In 1950 Watson and Crick understood that genetic information is coded in the DNA in the form of a double helix. Their original paper is available at: Watson, J.D. & Crick F.H.C. A Structure of Deoxyribose Nucleic Acid. Nature, 1953. 171:737-738 and earned them a Nobel Prize. James Watson also wrote a book titled “The Double Helix” which provides a highly biased view of the genesis of this discovery, which eulogizes the author and severely underplays the role played by the brilliant chemist Rosalind Franklin, is provided in the book. There is a YouTube documentary called “The Secret of Photo 51” that provides some additional background for the facts about her role in the discovery of the double helix:

Once scientists knew that DNA was the way information was maintained and transmitted, the next question was: How can one read the code? What is the code to translate the DNA sequence into proteins? In other words, “What is the genetic code?” The story of how the genetic code was deciphered is quite fascinating and involved the joint efforts of biologists, biochemists, chemists and physicists. Below I will discuss some of the salient aspects of how the code was deciphered. More information is at:

https://profiles.nlm.nih.gov/spotlight/sc/feature/deciphering

https://en.wikipedia.org/wiki/Nirenberg_and_Matthaei_experiment

https://www.acs.org/content/acs/en/education/whatischemistry/landmarks/geneticcode.html

Knowing who deciphered the code and how they did it is historically important because many people falsely believe that Francis Crick or James Watson deciphered the genetic code. This is incorrect. The major credit for cracking the code goes to Marshall Nirenberg, who discovered the first codon (three letters on DNA that code for an amino acid) in 1961. The complete code was deciphered by 1966 and Nirenberg shared Nobel Prize in Physiology or Medicine in 1968 for his seminal work on the genetic code. He shared the award with Har Gobind Khorana (University of Wisconsin), who mastered the synthesis of nucleic acids, and Robert Holley (Cornell University), who discovered the chemical structure of transfer-RNA. Collectively, the three were recognized “for their interpretation of the genetic code and its function in protein synthesis."

It was known that proteins in organisms were composed of 20 amino acids. Since DNA is a sequence made from 4 nucleotides (A: Adenine, C: Cytosine, G: Guanine and T: Thymine), it was clear that using two nucleotides to code for an amino acid is inadequate. Two nucleotides could only code for 4² = 16 amino acids. At a minimum, each amino acid must be represented by at least 3 nucleotides. Two physicists, George Gamow and Francis Crick, played a key (albeit futile) role in the search for this code. George Gamow was a theoretical physicist and cosmologist, who was intrigued by the challenge of deciphering the code. He believed that the best way to move forward to find the code was through collaborative effort, with the best scientists from different fields sharing their ideas and results. In 1954, he founded the "RNA Tie Club", with the aim of solving “the riddle of the RNA structure and to understand how it built proteins." The club had 20 regular members, one for each amino acid, and four honorary members, one for each nucleotide (see image above). The members all got woolen neckties designed by Gamow with an embroidered green-and-yellow helix. Many prominent scientists were members, eight of whom were or would become Nobel Laureates. Each member was identified by an amino acid. For example, James Watson was PRO for proline, Francis Crick was TYR for tyrosine and Sydney Brenner was VAL for valine.

Here is an image of some of the members:

Gamow’s genetic codes: Gamow came up with an ingenious (but wrong) genetic code, which he published as a paper in Nature: “Gamow, G. Possible relation between deoxyribonucleic acid and protein structures. Nature, 1954. 173, 318.” He proposed an overlapping triplet code. His reason to propose it was that it allows the highest information density, which is much beloved of cryptographers. In his idea, the double stranded helix of DNA acts as a direct template to translate nucleotides amino acids. So, for example, the sequence ATGCTA would contain the triplets ATG, TGC, GCT, CTA each of which would code for a different amino acid. The figure below is from his letter to Watson explaining his idea.

In Gamow's “diamond code”, double-stranded DNA are a direct template for assembling amino acids into proteins. The various combinations of bases along one of the grooves in the double helix form distinctively shaped cavities into which the side chains of amino acids fit and each cavity attracts a specific amino acid. Each of the cavities was bounded by the bases at the four corners of a diamond. When all the amino acids were lined up in the correct order along the grooves, an enzyme would come along to stitch them together.

As shown in the figure above, if the DNA helix is oriented vertically, the bases at the top and bottom corners of a diamond are on the same strand and are separated by a single intervening base; the left and right corners of the diamond are defined by that intervening base and by its complementary partner on the opposite strand. There are 64 possible codons, but not all of them are distinct. Gamow noted that most amino acid side chains are symmetrical, and he therefore postulated that the diamonds could be flipped end-for-end or flopped side-to-side without changing their meaning. For example, the triplet CAG becomes GAC when it is flipped end-for-end, and both of these codons must specify the same amino acid. Flopping CAG side-to-side changes the middle A into a complementary T, so that CTG and GTC are also members of the same family of equivalent codons. When all such symmetries are taken into account, how many distinct codons remain? Gamow counted them up and found the answer is 20—just the magic number he was looking for. In this code, certain amino acids were required to always appear next to each other.

To see more details about how this code gives 20 amino acids and what the restrictions are, look here.

Unfortunately for Gamow, even with the sparse protein sequence data available in the mid-1950s, Crick was able to show that the diamond code was ruled out by the experimental evidence.

Undaunted by this failure, Gamow proposed a "triangle code" that was also overlapping but had different constraints. In this code the 64 possible triplet codons again sorted themselves into 20 families.

Later Gamow suggested yet another overlapping-triplet code with an even simpler description: Each codon is defined entirely by its base composition, ignoring the order of the bases within the codon. Thus ACT, ATC, CAT, CTA, TAC and TCA are all members of the same codon family and specify the same amino acid. Remarkably, the number of codon families in this scheme again turns out to be exactly 20 (see here for details).

Still more overlapping codes came from Gamow and his friends. Richard Feynman also had a hand in working out one idea. Edward Teller proposed another—a fairly funky scheme in which each amino acid is specified by two bases in the DNA and by the previous amino acid. But overlapping codes were coming to the end of their string.

