The Prime Theorem Trishla Shah Prime

Prime numbers have always been of great fascination to people, whether they were mathematics or not. Nowadays, a is defined as any that can be divided only by 1 and itself and have no remainders. The first few prime numbers are 2, 3, 5, 7, 11. While it is a common misconception that 1 is also a natural number, it is mathematically impossible to have that. 1 is only divisible by 1, but in order to fulfill the requirements, it has to be both divisible by 1 and itself. Itself, being 1, cannot account for both numbers in this case. The largest prime known today has over 17000000 digits. However, we have not always known all this about primes. It seems like such a simple concept to us today, but it has been studied extensively throughout human history in various parts of the world and throughout all time periods. One question that was bound to come up was the idea of continuity. As numbers grow bigger the factors are increasing so the frequency of primes must go down. From 1 to 100 there are 25 primes. However from 101 to 200 there are 21 primes. As numbers get larger, the distance between primes gets larger as well.

Do primes go on forever? The answer seems to be yes, but proving that is something entirely different. This is the basis of the Prime Number Theorem. While the first proof of it came out in 1949, this question has intrigued and captivated since the beginning of time.

The definition of a prime number was defined in the Fundamental Theorem of Arithmetic. While the Theorem comes closely from Euclid’s Elements book series. Books VII to IX are about basic results in the theory of numbers. The Fundamental Theorem of Arithmetic is not fully written in the books, however there are two very important propositions that have a very close connection to it. They are known as VII.30 and VII.31 in Book 7. Then IX.14 is a uniqueness theorem in Book 9. The Fundamental Theorem of Arithmetic follows directly from these.

VII.30. If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers [i.e., if a prime number c measures ab, then c will measure a or b, where ”measure can be translated as ”divide, although repeated subtraction would be nearer to the spirit of the Greek word].

VII.31. Any is measured by some prime number.

From these two, the Fundamental Theorem of Arithmetic follows. The theorem states that a natural number p > 1 is called a prime number and its only are 1 and itself. It is easily found that 2 is the only even prime in the world, as any other even number would be divisible by 2 as well. It also goes on to say that any natural number that is not prime, such as 4, 6 and 8 is called a composite number. Number are called coprime is the only number that divides the two numbers is 1.

Numbers are coprime if the gcf (a.b)=1.

One of the most interesting things to come from this, aside from the Prime Number Theorem, is Goldbach’s conjecture. Christian Goldbach was a prominent in the 1700’s. He wrote in a letter to that ””at least it seems that every number that is greater than 2 is the sum of three primes. Euler went on to rephrase this asserting that ” all positive even greater than 4 can be expressed as the sum of two primes. This bold claim was left unproven until 1973 by a Chinese mathematician, Chen Jingrun. He even went on to add that any large even number is also at most the sum of a prime and a product of at most two prime numbers. Euclid was also able to prove that if every number in the form of 2n − 1 is a prime number then every number in the form of

2n−1(2n − 1) is a .

With all these advancements and interest in primes, it is clear that they have been studied for a long time. Ever since their first discovery with the ancient Greeks. They are recorded as the first ones because their finding were written down and managed to survive. It is very much possible that there was someone before them, who is unfortunately robbed of all credit. There is some current debate, a dissenting opinion in fact, saying that the ancient Egyptians also discovered and worked with prime numbers. Nonetheless, in Greece there were schools dedicated to the advancement of mathematical knowledge. One of these was where Pythagoras researched. He is well known for the pythagorean theorem, which states that the square of two sides of a right triangle are equal to the square of the third, and longest side. This is a crucial part of mathematics today. However, the greeks themselves were more interested in both perfect numbers and . Perfect number are those that are equal to the sum of all of their factors. The first perfect number is 6, as 1+2+3=6. The next perfect number is 28, and the gap between perfect numbers increases as they grow. Amicable numbers are pairs in which the the sum of the divisors of one equal the second number. The smallest pair is 220 and 284, in which the factors of 220 are 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284 and 284, where 1 + 2 + 4 +71 + 142 = 220. These discoveries started to pave the way for the eventual discovery of prime numbers themselves.

The next step in this was created by Eratosthenes. It was called the Sieve of Eratosthenes, and it is a way to find every prime in a set of numbers. It is very simple to do, however when the range gets rather large, it becomes a very painstaking task. First, write out every number in the range starting with 2 as the smallest number. For this example we will do from 2 to 50.

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Then start with 2, and cross out every second number from then on.

