthe below condition is satisfied [1]:

∀b ∈ N b|a ⇒ b = a

1

A composite number can be similarly defined and acts as the negation of the formaldefinition of the prime. In English, a composite number is a natural number that has atleast three distinct natural number divisors. More mathematically, the composite numbercan be defined as:

Definition 2. A natural number a is a composite number if the following condition is satisfiedbelow:

∃b ∈ N s.t. b|a ∧ b > 1 ∧ b < a

Now that we have defined the relevant definitions, we can begin to dive deeper into ananalysis of prime numbers.

1.2 Applications of Prime Numbers

To motivate a discussion of prime numbers, it is important to understand the usefulness ofthe study of prime number.

Until the mid 20th century, the study of prime numbers were seen as the epitome of puremathematics. However recently, around 1970, there were discoveries in two primary fieldswhich extensively use prime number theory:

1. Computer Science and Hash Tables

2. Public Key Cryptography

1.2.1 Prime Numbers and Hash Tables

To properly understand the usefulness of prime numbers in hash tables, we must first un-derstand what hash tables are. Formally, hash tables are data structures that map keys tovalues. These structures are used primarily in computer science and is used to keep a sortof ordering to sets. An enlightening analogy is to view hash tables as a sort of dictionary.Just like dictionaries, hash tables have keys (words) and values (the definitions of the words).

More specifically, hash tables use certain functions called hash functions to computeinto an index of buckets where values can be found. Ideally the hash function will assigneach key to a unique bucket, in which the memory is stored and can be retrieved. Thefunctionality is similar to that of a storage compartment. A visualization of a hash tableand its corresponding hash function can be viewed in the figure below.

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Figure 1: Visualization of Hash Table and Function adapted from [2]

In computer science, programmers tend to want to store each key and minimize hascollisions. Collisions tend to occur when the hash functions of, for example, various stringshave the same mathematical number assigned to them. A basic solution to create moreefficient hash tables is simply to use prime numbers in expressions for hash functions. Basicproperties of primality minimize clustering in hash tables. Using prime numbers in hashfunctions is often shown to yield successful results in practice. Especially in the face ofexpanding data, hash efficiency is needed.

1.2.2 Prime Numbers and Cryptography

Security of information has been an important issue for thousands of years. In particular,Cryptography is the art of writing or solving code in order to send messages and preventthird parties from determining messages. Originally from the time of ancient Rome, Cryp-tographers would use a private, or secret key to send messages. This was considered thetraditional approach to sending secret messages. Specifically, the traditional approach worksas follows: a key would be shared by communicators such that each could both encryptand decrypt messages. Although this methodology was pretty efficient for small number ofcommunicators, there were many inherent risks involved. Keys could be stolen or extracted,and there was a risk of losing the keys. Any compromising of the code would essentiallybreak the message.

The modern approach to cryptography is largely used in both web transactions and creditcard information. The modern method uses a combination of public and private keys. Ananalogy of this would be with a mailbox in the middle of the street. The mailbox itselfrepresents the public key. It can be accessed by anyone and everyone who knows the streetaddress. However, only the owner of the mailbox has the secret key, i.e. access to theactual contents of the mailbox. The following example is called the Double Lock Example,and is the basis behind the mathematical prime-factorization algorithm used in the modernapproach.

The primary example of the modern implementation is through the Double Lock Examplewhere user A sends a message to user B:

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1. A puts a message in a box. He then locks the box with his lock LA (only A has thekey to lock A).

2. B receives the box. He then adds his own lock LB to the box. Now there are two lockson the box. He then sends everything back to A.

3. A receives the box from B with the double lock. He then removes the lock LA andsends the back backs to B.

4. B finally receives the box and unlocks the remaining lock LB. User B can now readthe message

This simple example actually enumerates the mathematic version proposed by Rivest,Shamir, and Adleman in 1978 [3]. The process is as follows.