Patterns of mutations were one source of doubt. With an overlapping code, changing a single base in the DNA could alter three neighboring amino acids, but protein sequence data were starting to show instances of single amino acid replacements. It also turned out that there were known patterns of amino acid repetitions that the diamond code could not produce. For example, Sanger had figured out the amino acid sequence of insulin and it showed that any amino acid can sit next to any other. So, the “diamond code” and all overlapping codes were “beautiful but wrong” ideas. Then came a definitive proof. Sydney Brenner analyzed all the known protein sequence fragments and found enough nearest-neighbor correlations to rule out every possible overlapping code.

Crick’s Comma Free Codes: By the later 1950s, there was growing support for the idea of messenger RNA or mRNA, which is a single-strand molecule acting as an intermediary between DNA and the protein-synthesizing machinery. At the same time Crick was formulating the "adaptor hypothesis," the idea that amino acids do not interact directly with mRNA but are carried by small molecules that recognize specific codons. (Today, the adaptor molecules would be called transfer RNA or tRNA.) The codons were by then thought to be non-overlapping triplets of bases.

The process of gene expression was imagined as going something like this. First the appropriate segment of DNA was transcribed into mRNA by like replication. This was thought to be done by blind copying from DNA (with T replaced by U), without regard to the meaning of the sequence. Then the mRNA stretched out in the cytoplasm of the cell with its long row of codons exposed like a sow's nipples. Each adaptor molecule, already charged with the correct amino acid, poked around until it latched onto the right codon. When all the codons were occupied, the amino acids were linked together, and the completed protein was peeled off the template.

The adapter hypothesis seemed highly plausible because it seemed like the kind of chemistry that living organisms do. The non-sequential pattern-matching needed to line up adaptors on the messenger RNA is vaguely like an enzyme-substrate reaction or like the binding of antibody to antigen. And yet there was a serious problem with the vision of piglets suckling on RNA: A piglet might very well wind up between nipples.

Suppose somewhere in a messenger RNA is the partial sequence ... UGUCGUAAG.... (Note that in RNA uracil replaces the thymine of DNA, and so the code is written with U rather than T.) The intended reading is ... UGU, CGU, AAG..., but the RNA molecule has no spaces or commas to indicate codon boundaries. The sequence could equally well be read as ... UG, UCG, UAA, G ... or ... U, GUC, GUA, AG.... Each of these alternatives would have a different meaning. Furthermore, in the suckling-pig model of protein synthesis, adaptor molecules that attached to the messenger RNA in different reading frames might interfere with one another and prevent any protein at all from being produced. To see more details about the code proposed by Crick look here.

The Actual Genetic Code: As experimental tools grew more sophisticated, several research groups gradually pieced together the real code in the mid 1960s. Marshall Nirenberg and Johann H. Matthaei used a "cell-free" system: in a test tube they put together all the things they thought were needed for protein synthesis – RNA template, ribosomes, nucleotides, amino acids, stabilizing agents and energy. Using this system, they made a very simple nucleic acid, composed of a chain of only one single, repeated letter U, the nucleotide uracil. Using this nucleic acid, the system produced a protein that also contained a single letter but now written in the protein language: the amino acid F, phenylalanine. By showing that a strand of U triplets was indeed the template for the amino acid phenylalanine they had cracked the first letter of the code.

While Nirenberg was presenting this result at a conference in Moscow, he got a phone call from Matthaei (still working at the lab back home), who told him that CCC was the template for the amino acid proline, represented by the letter P.

Next, Har Gobind Khorana, at the University of Wisconsin, devised precise and intricate biochemical methods to produce well-defined nucleic acids, long strands of RNA with every nucleotide in exact position. The first one he made was a strand repeating the two nucleotides UCUCUC. This translated into a strand of amino acids, reading serine-leucine-serine-leucine... Similar synthetic RNA experiments were later used to decipher the rest of the genetic code. Robert Holley was a chemist at Cornell University but he learned about protein synthesis during a sabbatical year at Caltech in California. He discovered the special type of nucleic acid called transfer RNA, or tRNA for short. In 1965 Holley was able to work out its exact structure. This was the first time anyone had established the complete chemical structure of a nucleic acid that was biologically active. tRNA turned out to be the missing molecule that Crick had proposed in his ”Adapter Hypothesis” ten years earlier.

In 1968, seven years after the first letter of the code was presented in Moscow, Nirenberg, Khorana and Holley were awarded the Nobel Prize in Physiology or Medicine.

To the dismay of the members of the RNA Tie Club, especially the physicists, the actual code resembled none of their elegant theoretical notions. As the table assigning codons to amino acids was filled in, it became apparent that the magic number 20 held no magic after all. All the clever mathematical contrivances for getting 20 amino acids out of 64 codons turned out to be figments of the human urge to find pattern, not reflections of any natural order. The "extra" codons are merely redundant: Some amino acids have one or two codons, some have four, others have six. Three codons serve as “stop signs” and one as a “start sign.” At first glance the mapping between codons and amino acids appeared arbitrary, even haphazard. To the members of the “RNA Tie Club” the actual code was a great disappointment because it seemed to be a “frozen accident”, just one among many possibilities. It seemed that once a specific code was chosen, any deviation from it was, literally, punishable by death, since the organism using it would not be able to outcompete the ones who did use it.

Why this particular code? Why did nature choose only one code and not several other equally possible ones? There seemed to be no selective reason why one code would have a selective advantage over others. But if there was no selective advantage, why didn’t several codes coexist in different organisms? Crick’s solution was that all organisms on earth descended from a common ancestor, seeded as a bacterial clone derived from a single some extraterrestrial organism. He even suggested that these bacteria were deliberately seeded by some alien intelligence from space, an idea which he called “directed panspermia”.

However, there are in fact overt patterns in the code, suggesting why it may have been chosen from amongst many possibilities, because it is one of the few codes which are best able to resist change from mutations and speed up evolution at the same time! To read more about this, you should read the chapter on “DNA” in the excellent book by Nick Lane entitled: “Life Ascending, the Ten Great Inventions of Evolution.” You might also want to read the paper: Freeland SJ and Hurst LD, ‘The genetic code is one in a million’, Journal of Molecular Evolution, 1998, 47:238-48.