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Find the next non-crossed out number, 3 in this case, and cross out every third number from there on. If the number is already crossed out, skip it and go to the next one.

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Continue this until there are no numbers left that have multiples left, and all that remains are the list of all the prime numbers in that range.

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

While this method works entirely, it is not plausible to keep doing this as the numbers get larger and larger.

In fact, as the numbers get larger and larger, testing them to see if they are actually primes becomes more and more time consuming. recognized this and came up with he Little Theorem. It states that  p−1  if if there are two numbers a and p, and p is a prime number, then ax p will always have the remainder of 1. This became a crucial tool for future mathematicians to figure out the next so called prime number is

2 actually a prime number.

Through all this, there were any efforts to try and find a formula for prime numbers. One of them is as Fermat’s primes. He put forth the equation 22n + 1 However, Euler later found that certain numbers in that sequence are actually composite, such as 223 −1. This was a solid blow against the idea that all the numbers in 22n −1 were prime. Next came French mathematician Marin Mersenne. There was another equation during this time period to solve for all primes, but he showed otherwise. He was able to prove that the equation

2n − 1 did not account for every prime number, and in fact was only truly valid between the numbers n=2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257. Any n larger than 257 would be composite, not prime. Mersenne primes were names after him where such equation followed 2n − 1 as long as n is a prime number.

Next in line was Friedrich Bernhard Riemann in the 19th century. He put forth a hypothesis which was later known as the Riemann hypothesis. It is a crucial part of getting to the Prime Number Theorem and understanding prime numbers better, as it shows the distribution of prime numbers as a direct result.

∞ X 1 ζ(s) = ns n=1

Along with the help of Euler, the Euler-Riemann Zeta conjecture is an infinite series over which all prime number exist.

∞ X 1 Y 1 = ns 1 − p−s i=1 p

Since the left hand side is an infinite sum, the right hand side must also go towards infinity, and therefore shows that the distribution of all prime numbers must also diverge to infinity. The hypothesis is a conjecture stating that the Riemann zeta function is zero partly when ζ(s) is a negative even number. These are considered to be trivial zeros. However, the other numbers that would produce the Zeta function to be 0 are considered to be non trivial, and states that ”The real part of every non-trivial zero of the Riemann zeta function is 1/2.”

If this is correct then, there is a critical line, where all the non-trivial zeros lie, which are the primes in existence. Due to the importance of this, it is part of Hilbert’s eighth problem in David Hilbert’s list of 23 unsolved problems; it is also one of the Clay Mathematics Institute’s Millennium Prize Problems.

The next big player was Carl Friedrich Gauss, a German mathematician from the 9th century. He was considered to be a prodigy, as he was able to find a pattern in the prime numbers at the young age of 15. It was truly a spectacle, considering he came from a poor, working class family and worked his way up to being called the Prince of Mathematics by his death. Using a graph of prime numbers, he noticed that if they were increased by 10 then the probability of primes reduced to 2 in that instance. His method was turned into the formula

n π(x) ∼ log n

3 This was later even more simplified to

π(x) ∼ Li(n) where

Z n d x Li(n) = 2 lnx

However, he was unable to prove this before his death. The next step in this progression came with Peter Gustav Lejeune Dirichlet in 1838 when he was able to expand on Gauss’s statement. In his theorem on arithmetic progressions, he states that when there are two positive , there exists infinitely many primes in that outcome. From that, he output this equation

a + nd where a and d are coprimes and n is a positive number.

The next attempt to prove this was by Pafnuty L’vovich Chebyshev in his two papers in 1848 and 1850. He tried to prove the asymptotic law of distribution of prime numbers. He did this using the ζ(s) function, but he was only able to prove a slightly weaker version of the asymptotic law, stating that if the limit of (x)/(x/log(x)) as x goes to infinity exists, then it is equal to one. However, he was able to fundamentally prove that he ratio is bound above and below by two explicitly stated constants near 1, so long as x is large enough. He also determined the sum of prime numbers p that can be less than or equal to x. X θ(x) = log p p≤

His second relation is also used extensively in proving the prime number theorem X ψ(x) = log p pk≤x

This function is used to calculate the sum of all powers of prime numbers of pk ≤ x

While he was not able to prove the Prime Number Theorem, the estimates he made were strong enough to prove Bertrand’s postulate stating that there exists a prime number between n and 2n for an integer n ≥ 2.With all this build up, the stage was set for the first proof to come out.