1. B first chooses two relatively large primes for security. Usually two 200-300 digit primesare sufficient. B then calculates the two numbers N = pq and ψ(N) = (p− 1)(q − 1).

2. B then chooses a natural number that is coprime with ψ such that the number (call ite) is less than ψ. Furthermore, also calculate a natural number (call it d) such thatd× e ≡ mod(ψ(N)).

3. Make public the numbers N and e.

In order properly send a message to B, A needs to carry out a few necessary operations.He first needs to encode the message using numbers less than or equal to the number N. Athen sends the numbers M c(modN) and sends to user B each number M. As for decodingthe message, B simply calculates (M c)d(modN). B then obtains exactly M because of theFermat-Euler Theorem stated below:

Theorem 1.1. (M c)c ≡M (modN)

The public-private double key method using prime numbers is actually a very securecrytography methodology! For example, in order to decode the message as a third party,you would need to be able to calculate ψ(N). This is equivalent to factoring very largenumbers. As there is no very simple way to factor large numbers, even supercomputerstrying to crack a given algorithm using prime numbers would take years. Now that I havegiven you a taste of the applications of prime numbers, let us understand the history of theprimes in more detail.

2 Primes during the Greek Golden Age: Euclid to Er-

atosthenes

In this section, I will talk about the beginnings of prime numbers and the important theoremsthat were going on at the time.

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2.1 Early Prime Theory: Ancient Egyptians and the PythagorasSchool

It is largely accepted that prime numbers were first studied by the Ancient Greeks fromstarting from 500 BC - 300 BC. However, many scholars believe that the concept of primesexisted even during ancient Egyptian times, around 4000 years ago. In particular, one ofthe earlier written records of Egyptian manuscripts was the Rhynd Mathematical Papyrus.Interestingly enough, on this papyrus, Egyptians expressed the fraction 2/n as sums of unitfractions for n up to 11. In particular, it is largely assumed that the Egyptians understoodbasic primality and realized that it was harder to express the fraction 2/p as unit fractionswith p denoting a prime number. In fact, the Egyptians seemed to use the below formula tofurther express 2/p for low numbers of primes (i.e. 2, 3, 5, 7, 11):

2

p=

1

(p+ 1) /2+

1

p (p+ 1) /2(1)

However, prime number theory was first formally studied by the Pythagoras school whichexisted from 500 BC to 300 BC. In particular, Pythagoras and his students were interestedin mystical and seemingly magical properties of numbers. In determining the theory ofnumbers, they also understood basic primality, or determining whether an input is prime.In particular, the group made contributions in the theory of perfect numbers and amicablenumbers. Formally, perfect numbers are described as natural numbers whose proper divisorssum to the number itself. An example of a perfect number would be the number 6 becausethe proper divisors of 6 are 1, 2, and 3 (and 1 + 2 + 3 = 6). Furthermore, amicable numbersare formally defined as pairs of natural numbers where the proper divisors of one numbersum to the other number, and vice versa. A good example of this is the amicable pair 220and 280.

What is interesting is that during this time, the Greek mathematicians conjectured arelationship between certain prime numbers (called Mersenne prime numbers) and the formof perfect numbers. It is particularly interesting because the theory behind perfect numbersis still not well known. We still don’t know if there are any odd perfect numbers or if thereare even an infinite number of perfect numbers! To properly understand the below theorem,it is helpful to understand what a Mersenne prime is.

Definition 3. A Mersenne prime is a prime number of the form M = 2n − 1 where n iseither a composite number or a prime number.

It is usually understood that the only way a Mersenne prime can actually be prime iswhen the number n is a prime itself. Therefore, without loss of generality, we can replacethe n with the character p, denoting primes. The following theorem was conjectured duringthe time of Pythagoras, but it wasn’t until the time of Euclid that the theorem was actuallyproved:

Theorem 2.1. If a number is a Mersenne prime or is equivalent to 2p − 1, then thenumber 2p−1 (2p − 1) is a perfect number and all perfect numbers have the same form.