This is the insight that inspired Georg Cantor (19 February 1845 – 6 January 1918) to develop a theory for infinite sets. Cantor was a German Mathematician and is considered the father of The Calculus of Infinities Although we naively use just a single symbol “∞” to represent any infinite quantity, Cantor and others after him showed that there are many infinities of different sizes, indeed an infinite number, of “infinities”. Mathematicians call the number of elements in a set the “cardinality” of the set. For instance, the “cardinality” of a pile of apples = the number of apples in the pile. Cantor invented a system to compare infinities. Here is the basic idea: We start with the set of all integers Z = {1,2,3,4…}. Mathematicians use the symbol Z for this set. We need one more definition to move on: A set is called “countably infinite” if its elements can be mapped one to one onto the integers (i.e. on to the set Z). By definition, the set Z is “countably infinite”. Following Cantor, we will use the symbol ℵ₀ (or aleph-null) to represent the cardinality of the set Z. Thus, ℵ₀ represents the “infinity” of the number of elements in the set of integers. The genius of the pairing method is that it allows us to find other sets with cardinality ℵ₀. This can be done using Cantor’s Rule: Any set S whose elements can be mapped one to one onto the elements of Z has cardinality ℵ₀. If there is no such mapping: i.e. if S has elements left over after all the integers have been used up, the cardinality of S is greater than ℵ₀.

Let us consider some examples.

Consider the set E of all positive even integers: This is easy to map to Z. The E → Z mapping is, 2 → 1, 4→ 2, 6→ 3, etc. So, these two sets are of the same size and both have cardinality ℵ_0.

Next, consider the set P of all prime numbers. We already know from an earlier post that P has an infinite number of elements. What is the cardinality of P? It is easy to see that the cardinality is ℵ₀. This is because the prime numbers can be paired one to one with the integers. Thus, we can assign the integer 1 to the first prime number (the prime number 2), the integer 2 to the second prime number (the prime number 3), the integer 3 to the third prime number (the prime number 5) and so on. Since every prime number will thus be assigned a unique integer, with none left over, we have succeeded in a mapping P onto Z. By Cantor’s Rule, these two sets are of the same size and both have cardinality ℵ₀.

The set of points on a 2-d grid (see image above). Surely, you will say, there are more grid points in a 2-dimensional grid than the number of integers. Perhaps this set has cardinality greater than ℵ₀. Alas, no. The elements of this set can mapped onto Z the integers (see image above). So, this set also has cardinality ℵ₀.

How about the set Q of all rational numbers? Remember that a rational number is a number of the form p/q, where p and q have no common factor and q ≠ 0. Surely the cardinality of Q is greater than that of Z! Alas, Q also has cardinality ℵ₀. Here is the proof:

Consider the figure above, where we have labeled the grid points by an integer as well as by two grid coordinates (n,m). The number n is called the x-coordinate. It is the horizontal distance of the grid point from the origin. The number m is called the y-coordinate. It is the vertical distance of the grid point from the origin. The quantity (x+y) is sometimes called the “Manhattan Distance” between the origin and (x,y), because it is the distance you would have to walk to get from O to (x,y) in Manhattan, New York City. Since the numbers n and m range over all the integers, the construction shown in the figure assigns the rational number (n/m) to each grid point. This proves that the cardinality of Q is also ℵ₀. The acute reader may notice that we skipped some of the grid points (the ones shown as unfilled circles). This is to avoid mapping the same rational number to two different integers. For example, the point 3:(1,2) maps the integer 3 to the rational number ½. We then skip any grid point of the form (k,2k) for k=2,3,4,… since they would all represent the rational number ½. Indeed, in the figure above, we see that the grid point (2,4) is skipped. Similarly, all grid points (k,k) for k=2,3,4,… are skipped. They would all represent the rational number 1 = 1/1, which is already mapped at O as 1:(1,1).

You might be beginning to wonder if there are any sets which are not countable? Indeed, there are: Cantor’s showed that the set R of real numbers has cardinality > ℵ₀. In fact the cardinality of just rational numbers in the closed interval (0,1) is greater than ℵ₀..

Cantor’s Diagonal Argument: The set of Real numbers in (0,1) is uncountable.

Consider the set R of all real numbers between 0 and 1 written in decimal representation. Some of the elements of R have terminating decimal representations and others have non-terminating decimal representations. However, we can always write a terminating decimal as a non-terminating one by using an infinite sequence of 9s. For example, the terminating decimal number 0.1 can be written as 0.0999…. We will always use the non-terminating decimal representation. Suppose R is countable. Then there must be a way to label the elements of R using integers. Suppose that the following is one such representation:

R = {s₁, s₂, s₃, s₄, s₅, s₆, ….}, with

r₁ = 0.1721612…..

r₂ = 0.9212014…..

r₃ = 0.0981345….

r₄ = 0.4239190…..

r₅ = 0.2959856…..

and so on. We will now construct a number r₀ which is a member of R but is not in the list above. Here is how the construction goes: r₀ differs from r₁ in its first decimal digit, from r₂ in its second decimal digit, from r₃ in its third decimal digit and so on. For the mapping shown above, one possible value for r₀ is: r₀ = 0.345171…… Since r₀ is an element of R but not in the list, we have a contradiction. If R is countable, no such r₀ should exist. Hence R is uncountable and must have more elements than the set of integers. The cardinality of R is > ℵ₀. Since this is a subset of the real number set, the set of real numbers is also uncountable.