While Atle Selberg is credited for a proof of the Prime Number Theorem, he was not the first person to provide a proof of the prime number theorem. His was just the first one that was considered elementary. The original proof comes from the genius of Jacques Hadamard and Charles Jean de la Valle-Poussin in 1896. They solved it independently from each other. However, these proofs were highly complex and required complex analysis. Hadamard proved the Prime Number Theorem in 1896 saying that the number of primes ≤ n tends to ∞ as n . De la Valle-Poussin stated separately that π(x), the number of primes ≤ x, tends to x as x tends ln(n) loge x to infinity. With the combined efforts of complex analysis of the Rienmann hypothesis and the Chebyshev’s function they were able to prove the prime number theorem. However, due to the nature of it it is considered to be the more analytical one of the two. In the article Prime Number Theorem by D. Zagier, is it also stated that Donald J. Newman also provided a short proof of the Prime Number Theorem and his had an even simpler version of the Tauberian argument crucially needed in the Prime Number Theorem.

4 These days looking for the next prime is done by the computers, not a mathematician hunched over using the Sieve of Eratosthenes for hours. This first began in 1951 with two mathematicians, Jeffrey Charles Percy Miller and Donald J Wheeler. They were able to find various new primes, and broke the record with one with 79 digits.

Merely a year later, five new Mersenne primes were found by Raphael Robinson using the Standard Western Automatic Computer, which was one of the most advanced computers of that generation. The largest of these contained 687 digits. It was clear that the newer primes were going well beyond what the human mind could come up with entirely. There is also some speculation about this time if Alan Turing built the very first computer solely designed to find the next prime number. Then in 1996, a new movement, a revolutionary one, was founded. Computers were already at a point where they could find the next prime, all the needed was more power. The Great Internet Search allowed anyone would wanted to help in the search sign up for this program that would allow their computer to aid in the search for the next time. It was clearly working as before this program started 34 was the largest Mersenne Prime. On this date, it is currently up to the 48th Mersenne Prime. It is a honest wonder what computers with human ingenuity can do.

Selburg and Erd¨os

In 1921, Hardy delivered a lecture to the Mathematical Society of Copenhagen. He asked: No elementary proof of the prime number theorem is known, and one may ask whether it is reasonable to expect one. The mathematical world was understandingly shocked then in 1948 when Paul Erd¨osannounced that he and Atle Selberg had found a truly elementary proof of the prime number theorem which used only the simplest properties of the logarithm function. This lead to one of the biggest disputes in mathematical history. Selberg is said to have found a simpler way to do what Erd¨oshad shown him and put forth his paper, An elementary proof of the primenumber theorem, in the Annals of Mathematics. These papers were brilliantly reviewed by A.E. Ingham. On the other hand Erd¨osthen put forth his paper,On a new method in elementary which leads to an elementary proof of the prime 3 number theorem, the Bulletin of the American Mathematical Society informed Erd¨osthat the referee does not recommend the paper for publication. Erd¨os immediately withdrew the paper and had it published in the Proceedings of the National Academy of Sciences.

Selberg and Erd¨oscould not have been more different either. While they were both born in highly math centered household, they grew up in very different lifestyles. Erd¨oswas born on March 26th, 1913 in Budapest, Austria- Hungary. His mother and father were both math teachers and they always pushed him to pursue mathematics partly because he was their only child to survive past adulthood.. The stories go on to say that even at the age of 5 he was able to ask people for their age and mentally calculate the number of seconds they existed on the Earth. He was incredibly skilled with mental math. At the age of 5 he was able to show that 250 less than 100 was minus 150. He had an incredibly aptitude for mathematics and was able understand concepts such as negative numbers from such a young age.

Selberg on the other hand was born on June 14, 1917 in Langesund Norway. His mother was a teacher and his father was a prominent mathematician as well. Unfortunately, he was born in the era of World War Two. His forced his work to be done in much seclusion, which may have impacted his personality to be rather reclusive. He was never big on collaborations. From a young age he also showed a massive mathematical aptitude. At the age of seven he showed that the difference between every consecutive squares is always an odd number.This may have played a part as to why he decided to publish on his own, rather than with Erd¨os. When he was doing his own work, he was unable to share it with it anyone, due to the fact that Germans had already invaded his home. It was a true shame. His was a huge impact on his working style.

While Erd¨osloved working with others, Selberg did not. Erd¨oshad over 500 coauthors on his works through the years. He was not the type to shy away from sharing credit and he was said to be a very pleasant man.