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2.2 Euclid and his Contributions to Prime Number Theory

The result from theorem 2.1 was particularly important and gave mathematicians a stronginterest in number theory. How can prime numbers, so unique in their properties, make upa number so perfect that its divisors sum to the number itself?

However, later on, a greater mathematician came along named Euclid (325 BC - 265 BC).Euclid made very important contributions to the field of geometry. In fact, his most famoustext The Elements was used as the primary mathematical geometry textbook for thousandsof years until the early 20th century! Not surprisingly, he is often called the father ofgeometry [4]. Perhaps his most famous contribution to geometry was his development ofvarious inferred values for what is commonly called today as Euclidean Geometry from ajust a minor set of axioms.

2.2.1 Euclid’s Proof of the Infinitude of Prime Numbers

Euclid also made many important contributions to prime number theory. In fact, he provedone of the properties of primes that relates directly to the prime number theorem conjecturedin the 17th century AD, which is that there are infinitely many prime numbers. What isinteresting about this theorem is that it was the first documented proof by contradiction. Amodified version of the proof of Euclid’s theorem is shown below [5]:

Proof. Let us suppose that there are finitely many primes. In other words, the primes canbe defined as being in a finite set: {p1, p2, ..., pn}. We can then construct a new naturalnumber; call this number M. We can define M as:

M = p1 × p2 × p3 × ...× pn + 1

Because M is greater than any one of pi, it is obviously not one of the primes in the finiteset. Therefore, M cannot be prime which implies that it must also be divisible by at leastone of the primes in the finite set. However, dividing M by any such pi gives us a remainderterm of 1. This contradicts our original assumption that the set of primes is a finite set.Therefore, we cannot say that the set of primes is finite. There are infinitely many primenumbers.

It is important to note that during the time of Euclid, mathematicians did not haveour modern understanding of infinity. Thus, in Euclid’s conclusions, it would have beenmeaningless to say that there are an infinite number of primes. Rather, he simply statedthat the number of prime numbers are more than any finite set of primes.

2.2.2 Euclid’s Contribution to the Fundamental Theorem of Arithmetic

Probably Euclid’s most famous contribution in the field of prime number theorem was thefundamental theorem of arithmetic. The statement of the fundamental theorem of arithmeticis given below:

Theorem 2.2. Every integer that is greater than 1 can be expressed as the product of twoor more prime numbers or is prime itself.

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This statement is simple enough, but has profound impact later advancements of numbertheory. For example, the relationship between the Zeta function and primes is proved througha restatement of the fundamental theorem of arithmetic.

Although the modern proof is more generic and much more simplified, Euclid used whatis currently known as Euclid’s lemma to prove theorem 2.2. In general terms, Euclid’s lemmais stated as:

Lemma 2.3. If a prime number p divides the product of two natural numbers a and b, then p divides a or p divides b or p divides both a and b.

In a way now, once Euclid proved the fundamental theorem of arithmetic to be true,the core importance of prime numbers was established. In essence, any natural number canbe constructed through the factorization of primes in one unique way. The impact of thefundamental theorem of arithmetic is not only mathematical, but also philosophical. Sinceany positive integer can be expressed as the product of primes, primes can be consideredthe atomic elements of numbers. By understanding how primes behave, we can understandmore about mathematics and thus, our universe.

2.3 The Sieve of Eratosthenes

From around 276 BC till about 195 BC, a mathematician named Eratosthenes arose. Eratos-thenes was actually an interesting character, and by current terms would be characterizedas a Greek ”renaissance man”. Not only was Eratosthenes a mathematician, but he wasalso a geographer, poet, astronomer, and music theorist [6]. Unrelated to mathematics, oneof his most important and long-lasting contributions to geography and navigation was hiscalculation of the circumference of the earth.