Using similar beautiful and simple arguments, Cantor developed the theory of infinite sets. Among his many astonishing results was the proof that the cardinality of R is 2^ℵ₀. He also showed that given a set with cardinality ℵ_i, one can construct a set with cardinality 2^ℵ_i. Hence, there is an infinite hierarchy of sets with increasing cardinalities. One of his most famous conjectures was the Continuum Hypothesis, which states that there are no sets which have cardinality strictly between ℵ₀ and 2^ℵ₀. In other words, it is impossible to find a set S such that ℵ₀ < Cardinality(S) < 2^ℵ₀. Cantor believed his hypothesis was true and tried very hard to prove it, with no success. Kurt Gödel showed in 1940 that the continuum hypothesis (CH for short) cannot be disproved from the standard Zermelo–Fraenkel set theory (ZF), even if the axiom of choice is adopted (ZFC). Paul Cohen showed in 1963 that CH cannot be proven from those same axioms either. Hence, CH is independent of ZF. Both of these results assume that the Zermelo–Fraenkel axioms themselves do not contain a contradiction, which assumption is widely believed to be true. These results seem to show that under very general axioms of set theory, it is not possible to either prove or disprove the Continuum Hypothesis. However, not all Mathematicians agree. The search for a proof of the continuum hypothesis or its refutation remains an active area of research, and the problem remains the first of Hilbert’s 23 outstanding unsolved problems in mathematics.

We end with a word of caution. The ideas presented here were highly controversial when they were originally developed by Cantor and for many years afterwards. To some extent they remain controversial today. To find out more, we recommend looking here and here and here. They have a lot of information and links to papers and books which will tell you more about this fascinating field. Indeed, if a set theorist were to meet God, probably the first question she would ask would be: “Is the Continuum Hypothesis true?”

Prisoner’s Dilemma:

This is one of the most famous problems in Game Theory:

Two prisoners are separated and questioned. There is no real evidence against them so if they both do not confess, they will have to be released. This is the situation shown as AA: payoff (0,0). Of course they are not told this. Instead, they are told that if they both confess, they will both get a light sentence – BB: payoff (-1,-1). If one confesses and the other does not, then the one who confesses will be released and rewarded with cash, while the other will get a stiffer sentence - AB payoff (-2,1) or BA payoff (1,-2).

For both prisoners, Strategy B dominates Strategy A, because their payoff is better than with strategy A. Hence, in a rational world, both will choose B. However, that is worse than both choosing A, which would set them free.

Individuals rationally pursuing their interests end at a bad outcome for both.

However, as we will see, cooperation can appear if both players play many turns.

General Formulation:

The general matrix below represents a prisoner’s dilemma where

C = cooperate = do not confess, D = defect = confess, R = reward for cooperation, S = sucker’s payoff, T = temptation payoff, U = un-cooperative payoff.

This game is a Prisoner’s Dilemma when T>R>U>S and R≥(S+T)/2

Iterated Prisoner’s Dilemma:

Finite number of games: If the number of games played is finite, say 100, then at the 100 th game, there is no future so both will play D. But on the 99 th game, they will also play D since the outcome of the final game is known... and so on. Thus, all games will be DD (assuming the players are both completely rational and know there will only be a finite number of games).

2. Unknown number of games: Let p = probability that the next play will happen. Suppose the first player plays C and continues to play C until the second plays D. After that, both play D (do not cooperate).

If the second player never plays D, the second player’s payoff is:

X = R + Rp + Rp² + R p³….. = R/(1-p) Eq. 1

If the second player plays D at the m’th turn, the second player’s payoff is:

Y = R + Rp +Rp² + Rp³ + .. Rp^m-1 + Tp^m + U p^m+1+ U p^m+2 + ….

= [R(1-p^m) + (1-p)p^mT + p^m+1U]/(1-p) Eq. 2

X > Y if

R > R(1-p^m) + (1-p)p^mT + p^m+1U

which can be simplified to

p > (T-R)/(T-U) Eq. 3

In other words, if the probability to continue play is greater than some threshold, Player 2 will never play D. On the other hand, if T-R approaches zero, so that the value of mutual cooperation and the temptation to defect are the same, then p=0 and the game will end with both not cooperating!

Sequential Iterated Prisoner’s Dilemma in Practice (Axelrod, 1980) (from Wikipedia)

In 1980, Robert Axelrod, then a professor of political science at the University of Michigan, held a tournament for the prisoner's dilemma. He invited a number of well-known game theorists to submit strategies to be run by computer. The programs played games against each other and amongst themselves repeatedly. Each strategy specified whether to cooperate or defect based on the previous moves of both the strategy and its opponent. Fourteen programs were submitted and they played against each other for a long time. The average score per game was recorded.

Some of the strategies submitted were:

• Always defect: This strategy defects on every turn. This is what game theory advocates. It is the safest strategy since it cannot be taken advantage of. However, it misses the chance to gain larger payoffs by cooperating with an opponent who is ready to cooperate.

• Always cooperate: This strategy does very well when matched against itself. However, if the opponent chooses to defect, then this strategy will do badly.

• Random: The strategy cooperates 50% of the time.

The winning program (submitted by Anatol Rapoport) was Tit-for-Tat with the following simple rule:

• Start by choosing C

• Thereafter, on your turn do whatever your opponent did on his previous turn.

In other words: Tit-for-Tat or “Do unto others what they do unto you!”

Axelrod published a program that beat Tit-for-Tat and asked the CS community to try to find others. This time, sixty-two programs were entered, some of which could beat Tit-for-Tat. But Tit-for-Tat again had the highest average score against all the other programs. The reason this happened was the programs that beat Tit-for-Tat did very poorly against one another!

Axelrod analyzed the programs that did well (top 5-6 winners). They all had the following properties:

· They were Nice: Started by cooperating and never defected first.

· They were Retaliatory: Punished reliably if opponent defected.

· They were Forgiving: Willing to cooperate again if opponent did so.

· They were Clear: Pattern of play was consistent and easy to predict (not random).

This seems like a perfect set of rules for governments, businesses and social interactions.

Suggested Level: Advanced High School Senior and up

Solutions to Challenges in this topic are here:

All numbers are made up of prime numbers. Prime numbers are the atoms of arithmetic, since every number can be written uniquely as a product of primes. We know that there are an infinite number of primes, but where are they? Are there huge regions of numerical space in which no primes are present at all? And are there other regions where large numbers of primes are concentrated? The Riemann Hypothesis is a conjecture about how the prime numbers are distributed throughout the infinite space of numbers. It answers the question, “How many primes are there less than or equal to some given number?” If we could answer that question as precisely as possible, the result would obviously have very important implications for our understanding of the numbers. Because the answer would tell us so much about the numbers, the Riemann Hypothesis is considered to be the most important unsolved problem in pure mathematics. Over 500 mathematical results have been proved to be true if the Riemann Hypothesis is true. Fame and fortune awaits anyone who can resolve the Riemann Hypothesis. The Riemann Hypothesis was the 8th of Hilbert’s 23 challenge problems and is also one of the Clay Mathematics Institute’s 7 millennium prize problems, with a $1 million prize for its proof.