5 Selberg, due to his isolated childhood hard a hard time working with others. It is not that he considered them to be distrustful, he just did not see the need to have anyone else’s help. He was a very independant person, something he had to be due to World War Two. Working along then, made him work alone for the rest of his life. He has only ever had one other coauthor, Sarvadaman D. S. Chowla. Selberg was rather fond of Chowla and the story goes that Chowla was highly nervous around him in the beginning, but in the end they were the closest of friends. One story states that a post doc once asked Chowla for help to evaluate an integral and Chowla said he would have the answer in half and hour for five dollars. He did in fact return with the answer and when asked how he did so, he responded that he took it to Selberg who answered it in a heartbeat.

It is easy to say that they are so different solely because of their preferences, but according to Tim Gower’s essay The Two Cultures of Mathematics, they have entirely different mindsets on mathematics itself. Gower says that there are two driving reasons behind the furthering of mathematics itself:

1. ”The point of solving problems is to understand mathematics better.”

2. ”The point of understanding mathematics is to become better able to solve problems.”

It is clear to see that Erd¨osis in the first type and Selberg is the second type. Erd¨osalways preferred to work with more people, believing that the more people he worked with, the better he would be at understanding the ideals. the more people he worked with would also lead to the further sharing of ideas to the mass public and other mathematicians as well. There is no doubt that Erdo cared more about understanding mathematics than just being able to solve problems. Selberg on the other hand, cared more about furthering his own knowledge. Aside from Chowla, he was not taken to anyone, and he cared not much about others, as shown with his dispute with Erd¨os.By using his mind, and his mind alone, he is able to become a better solver, because he has to solve everything by himself all alone. He was able to develop his own mind, at the price of being a loner.

In terms of work, Erd¨osworked with combinatorial and probabilistic number theory, combinatorial geometry, probabilistic and transfinite combinatorics and graph theory. Selberg has dealt with positive density of zeroes of the Zeta-function which has lead Weyl be fond of Selberg. This is very important during the Prime Number Theory Controversy. Selberg was best known for his analytic number theory. It is important to note that while Selberg did get the Field’s Medal, it was not solely for the his proof on the Prime Number Theorem. He produced many brilliant works during his lifetime, and this was just one that put it over the edge. He was a brilliant man and he did deserve that Field’s Medal. It is also important to note that even with everything that both men did, not a simple mention of the Prime Number Theorem is ever made without both Erd¨osand Selberg mentioned. These men will go down in history in mathematics.

Outline of Selberg’s Proof

Selberg’s Proof of the Prime number theorem consists of four main parts.

Part A

In Part A, Selberg begins by giving an explanation and the proof of his formula:

X x Θ(x)logx + Θ logx = 2xlogx + (x) p p≤x

If we go further into the depth of this equation, we can see that Selberg came up with a function that included the M¨obiusfunction. This function was as follows:

6 x λ (x) = µ(d)log2( ) d d

Selberg defined this new function that was able to start the proof for his Prime Number Theorem.

X Θn(x) = λd(x) d/n

X Θn(x) n≤x yields two seperate results. The first is

X Θ(x)logx + logplogq + (x) pq≤x

The second result that is yielded is by using Dirschlet’s estimate. This result comes out to be

2xlogx + (x)

Since both are results of the same summation, we can see that both are equivalent. If we return to the first result given by the summation, we can see that

X X X Θ(x)logx + logplogq + (x) = log2p + logplogq + (x) pq≤x pq≤x pq≤x

Now to understand how this solution was reached, the next portion will define how

X log2p = Θ(x)logx + (x) pq≤x

For this, it can be said that

X X log2p = (Θ(n) − Θ(n − 1))logn pq≤x n≤x

It can also be said that if n is prime, n − 1 is not. This is always be true because n − 1 will be an even number, and even numbers can never be prime. We can see an example of the above solution by looking at the equation if n = 5. If n = 5, then n − 1 = 4. This would mean that:

Θ(4) = log(2) + log(3)

Θ(5) = log(2) + log(3) + log(5)

7 this means that Θ(5) − Θ(4) = log(5). If one were to look at this example in a conceptial manner, it could be seen that (Θ(n) − Θ(n − 1))logn = log2n and because n is a prime number that can also be represented as p, (Θ(n) − Θ(n − 1))logn = log2n = log2p.