However, relating to prime number theory, Eratosthenes was famous for producing thefirst prime number algorithm for detecting small primes up to a certain number. The simplealgorithm which he produced is commonly coined the Sieve of Eratosthenes. As a testamentof the usefulness of Eratosthenes’s sieve, many computers still run this algorithm for primenumber that are relatively small (usually under 30 million). Because of the simplicity of thealgorithm, this methodology is often used to teach basic computer science students abouttime complexity of different functions. In general, the time complexity of calculating allprimes under a certain number n is O(nlog(logn)). This directly relates to the fact that theprime harmonic series asymptotically approaches a time complexity of log log n [6].

The steps needed to determine all the prime numbers below a certain value n are enu-merated below:

1. Create a list of consecutive natural numbers from the number 2 all the way until youfinal value n.

2. Initially let p = 2 be your first prime number and add it to your list of prime numbers.Mark the number 2 off of your list of numbers.

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3. Starting from p, enumerate its multiples counting to the final number n in multiplesof p while marking off the multiples. As an example, for the first iteration, you’d wantto count in multiples of 2 and marking them off all the way until the final number n(2, 4, 6, ..., n).

4. After reaching the end, go back to the beginning of the list of numbers and find thefirst number that is greater than the original p. Once you find this, add this numberto your list of primes and mark it off your list of consecutive numbers. This numberwill be your new p. Keep repeating steps 3 and 4 until there are no numbers left tomark off anymore. The resulting list of primes is all the primes under the number n.

3 The Prime Number Theorem: the revival of num-

bers after the Dark Ages

Between the Greek golden age of mathematics (around 300 BC) until the end of the DarkAges in prime number history (around 1700 AD), there were not many advancements inprime number theory primarily because of a general noninterest in the field. However, withthe first of the 18th century, mathematicians such as Legendre, Gauss, and Euler beganto revive the topic of number theory. In particular a fundamental problem began to arise:how can the distribution of prime numbers be understood? In this section, we give anintroduction of the Prime Number Theorem and its far reaching consequences.

The prime number theorem is very interesting and tells us a lot about the distributionof prime numbers. This can be related to the Riemann zeta function, which is an importanttopic in this paper. To properly understand the prime number theorem, it is important tofirst understand the prime counting function π(x). The prime counting function essentiallycounts the number of primes that are less than or equal to the number x. As an example,π(6) = 3 because there are three prime numbers less than or equal to the number 6, whichare: 2, 3, and 5. The prime counting function is primarily used as a tool to better understandthe distribution of primes. However, an alternate counting function called the Chebyshevfunction is often used in number theory proofs primarily because it is easier to manipulate.The Chebyshev function will be explained in further detail in a later section.

As a formal mathematical statement, the prime number theorem states:

Theorem 3.1. If π(x) ≤ x then x−1π(x) ln(x)→ 1 as x→∞ or π(x) is asymptoticallyequal to x

lnx.

Asymptotically equal to is a mathematical term that is best shown in example. Whentwo functions f(x) and g(x) are asymptotically equal to each other, they are shown to beso by f(x) ∼ g(x). Technically, when the two functions are asymptotically similar to eachother, the following statement holds true:

limx→∞

f(x)

g(x)= 1

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The first recorded property of π(x) was made by Euclid, although he did not explicitlymake any reference to the prime density function. In essence, Euclid’s theorem proving thatthere are an infinite number of primes is equivalent to saying that π(x)→∞ as x→∞.

Although this statement tells use the general nature of π(x), the bounds on π(x) are nottoo precise and don’t give us too much information on its behavior as x grows large. A moreprecise result was actually developed by a mathematician named Euler in 1737 which showedthat the sum of the reciprocals of prime numbers is a divergent series [7]. In particular, Eulerused a reflection of the fact that the zeta function has a simple pole at s = 1 along with thefact that the harmonic series diverges to prove that the reciprocals of the primes diverges.In other words, with p denoting the primes:∑

p

1

p=

1

2+

1

3+

1

5+ ... =∞ (2)

This relationship is actually a landmark expression because it was one of the first state-ments showing actually how fast π(x) grows as x grows large!