Early Attempts To Understand the Distribution of Prime Numbers

Can we find a formula to predict the distribution of prime numbers? This is the question that the mathematician Karl Frederick Gauss turned to in his youth.

Karl Frederick Gauss

Called the “Prince of Mathematics,” Gauss made profound discoveries in many areas of mathematics, many of which he never published. For example, he discovered non-Euclidean geometry that would later be used in General Relativity. His motto regarding his mathematical research was: “Pauca, sed matura—Few, but ripe.” Besides mathematics, he was very good at languages and by 19 had already mastered German, English, Greek, Latin, Danish, and French. Later in life he also mastered Russian and Swedish. He called mathematics the “Queen of the Sciences” and number theory the “Queen of Mathematics.” As a boy, Gauss was obsessed with prime numbers and spent his free time calculating them.

From calculating prime numbers, Gauss observed that they thinned out according to a pattern. We can see how prime numbers thin out by graphing the Prime Counting Function.

The Prime Counting Function

For any number N, the prime counting function counts the exact number of primes less than or equal to N. For example, if we wanted to know the number of primes less than or equal to 100, we’d feed 100 to the prime counting function and get 25. The Prime Counting function looks like a staircase since for some numbers we don’t have any more primes than we did for previous numbers. For example, there are 6 primes up to and including the number 13: 2,3,5,7,11,and 13. So, if we feed 13 to the prime counting function, we get 6. But since 14,15, and 16 are not prime, we also get 6 if we feed 14, 15, or 16 to the function.

Gauss observed that the probability of selecting a prime at random from numbers 2 to N is approximately 1/ln(N). It’s as if Nature is throwing a Log-weighted coin. The expected number of primes between 2 and N would then be N/ln(N) As seen in the chart below, Gauss’s approximation seems to underestimate the true number of primes.

Gauss refined his estimate of the number of primes that are less than or equal to some number N by noting that a better approximation is

A key step that Gauss took, which brings to bear the tools of calculus to solve questions in number theory, was to write the approximation as an integral, the logarithmic integral function, denoted li(N) and defined as

The chart below compares the li(N) function and the simpler approximation to the true number of primes less than or equal to N.

Gauss’s log integral approximation seems to overestimate the number of primes while the simpler approximation seems to underestimate them. By the time he was 70, Gauss had computed primes for the first 3 million numbers and li(N) had always overestimated the true number of primes. Gauss believed li(N) always would overestimate them, but he had no proof for that belief.

Gauss conjectured that the relative error between N/ln(N) or li(N) and the prime counting function would decrease as N became larger and larger. As we can see, the percentage error between li(N) and the true number of primes does indeed get smaller and smaller as N increases. This is the Prime Number Conjecture:

as x becomes larger and larger. The numerical evidence in the table below is very suggestive that’s it true but of course that’s not proof. The prime number conjecture could fail for much larger numbers.

It was Gauss’s student, Bernard Riemann, who made decisive progress in understanding the distribution of the prime numbers. Riemann began by studying theology, but Gauss recognized Riemann’s extreme mathematical talent and convinced him to switch to mathematics. Riemann made fundamental discoveries in complex analysis and non-Euclidean geometry and put the integral on rigorous foundation with Riemann sums. However, Riemann became most famous for the Riemann hypothesis, which came out of a short paper that Riemann wrote on the subject that had so long intrigued Gauss: how many primes are there less than a certain number? To understand what foundation Riemann built upon, we have to go back to Euler and the solution of the Basel problem.

The Basel Problem

The Basel problem was to find the sum of the reciprocal of all the squared natural numbers, i.e.,

The problem was named after the city of Basel, which was Euler’s and the Bernoulli brother’s hometown. The Bernoulli’s were not able to solve the problem, but Euler managed to do it around 1735. Euler’s solution is ingenious, leading to a solution not only to the original problem but also the to the sum of all reciprocals of the natural numbers taken to any even power.

To solve the problem, Euler started by writing the standard Maclaurin series for the sine function, which should be familiar from elementary calculus:

Any function can also be factored into a product of where its zero, providing an alternative way to write the function. For example, the function

and written in two way, since the function is zero at either x = 1 or x = -2. Since we know that sin(x) is zero at

we can write sin(x) in factored form as an alternative

Now let’s multiply this out, focusing on the cubic terms

If we equate the terms multiplying the cubic power of x in both the McLaurin series version of sin(x) and the factored form of sin(x), we get

or

From this relation, we can derive others that will prove useful later. Note that

and subtracting the equations

Also, we can see that

And finally, we can then get the sum of the alternating series:

Euler’s Product Formula

While he was working on the Basel problem, Euler realized that if he generalized it, writing it as sum to an arbitrary power of x, the new function, the ζ(x) function (zeta function), amazingly encodes all the primes. The ζ(x) function, a generalization of the Basel problem, is

For the zeta function, Euler derived closed-form solutions in terms of π for all even x: when x = 2, the sum is Π²/6. When x is 4, the sum is Π⁴/90 and when x is 6, Π⁶/945, and so on. The function is known to converge for all odd powers of x as well, although it is not known whether it converge to any nice formulas as the even powers do. However, the zeta function does not converge when x = 1, i.e., in that case it grows without bound, never approaching a particular value.

Since every natural number n can be written as a unique product of powers of primes, Euler noticed that ζ(x) could alternatively be written as

But each sum in the infinite product in the equation above is the sum of a geometric series. For example,

and

So, Euler showed that the zeta function can be written alternatively as a function of all the prime numbers.

Riemann’s Realization

Riemann realized from Euler’s work that the zeta function contains the secrets of the primes, which must hidden in the function somehow.