If n is a composite number, such as n = 6, it can be shown that Θ(5) = log(2) + log(3) + log(5) Θ(6) = log(2) + log(3) + log(5) this means that Θ(6) − Θ(5) = , which means it is log(6). If the previous two results are combined, it can P 2 be seeen that n≤x(Θ(n) − Θ(n − 1))logn is equivalent to log p if n is prime and is otherwise. If the original definition is revisited and x = 4 is plugged in, one achieves the following:

(Θ(1) − Θ(0))log(1) + (Θ(2) − Θ(1))log(2) + (Θ(3) − Θ(2))log(3) + (Θ(4) − Θ(3))log(4)

After a series of integration by parts, the following solution is determined:

−Θ(0)log(1) + Θ(1)(log(1) − log(2)) + Θ(2)(log(2) − log(3)) + Θ(3)(log(3) − log(4)) + Θ(4)log(4)

X = Θ(n)(logn − log(n + 1)) + Θ(m)logm n≤m−1

Note: m is the integar part of xm = [x]. P After the summation, it can be determined that n≤m−1 Θ(n)(logn − log(n + 1)) = (x) and Θ(m)logm = Θ(x)logx + (x)

This relation is shown to be the breakthrough discovery that Selberg uses as a the foundation for the rest of his proof of the Prime Number theorem.

Part B

The main outcome derived from Part A was Selberg’s relation:

X x θ(x)logx + θ logp = 2xlogx + O(x) p p≤x

This relation yields for the function R(x):

X x R(x)logx = − R logp + O(x) p pq≤x

By applying a couple partial summations to Selberg’s relation, we can also obtain:

X X logplogq   x  logp + = 2x + O logpp logx p≤x pq≤x

8 This relation becomes useful for two reasons. The first is that by applying Mertens relation we get:

X logplogq  x  R(x)logx = R + O(xloglogx) log(pq) pq pq≤x

The second is that when we combine the two expressions for R(x)logx, we get:

    X x X logplogq x 2 |R(x)| logx ≤ logp R + R + O(xloglogx) p log(pq) pq p≤x pq≤x

After From there, after applying one more partial summation and simplifying, we derive the inequality:

  X x |R(x)| logx ≤ R + O(xloglogx) n n≤x from Selbergs formula.

Which can also be rewritten as:

    1 X x loglogx |R(x)| ≤ R + O x logx n logx n≤x

Part C

After acquiring the following new inequality, we can establish the existence of a constant K > 0 so that for K K δ any δ > 0 and any x > e δ , the interval (x, xe δ ) contains a sub interval of the form (y, ye 2 ) so that for every z in this last interval, we have:

R(z) < 4δz

Part D

Showing that if a < 8 is a positive number then the inequality R(z) < ax for x large enough, leads to the inequality:

 2  a R(z) < a 1 − x 300K for x large enough.

With that estimate, he shows the elementary version of the proof of the Prime Number Theorem.

9 Proof of the Basic Formulas

We write, when x is a positive integer and d a positive integer,

x λ = λ = µ(d) log2 d d,x d and if n is a positive integer,

X θn = θn,x = d/nλd

Then we have,

 2 log x, for n = 1,  2 log p log x , for n = pα, α ≥ 1, θ = p n α β 2 log p log q , for n = p q , α ≥ 1, β ≥ 1,  0, for all other n

Then if n = p1p2...pk,

θn,x = θ n ,x − θ n , x pk pk pk

From that, the rest of the equations follow.

2 x  P It is clearly shown that λd,x = µ(d) log d and if θn = θn,x = d/nλd P 2 x  Then, θn = d/nµ(d) log d From that the conditional equation above is made for the first 3 conditions. For the fourth one, it is said in a previous part that it is to be assumed that n can be square free, therefore the final condition of θ = 0 can be taken from that ideal that n = p1xp2...pk.

Work Cited

1) http://mathworld.wolfram.com/PrimeNumberTheorem.html

2) http://citeseerx.ist.psu.edu/viewdoc/pdf

3)http://www.ams.org/notices/200906/rtx090600692p-corrected.pdf

4)http://www.math.columbia.edu/ goldfeld/ErdosSelbergDispute.pdf

5) http://kobotis.net/math/MathematicalWorlds/Fall2015/131/Projects/PNT/SelbergPNT1949.pdf

6)http://kobotis.net/math/MathematicalWorlds/Fall2015/131/Projects/PNT/PNT.pdf

7)http://www-history.mcs.st-and.ac.uk/HistTopics/Primenumbers.html

8)http://people.math.umass.edu/ tevelev/4752014/laitinen.pdf

10 9)https://primes.utm.edu/notes/proofs/infinite/

10)https://primes.utm.edu/notes/faq/WhyCalledPrimes.html

11)https://en.wikipedia.org/wiki/Primenumbertheorem

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