3.1 Gauss, Legendre, and Proofs of the Prime Number Theorem

Near the end of the 18th century, mathematicians like Gauss and Legendre hypothesizedand tested results of π(n). In particular, Legendre hypothesized that for large n, the primecounting function was asymptotically similar to the following expression:

π(x) ∼ n

ln(n) +B,B = −1.08366 (Legendre’s constant) (3)

During this time, Gauss was only 15 years old. However he, as a budding mathematician,proposed the following idea closely relating to the prime number theorem that:

π(x) ∼ n

ln(n)(4)

However, later on, Gauss refined his estimate such that π(x) was asymptotically similarto the logarithmic integral, or:

π(x) ∼ Li(n) =

∫ n

2

dx

lnx(5)

In fact, this statement is very interesting and is very relevant to the Riemann Hypothesisthat is still unsolved today. In particular, solving the Riemann hypothesis would let usquantify a remainder term for a certain value of x. The logarithmic integral is a commonlyoccurring function in both number theory and physics.

Despite both Legendre and Gauss working on the prime number theorem, it wasn’tuntil 1896 that the proof was finally developed independently by Jacques Haddamard andde la Vallee Poussin. Although those two mathematicians were accredited with proving thetheorem, Riemann, in his quest to prove the prime number theorem, actually developed manycomplex variable methods and came up with the Riemann zeta function as a tool. Although

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at first it may seem unclear how complex analysis plays a role in the distribution of complexnumbers, every proof of the prime number theorem from Hadamard up until 1949 has usedcomplex variables in an essential way. The most recent proof using Newman’s method (1980)uses the most extensive properties of complex analysis and is a huge simplifications of theoriginal proofs.

Below, I will give a short biography on one of the mathematicians who first proved theprime number theorem: Jacques Hadamard and then go on to introduce one of the mostimportant functions in analytic number theory, the Riemann zeta function.

3.2 A Short Biography on Jacques Hadamard

Jacques Hadamard, to say the least, led an extremely tragic life. However, in spite of allthe hardships that he endured, he threw himself onto mathematics and was able to discoversome important things. Jacques Hadamard was born in France in 1864. When he was just 6years old, the Franco-Prussian war began, and famine was widespread across the region [8].In fact, Hadamard’s family even had to eat elephant meat to survive from the nearby zoo.

When Hadamard was just a child, he was good and excelled in all subjects except inmathematics. In fact, Hadamard was so shoddy in mathematics that he even wrote ina journal in 1936 that ”...in arithmetic, until the fifth grade, I was last or nearly last.”However, later on, due to the influence of a few good teacher, Hadamard began to developa strong interest in mathematics. Hadamard obtained his doctorate in 1892 on functionsdefined by Taylor series. In fact, he really got interested in complex analysis and worked onfunctions of a complex variables. Hadamard was one of the first to look at analytic functions.

Despite all the success that Hadamard received in the field, many of those closest to himdied. In 1874, both of his younger sisters died. After marriage, Hadamard lost all three ofhis sons in WWI and WWII. After his grandson died in a mountaineering accident, it almostseemed as if Hadamard gave up on life, and mathematics.

Jacques Hadamard was most famous for discovering the proof for the prime numbertheorem using constraints on the Riemann zeta function. In order to have a better under-standing of how prime numbers actually work, it is important to have an understanding ofthe Riemann zeta function, which is what the next section talks about.

4 The Riemann Zeta Function

4.1 And Introduction to the Riemann Zeta Function

The Riemann Zeta function was first coined by the mathematician Riemann because he wasthe first to do extensive analysis on it. It is important to note that when relating to the zetafunction, complex numbers are represented as s = σ + it instead of the normal z = x + iy.By convention, we stick with s. The Riemann zeta function can be defined in terms of theanalytic continuation (this concept will be further explored in the next section) of the infinitesum:

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Definition 4.

ζ(s) =∞∑n=1

n−s =1

1s+

1

2s+

1

3s+ ... σ = R{s} > 1 (6)

The zeta function can also be defined in terms of an integral and the gamma function:

Definition 5.