However, a look at Euler’s product function above presents a challenge. We know that this function converges to some non-zero value for all x > 1. The sum diverges for all x<=1. How does that help? It’s not at all clear how this function can tell us anymore about the primes than we already know.

Analytic Continuation Comes to the Rescue

Riemann was a master of complex analysis, the calculus of complex numbers. A complex number is a number a + ib that is a combination of a real part a and an imaginary part b, where

Riemann’s strategy to uncover the hidden structure of the primes in the zeta function was to generalize it to be a function of complex rather than real numbers. To denote that the zeta function is defined over complex numbers, we can change the argument x to s, where s is some complex number of the form s = a + ib. The new zeta function is then

If we do that, we have a complex-valued zeta function. It can be shown to converge for all s = a + ib for a > 1. For a <=1, the zeta function diverges. Unfortunately just extending to complex numbers doesn’t help much.

However, Riemann performed some mathematical wizardry, analytically continuing the zeta function so that it is defined over the larger domain of complex numbers that includes a <=1. To understand analytic continuation intuitively, it may be helpful to recall the parable of the three blind men trying to understand an elephant. One man can only feel the elephant’s trunk, and so thinks the elephant is the same as its trunk. Another man feels the elephant’s tail and thinks the tail is the elephant. The third man feels the elephant’s ears and thinks the ears are the elephant. The men need to find a way to piece together their limited observations of the elephant to understand what it is in its entirety.

Functions in the complex plane can behave the same way. We may see one form of a function over a limited domain of the complex plane. But that’s just a limited piece of a larger function that takes a different form over a broader domain of complex numbers. But it’s still the same function. Finding the analytic continuation of the zeta function is analogous to the blind men feeling different parts of the elephant and piecing them together.

Analytically Continuing the Zeta Function To Be Defined Between 0 and 1

We can extend the zeta function to a larger domain of complex numbers by defining the eta function

It can be shown that this function converges for all complex numbers s = a + bi, where a>=0 and a ≠ 1. Notice that

Solving for ζ(s)

We have found the more general zeta function that is defined over the larger complex domain s = a + bi in terms of the eta function, defined for a>=0 and a≠1. Originally, zeta was defined for a > 1 and now the extended version is defined for a >= 0, a larger domain.

Since η(s) is defined over that larger domain, this extended zeta function will give the same values as the original zeta function for a >1. For example, let’s use it to calculate ζ(2) which we already know to be Π²/6.

Plugging 2 into the extended zeta function, we see the extended zeta function agrees with the original zeta function.

The Zeros of the Extended Zeta Function

How does the extended zeta function help? Riemann deduced that the zeros of the extended zeta function, which are complex numbers that make the extended zeta function zero, amazingly determine where the prime numbers are. The extended zeta function has an infinite number of complex zeros in the extended strip

for all imaginary numbers b. These zeros are called the critical zeros. If we know the values of the critical zeros, we can determine where the primes are.

How Do We Find the Zeros of the Zeta Function?

There are many methods to calculate the zeros of the zeta function. For example, we can use the eta function to find the zeros of the zeta function. As the expression we already derived shows below, whenever the eta function is zero the zeta function will be zero as well, i.e, if η(s) = 0, then ζ(s) = 0.

So, we have reduced the problem to finding the zeros of the eta function. The eta function can be written as

Looking at the general term, we see that

Recall Euler’s relation

Then

Thus, we can write the eta function as

when the real and imaginary parts of η(s) equal zero. The eta function will be zero when the real and imaginary parts are both zero.

and

Notice also that if s = a + bi is a zero, then s = a – bi is also a zero. That’s because the sign of b doesn’t matter since, by properties of the sine and cosine

and

Thus, critical zeros of the zeta function always occur as complex conjugates.

The chart below graphs the zeros for the real and imaginary parts of the eta function. The red line is the graph of all the complex values in which the cosine terms sum to zero, and the blue line depicts all the complex values in which the sine terms sum to zero. Where they intersect is where both the real and imaginary components are zero, i.e., where the eta function is zero. The intersection of the curves defines the critical zeros. The chart below shows the first three critical zeros.

Riemann calculated the first few zeros of the zeta function by hand and noticed that the real part was ½ in each case. Riemann conjectured that all the zeros have real part equal to ½--that’s the Riemann hypothesis. To date, over 12 trillion zeros have been calculated and all of them have real part equal to ½. However, no one yet knows whether it’s always true that the zeros have real part equal to ½. We’ll explain why it’s so important to settle this question in the following sections, when we learn how the zeros determine where the prime numbers are.

More General Analytic Continuation: The Functional Equation

The zeros we found in the strip

are termed the “critical zeros.” There are also additional infinite zeros on the negative real axis called the “trivial zeros.” Given how we’ve extended the zeta function so far, we cannot see these zeros. However, Riemann found an even more general analytic continuation of the zeta function that is defined for all s = a + bi where a ≠ 1.

This general zeta function is written as

This equation is called the “functional equation” because it relates ζ(s) to ζ(1-s)

Using this equation, we can trivially see that ζ(s) also has zeros at s = -2,-4, -6, -8, …. besides the critical zeros we already found, since

All the zeros of the zeta function in the complex plane are depicted below.

How Many Primes Are There Less Than Or Equal to any Chosen Number?

Armed with knowledge of the zeros of the zeta function, Riemann was able to derive a formula that related the prime numbers to the zeros of the ζ(s) function. Let us define π(x) to be the number of primes less than or equal to x. Riemann derived the following formula for π(x)

where

The function η(n) is the mobius function. It takes on values of -1, 0, or 1 depending on what n is. Notice that the logarithmic integral li(x) appears in the formula, but in a more complex way than known to Gauss. The critical zeros appear in the formula as well, denoted as ρ. Each critical zero is the exponent of x in the li(x) function. If we take just one term in the infinite sum, n = 1, and drop the critical zeros summation term and both ln(2) and the integral in J(x), both of which are small, we can recover Gauss’s approximation discussed earlier.

since μ(1) = 1. Gauss’s approximation is therefore a special case of the Riemann formula.