ζ(s) =1

Γ(s)

∫ ∞0

xs−1

ex − 1dx (7)

Γ(s) :=

∫0∞e−tts

dt

tRe{s} > 0 (8)

However, what does each term in the series actually mean? Using our tools in complexanalysis, we can rewrite 1

ns as

n−s [cos(−t log n) + i sin(−t log n)]

It is extremely important for us to ask now: Whether, or for which s in the complexnumbers does the sum involving the zeta function actually converge?

Figure 2: Zeta Function for different values of s adapted from [8]

It is important to note that in the right graph of 2, we can clearly see a discontinuityat s = 1. In fact, give Riemann’s definition of the zeta function and the fact that it can beanalytically continued about the whole complex plane, the function is only non-analytic atthat single point: s = 1 which acts as a simple pole with a residue of 1. Notice that thefunction having real part of s greater than 1 does not cross the zero axis at all. We can,with Riemann to prove, show that the zeta function will not have zeros with the real partof s above the value of 1.

In contrast, if we look at the left graph, we can clearly see that the zeta function crossesthe zero axis multiple times. In fact, if one views the graph very carefully, they can probablyfind a pattern in when the zeta function crosses the zero axis. Generally, the zeta functionwill cross the zero axis periodically every negative even integer. These zeros are called thetrivial zeros and will be discussed in a future section.

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4.2 Analytic Continuation of the Zeta Function

Analytic continuation is generally a technique used to extend the domain of a given analyticfunction. This is a very useful technique actually because sometimes, functions may have apower series with a larger than expected radius of convergence, allowing us to continue thefunction in a region that was originally prohibited. In fact, Riemann showed that the zetafunction could be analytically continued about the entire complex plane with just one pointwhere the function is not defined: when s = 1. Therefore, the zeta function is a meromorphicfunction on the whole play and is analytic expect for s = 1 where it has a simple pole withradius 1.

The derivation of this result comes from 4 types of functions that play an important role.They are enumerated below:

1. Theta function: θ(t) :=∑∞

n=−∞ e−πn2t t > 0

2. Its Mellin transform:∫∞0θ(t)ts dt

ts ∈ C

3. Γ(s) :=∫∞0e−tts dt

tRe{s} > 0

4. Fourier Transforms

In specific terms, the Mellin transform g(s) of f is defined by:

Definition 6.

g(s) :=

∫ ∞0

f(t)tsdt

t(9)

An example of this would be that the Mellin transformation of e−t is Γ(s).

The important thing to know is simply the result of the 2nd Riemannian proof whichshows that the zeta function can be analytically continued throughout the complex plane.It is not that important to understand the proof behind it. However, one may ask: what isthe relationship between the zeta function and the distribution of primes? Well, the nextsection will explain.

4.3 Relationship between the Riemann Zeta Function and thePrimes

To properly understand where Euler first made his contribution in this field, it is importantto understand how Riemann often wrote the harmonic series. Riemann had a new way ofwriting the harmonic series as:

1

1+

1

2+

1

3+

1

4+ ... =

(1 +

1

2+

1

4+ ...

)(1 +

1

3+

1

9

)(1 +

1

5+ ...

)... (10)

Notice that this representation of the harmonic series is particularly insightful because ituses the fundamental theorem of arithmetic to relate the inverses of primes and their powersto the harmonic series! Similarly, Euler was able to then define the zeta function (at leastwhen σ > 1) in terms of absolute convergences:

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ζ(s) =1

1s+

1

2s+

1

3s+ ... =

(1 +

1

2s+

1

4s+ ...

)(1 +

1

3s+ ...

)(1 +

1

5s+ ...

)... (11)

Therefore, we can see that:

ζ(s) =∏p

∞∑k

p−ks =∏p

(1− 1

ps

)−1(12)

This expression is called the Euler property. Its proof only uses two major concepts:

1. Formula for infinite geometric series

2. Fundamental theorem of arithmetic

As you can see, this equation directly links the Riemann zeta function to an applicationof understanding the primes. In fact, it is primarily because of the Euler product that manymathematicians believe that embedded in the properties of the zeta function is informationabout the distribution of prime numbers.