To see how the zeros determine where the primes are, we can compare the true number of primes to Riemann’s formula using just 10 J(x) terms and either 10, 25, 50, or 100 zeros and their complex conjugates. Truncating Riemann’s formula implies that we are approximating where the primes are, but, as can be seen, even using just 50 zeros reproduces the true prime counting function very well.

A Better Way To Pick Out The Primes Using the Zeros of ζ(s)

About 35 years after Riemann, Hans von Mangoldt derived a simpler formula for the primes, a formula that is preferred in modern textbook treatments of the Riemann hypothesis. By looking not at the number of primes, but rather at the sum of the natural logs of the number of primes, von Mangoldt was able to produce a simpler formula. Let ψ(x) denote the sum of the natural logs of each prime or power of a prime less than x:

The function ψ(x) defines a step function like the prime counting function, with the difference that it pulses up by ln(p) for each prime or power of prime it counts. Von Mangoldt showed that this formula could be written as:

This function says that ψ(x) rises linearly with x, but then has some correction terms, with the most important being the last term, which depend on the critical zeros ρ. Although not obvious, the second to the last term arises from the trivial zeros. The term

actually depends on all the zeros, but the trivial zeros have been written separately in the formula, so that the ρ terms refer only to the critical zeros in Von Mangoldt’s formula. To see this, let’s take the summation term, but only include the trivial zeros in it, the negative even numbers. Then that summation term would be written as

We’ll prove that

The Maclaurin series of ln(1+y) is

If we substitute

we get

Thus, the infinite trivial zeros can be compressed into the simple expression

Picking Out the Primes

To show how we can use the von Mangoldt function to pick our the primes or power of primes, note that

The derivative of the von Mangoldt pulses whenever it encounters a prime or power of a prime but is zero otherwise. Thus, we can use the derivative of the von Mangoldt function to pick out where the primes are.

Taking the derivative of ψ(x), we get

This function will pulse at every prime or power of primes. We can write this formula more simply by taking one x^ρ-1 term in the sum and noting that every zero s = a + bi is matched by another zero that is its complex conjugate a - bi in the infinite sum. Taking as an example one critical zero a + bi and its complex conjugate a - bi, we see that

We can write

But we know that

and thus,

a simpler formula that makes clear the role of the the real part a and the imaginary part b of the critical zeros. The Riemann Hypothesis states that a is always 1/2 in the formula above.

The Music of the Primes

The last term in ψ'(x) is a series of waves. Let’s look at the equation for the wave resulting from the first critical zero 1/2 + 14.134i

It looks like this:

As we can see, the first zero defines a wave that goes out to infinity. Notice that its frequency slows down, since the imaginary component of the zero is multiplied by ln(x), which grows more slowly than x. The amplitude of the wave also gets smaller over time, at the rate 2/x^1/2

The chart below depicts the wave produced by the 50th zero. We see the same pattern: the wave starts off with very high frequency, because of the constant 143.118 in the cosine term. However, the frequency slows down because the constant is scaled by ln(x). The amplitude of the wave also slows at the rate 2/x^1/2

The essence of the Riemann hypothesis is that the amplitudes of all the waves slow down at exactly the same rate, 2/x^1/2. That’s only possible if all zeros have real part equal to ½.

Mathematicians sometimes refer to the “music of the primes” because each wave created by the infinite imaginary parts of the zeros are playing at a different frequency, much like musical instruments emit sound waves at different frequencies. The waves of the zeta orchestra come together to create a symphony of the primes and powers of primes. However, if the Riemann hypothesis is true, all the zeta waves have the same amplitude at each point: none of the zeta instruments are playing any louder than any other. The amplitude of every wave slows down at exactly the same rate, 2/x^1/2.

We can see how the zeta waves come together in a symphony, canceling each other out except where primes and powers of primes reside, by graphing them for different numbers of zeta zeros.

Suppose we use only the first zeta zero and its complex conjugate in the sum. No pattern emerges yet.

With ten pairs of zeros included in the sum, we still don’t see a clear pattern.

Using 50 pairs of zeta zeros in the sum, our orchestra seems to be picking out primes and powers of primes.

Let’s add more instruments to the orchestra to confirm, including 100 pairs of zeta zeros in the sum. One hundred pairs of zeros is enough to see the primes below 20 picked out.

Do the Trivial Zeros Matter?

We focused on the wave component of Mangoldt’s function, which depends on the critical zeros. It’s sometimes thought that the trivial zeros play no role, but, as we have seen, that’s not really true. As we will recall, the trivial zeros contribute the term

in Von Mangoldt’s formula. That term is very tiny however. At x = 2, it’s only 1/6 and becomes smaller and smaller as x increases. The effect of the trivial zeros is indeed trivial, but not zero.

Why Does the Riemann Hypothesis Matter For Pure Mathematics?

Using methods first developed by Riemann, mathematicians finally proved that the prime number conjecture was valid at the end of the 19^th century. Recall that the prime number conjecture is that

as x becomes larger and larger. That means that the true number of primes less than x will get closer and closer to Gauss’s approximation li(x). However, the prime number conjecture doesn’t really pin down the structure of the primes precisely. There could still be pockets for very large x in which the density of primes suddenly increases and other regions where they become scarcer than expected. However, if the Riemann hypothesis is true, we can say precisely how good an approximation li(x) will be to the true number of primes less than x. The error between π(x) and li(x) will always be less than or equal to

if the Riemann Hypothesis is true. The following table shows highly tightly bound the primes are to the li(x) function if the Riemann Hypothesis is true.

The Quest Continues

The mathematician G.H. Hardy, working fifty years after Riemann, was obsessed with proving the Riemann Hypothesis. Besides mathematics, Hardy played cricket and real tennis (as opposed to lawn tennis) and was an expert in ball games. He believed in pursuing mathematics for its own sake, a philosophy he explained in “A Mathematician’s Apology”. Hardy proved that there are an infinite number of zeros of the zeta function that have real part ½ , but that of course doesn’t mean that all of them do.

Mathematicians continue their attempts to prove the Riemann Hypothesis. One empirical way to proceed is to calculate the zeros by computer. If one zero is found that has a real part not equal to 1/2, that would disprove the Riemann hypothesis. However, so far the Riemann hypothesis has been verified computationally for over 12 trillion zeros.

Like Hardy, pure mathematicians work on the Riemann Hypothesis for its own sake, not because there are necessarily any applications. However, the resolution of Riemann’s hypothesis will probably open up new vistas in mathematics, many of which may inspire new applications. Hilbert once said that if he woke up after 500 years, his first question would be to ask whether the Riemann Hypothesis had been proved. In his retirement address (and on his tombstone), Hilbert summed up the motivation of pure mathematicians and scientists by saying: “Wir mussen wissen, wir werden wissen”—We must know, we will know.

This is a problem that is extensively discussed on the internet. Here we will provide a solution to the general case which is from a paper by Freeman Dyson.

This is the way the problem is generally posed: You have four to twelve coins and a weighing balance but no weights. All of the coins have the same weight except one which is either lighter or heavier but you do not know which). In how many weightings can you identify the defective coin?

The usual solution proceeds as follows:

Only three weightings are necessary for 4-12 coins. We will sketch the cases with 4 and 12 coins.

4 Coins:

· Label the 4 coins c1, c2, c3, c4. Weigh c1 against c2.

• If they balance, the defective coin is either c3 or c4. In this case, weigh c3 against c1.

• If they balance, c4 is defective and weighting it against one of the others will determine if it is heavier or lighter.

• If c1 and c2 do not balance, weigh them one by one against c3 to identify both the defective coin and whether it is heavier or lighter.

12 coins:

• If there are 12 coins, compare the weights of two sets of 4 coins.

• If they do not balance, label the coins h1,h2,h3,h4 and l1,l2,l3,l4 depending on which 4 were higher or lower weight.

• Now weigh h1,h2,l1 vs h3,h4,l2. If they balance, the defective coin is either l3 or l4 and can be found in one weighting.

• If they do not balance, say h1, h2, l1 is heavier, then the defective coin is either h1, h2 or l2. Now we compare h1 and h2. If they balance, defective coin is l2. If they do not balance, the heavier of the two coins h1 or h2 is defective.

• If instead the initial sets of 4 coins balanced, then they are good coins, and the defective coin is in the remaining four. Label these c1,c2,c3,c4 and let us use a coin labeled c from the initial weighing to find the defective coin within these 4 coins as follows:

• Weigh (c1,c2) vs (c3,c). If they balance, then c4 is defective and it can be weighed against c to see if it is lighter or heavier. If (c1,c2) vs (c3,c) do not balance, we can assume without loss of generality that (c1,c2) are heavier then (c3,c). Now weigh c1 vs c2. Whichever is heavier is the defective coin. If c1 vs c2 balances, then the defective coin is c3 and is lighter.

https://en.wikipedia.org/wiki/Freeman_Dyson

Dyson’s General Solution for any number of coins: The general solution was provided by Freeman Dyson in a 1946 paper. Reference: The Mathematical Gazette, (Oct., 1946, Vol. 30, No. 291, pp. 231-234.

The method used by Dyson is an example of optimum coding. Dyson first showed that if M = (3ⁿ-3)/2, then the defective coin can be found in n weightings (including determining whether it is heavier or lighter). We will sketch the proof for n = 3, M = 12. where in three weightings it is possible to identify a defective coin in up to 12 coins. We start by considering all possible labels in base 3 from 001 to 221, skipping the mappings 000, 111 and 222. Now we start by assigning sequential labels for the 12 coins: 001, 002, 010, 011, 020, 021, 022, 100, 101, 102, 110. Next, we convert these to “cyclic labels”, where cyclic means that the labeling for each of the coins read from left to right must have 01, 12 and 20. Such cyclic labels are generated by swapping 0 and 2 in the original labels. This converts the labeling of the twelve coins to: 001, 220, 010, 011, 012, 202, 201, 200,122, 121,120,112 where the ones converted to cyclic order are in bold. We now group these labels into 4 sets of 3 so that each of three labels in a set are obtained by permuting 0 to 1, 1 to 2 and 2 to 0. The four sets are (001, 112, 220), (010,121,202), (011,122,200) and (012,120,201).

As an example, we show how to find the defective coin among six coins. For this we choose two sets of three cyclic permutation labels to label the coins.

Label of Coin 1: 001

Label of Coin 2: 112

Label of Coin 3: 220

Label of Coin 4: 010

Label of Coin 5: 121

Label of Coin 6: 202

The column labels correspond to the three weighting rules for the coins: If the label is 0, 2 the coin is placed in the left or right pan respectively. If in the weighting, the left/right hand pan is heavier, we call the result 0/2 respectively. If the pans balance, call the result 1. After the three weightings, the result will represent either the code of the coin or its complement (obtained by switching 0 and 2). If the code matches the assigned code, the coin was heavier. If the code is the complement of the assigned code, the coin was lighter.

Let us see how this works. Suppose coin 1 is heavier. Going column wise, in the first weighting, we put coins 1,4 in the left pan and 3,6 in the right pan. The left pan will be heavier, so we get 0 as the result. In the second weighting, we put coins 1,6 in the left pan and 3,5 in the right pan. The left pan is heavier, so we get 0 as the result again, and the two results together are 00. In the third weighting, we put 3,4 in the left pan and 2 and 6 in the right pan. The coins will balance, so we get the result 1. The complete result is 001, which is the label of coin 1. It also shows that this coin is heavier. If coin 1 were lighter, it is easy to see that the result would have been 221, which is the complement of 001.

Dyson also showed that for M in the interval 3 < M < (3ⁿ-3)/2, then n weightings are sufficient. The Table above summarizes some of the results. For details, please read his paper referenced above.

Solution: We first consider pairs. Choose two people. The probability that they both will have the same birthday on a specific day of the year is

The probability that they will have the same birthday on some day of the year is:

There are:

ways to choose the pairs. Hence the average number of pairs with the same birthday will be the product of these or

This number is greater than 1 if n is greater or equal to 28.

A similar argument shows that the average number of triples (three people with the same birthday) from among n people will be:

This exceeds 1 if n is greater than or equal to 94.

The general solution for the average number k of people who have the same birthday among n people is:

.