4.4 Prime Number Theorem and its Relationship to the ZetaFunction

Although both Poussin and Hadamard developed the proof of the prime number theoremseparately, both constrained the parameters of the zeta function and analyzed when thefunction equaled 0. Interestingly enough, with the aid of complex analysis, both used as amain step of the proof that:

Theorem 4.1. ζ(s) 6= 0 ∀ complex values s = 1 + it where t is greater than 0.

In fact, with these steps, both mathematicians used the Mangoldt function Λ(n) todetermine the Chebyshev Function ψ(x), which is defined as the summation for when n ≤ x.This function is actually called the second Chebyshev function, and its explicit formula isgiven below:

ψ(x) = x−∑ρ

xρ

ρ− ln(2π)− 1

2ln(1− x−2

)(13)

In fact, an equivalent statement of the prime number theorem is that the second Cheby-shev function is asymptotically equal to the function x. It is important to now note thatthe sum in the equation above refers to the sum over the nontrivial zeros ρ of the Riemannzeta function for zeros in the critical strip (this will be discussed later as well), or where0 < Re [ρ] < 1 [10]. The Chebyshev function is often used in place of the prime countingfunction π(x) simply because it’s easier to compare the Chebyshev function to a linear func-tion x than it is to involve a natural logarithmic function. In fact, many proofs in primenumber theory often use the Chebyshev function instead of the prime counting function.

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A graphical representation of the Chebyshev function is shown below. Note that at fardistances away, the function becomes indistinguishable from the function x.

Figure 3: Chebyshev Function as Sum of Nontrivial Zeros adapted from [10]

4.5 Zeros of the Riemann Zeta Function

Now that we have established the importance of determining the zeros (as it directly influ-ences the distribution of the primes), we need to know where the zeros actually are.

The functional equation of the Riemann zeta function was first developed by Riemann.Through his first Riemannian proof, Riemann not only derived a functional relationship forthe zeta function, but also determined where the trivial zeros lie and proved that the non-trivial zeros lie in the critical strip 0 < Re [s] < 1. It is not derived in this paper, but thefunctional equation of the zeta function relates ζ(1− s) to ζ(s). The functional equation isbelow:

ζ(1− s) = 2(2π)−s cos(πs

2

)Γ(s)ζ(s) (14)

In words, the functional equation relates the zeta function’s value at a particular complexnumber s to its value at the point given by reflecting s across 1/2 ∈ C.

Essentially, with a good understanding of the components of the functional equation,Riemann determined through his first Riemannian proof that:

ζ(1− s) = 0 ∀ odd integers at least 3 (15)

This implies the following relationship:

Theorem 4.2.ζ(s) = 0 & Re(s) < 0⇒ s = −2,−4,−6, ... (16)

The result of this implies that all of negative zeros of the zeta function are located atevery even negative integer. Because of the periodic nature of the occurrence of the zeros,these particular zeros are often call trivial zeros. All other zeros of the zeta function arecalled non-trivial zeros. A visualization can be seen below. The shaded region is, again,called the critical strip. Riemann’s proof shows us that all non-trivial zeros are located inthis particular area.

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Figure 4: Graphic of Zeros of Zeta Function adapted from [10]

4.6 The Riemann Hypothesis and Future Steps

The Riemann hypothesis, conjectured by the mathematician Riemann in 1859, is one of thegreatest unsolved problems in prime number theory and in the greater subject of mathemat-ics. Simply stated, the hypothesis states that the non-trivial zeros of ζ(s) have real part 1/2.Mathematically it means that:

Proposition 4.3. ζ(s) = 0, 0 ≤ Re(s) ≤ 1 ⇒ Re(s) = 1/2.

As you can see in figure 4, all of the first few non-trivial zeros lie on the line σ = 1/2.In fact, the first 10 trillion or so non-trivial zeros all lie on the line. However, to this daymathematicians have not yet proven this property to be true for every zero [12].

In particular, proving the Riemann hypothesis inherently implies results relating to thedistribution of primes, and is thus considered one of the most important unsolved problems.In fact, it is on the list as one of the Millennium Prize Problems. Thus, anyone proving ordisproving the hypothesis would be able to claim one million dollars!

Proving the Riemann hypothesis has a large impact on understanding the bounds onthe prime number theorem better. In proving the prime number theorem, Hadamard andPoussin actually showed that the expression

π(x) = Li(x) +O(xe−a

√log x)

(17)

where the third term describes a remainder term. In 1976, a mathematician namedSchoenfeld actually restated the Riemann hypothesis. According to him, the Riemann hy-pothesis is equivalent to the following expression, giving us more information on the boundsfor the remainder term. The hypothesis is restated as:

Proposition 4.4.

|π(x)− Li(x)| < 1

8π

√x log(x) x ≥ 2657 (18)

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In general terms, knowing the behavior of the zeros of the zeta function contributes toknowing the deviations for the expected location of primes.

Given that the study of primes has come so far since the time of Pythagoras and theancient Greeks, it is not a surprise that we now know a good amount about the behaviorof primes. However, there still exists fundamental problems that we have not discoveredsolutions to yet. Yet, given the rate at which mathematics is being innovated and challenged,I believe that in the nearby future, the Riemann hypothesis and other unsolved problemslike the twin prime conjecture will be solved, and we will gain a better understanding ofnumbers, mathematics, and our universe.

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References

[1] Curgus. ”Prime and Composite Integers”. (n.d.): n. pag. Web.http://myweb.facstaff.wwu.edu/curgus/Courses/302_200902/Prime&Comp.pdf

[2] Alhanjouri, Mohammed, and Ayman M. Al Derawi. ”A New Method of Query overEncrypted Data in Database Using Hash Map”. International Journal of Computer Ap-plications 41.4 (2012): 46-51. Web.http://research.ijcaonline.org/volume41/number4/pxc3877580.pdf

[3] Lessandro. ”Prime Numbers and Cryptography”. (n.d.): n. pag. Web. http://www.math.unipd.it/~languasc/lavoripdf/R8eng.pdf

[4] Ten Facts about Euclid (n.d.): n. pag. Web. http://www.10-facts-about.com/Euclid/id/382

[5] Why are there infinitely many prime numbers? (n.d.): n. pag. Web. http://www.math.utah.edu/~pa/math/q2.html

[6] ”Eratosthenes of Cyrene”. N.p., n.d. Web. 25 Nov. 2014. http://www-history.mcs.

st-andrews.ac.uk/Biographies/Eratosthenes.html

[7] Maynooth. The Prime Number Theorem. Maynooth College, 1982. Web. http://www.math.uiuc.edu/~r-ash/CV/CV7.pdf

[8] Jacques Salomon Hadamard. Hadamard Biography. N.p., n.d. Web. 25 Nov. 2014. http://www-history.mcs.st-and.ac.uk/Biographies/Hadamard.html

[9] Vericat, Fernando. ”A Lattice Gas of Prime Numbers and the Riemann Hypothesis”.Physica A: Statistical Mechanics and Its Applications 392.19 (2013): 4516-522. Web.

[10] ”Mangoldt Function”. from Wolfram MathWorld. N.p., n.d. Web. 25 Nov. 2014. http://mathworld.wolfram.com/MangoldtFunction.html

[11] ”Riemann Zeta Function”. Wolfram MathWorld. N.p., n.d. Web. 26 Nov. 2014. http://mathworld.wolfram.com/RiemannZetaFunction.html

[12] Problems of the Millennium: The Riemann Hypothesis. (n.d.): n. pag. Web. http:

//www.claymath.org/sites/default/files/official_problem_description.pdf

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