Atlas / Mathematics / The Number Theory Thread

Field · Emerged 1737 – 1896

Analytic Number Theory

How are the primes distributed, and why can calculus answer questions about whole numbers?

5 chapters4 min read7 turning points 2 open problems

Branched from: Elementary Number Theory + Complex Analysis
Branched into: Arithmetic Geometry
Figures: Leonhard Euler, Carl Friedrich Gauss, Adrien-Marie Legendre, Peter Gustav Lejeune Dirichlet, Bernhard Riemann, Jacques Hadamard, Charles-Jean de la Vallée Poussin, Atle Selberg, Paul Erdős, Yitang Zhang, James Maynard

In brief

Analytic number theory uses the tools of calculus and complex analysis (limits, infinite series, functions of a complex variable) to study whole numbers, above all the primes. The primes look random one at a time, but in bulk they follow precise laws. Near a large number $x$ , roughly one number in $\ln x$ is prime.

Its central object is the Riemann zeta function. The most famous open problem in mathematics, the Riemann hypothesis, is a statement about where that function's zeros lie, and it controls how closely the primes follow their predicted pattern.

Key ideas

Prime-counting functionEnters 1896: $\pi(x)$ is the number of primes up to $x$ . The prime number theorem says $\pi(x) \sim x / \ln x$ : the ratio tends to 1 as $x$ grows.
Riemann zeta functionEnters 1859: $\zeta(s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$ , extended by Riemann to almost all complex numbers $s$ . Its zeros encode the fine distribution of the primes.
Euler productEnters 1737: $\zeta(s) = \prod_{p} \frac{1}{1 - p^{-s}}$ , a product over all primes. It is the bridge between a smooth analytic function and the primes, which is why analysis can say anything about them.
L-functionsEnters 1837: Generalisations of the zeta function built from arithmetic data. Dirichlet introduced them to count primes in arithmetic progressions. They are now central throughout number theory.
Critical lineEnters 1859: The vertical line of complex numbers with real part $\tfrac12$ . The Riemann hypothesis says every nontrivial zero of $\zeta$ lies on it.

Chapter I

Counting Primes with Calculus

Euclid proved the primes never end, but not how common they are. Leonhard Euler found the first link between primes and analysis in 1737. The infinite series $1 + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$ equals a product over all the primes,

\sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}},

a consequence of unique factorisation written in the language of calculus. From it he deduced that the reciprocals of the primes, $\frac12 + \frac13 + \frac15 + \frac17 + \cdots$ , add up to infinity. The primes are rare, but not too rare.

Half a century later, poring over tables, the teenage Gauss and Legendre each guessed the law: up to $x$ , about $x/\ln x$ numbers are prime. In 1837 Dirichlet proved that primes are spread across all compatible arithmetic progressions, using new functions, now called L-functions, and the full machinery of analysis. The methods of elementary number theory had been left behind.

Chapter II

Riemann's Eight Pages

In 1859 Bernhard Riemann, better known for his work in geometry, wrote his only paper on number theory. Using the complex analysis he had developed in his thesis, he extended Euler's function to complex numbers, where it becomes the zeta function $\zeta(s)$ , and showed that the exact count of primes is determined by the locations of its zeros. The primes are, in a precise sense, a sum of waves whose frequencies are those zeros.

He also computed a few zeros and found them all on one vertical line. It was "very probable", he wrote, that all of them lie there, but he had put aside the search for a proof "after some fleeting vain attempts". That remark is the Riemann hypothesis.

Chapter III

The Theorem, Twice

Riemann's programme took nearly forty years to carry out. In 1896 Jacques Hadamard and Charles-Jean de la Vallée Poussin, working independently, proved that $\zeta$ has no zeros on the line with real part 1, and deduced the prime number theorem. Both lived past ninety, which prompted a joke that proving it grants long life.

Many believed the theorem was inseparable from complex analysis. In 1948–49 Atle Selberg and Paul Erdős found a proof using only elementary estimates, and then fell out bitterly over who deserved the credit.

Chapter IV

A Closer Look: How Good Is the Prediction?

The prime number theorem predicts that there are about $x / \ln x$ primes up to $x$ . Gauss's refinement, the logarithmic integral $\operatorname{li}(x) = \int_0^x \frac{dt}{\ln t}$ , adds up the "probability" $1/\ln t$ that each number near $t$ is prime. Compare both with the true counts:

$x$	primes up to $x$ , $\pi(x)$	$x / \ln x$	$\operatorname{li}(x)$
$10^6$	78,498	72,382	78,628
$10^9$	50,847,534	48,254,942	50,849,235

Both estimates have the right growth, but $\operatorname{li}(x)$ is far better. Up to a billion it is off by about 1,700, where $x/\ln x$ is off by more than two and a half million.

How large can the error $\pi(x) - \operatorname{li}(x)$ get? That is exactly what the Riemann hypothesis controls. If it is true, the error never grows much faster than $\sqrt{x}\,\ln x$ , about the size of the fluctuations in a random walk of $x$ steps. At $x = 10^9$ that scale is about 660,000. Lowell Schoenfeld's explicit form of the bound, $\frac{1}{8\pi}\sqrt{x}\,\ln x$ , is about 26,000. The actual error is 1,701, well inside it. Riemann's explicit formula shows where the error comes from: each zero of the zeta function contributes a wave to the count of primes. Zeros on the critical line produce waves of the smallest possible size. A zero off the line would produce a larger wave, and the primes would be measurably less regular than they appear.

The table also hides a surprise. In every computed case $\operatorname{li}(x)$ overestimates $\pi(x)$ , and for a century it was assumed to always do so. In 1914 Littlewood proved that the two swap places infinitely often, but the first crossing lies beyond the range of any computer: somewhere below about $10^{316}$ , by current bounds.

Chapter V

Gaps and the Fog

The biggest questions remain. The Riemann hypothesis has resisted more than 160 years of attempts, even as trillions of zeros have been checked. The twin prime conjecture, that primes 2 apart never run out, looked hopeless until 2013. Then Yitang Zhang, a little-known lecturer who had once worked in a sandwich shop, proved that some fixed gap below 70 million occurs infinitely often. James Maynard and a worldwide online collaboration soon brought the bound down to 246. The last step, from 246 to 2, needs an idea no one yet has.

Turning points7

1737PROOF
Euler links the primes to an infinite series
Leonhard Euler shows that the sum $1 + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$ equals a product over all primes, and deduces that the sum of the reciprocals of the primes diverges, a stronger form of Euclid's theorem. For the first time the primes are studied through the analysis of a function.
Leonhard Euler
› Sources (1)
- Euler, L. (1744). Variae observationes circa series infinitas. Commentarii academiae scientiarum Petropolitanae 9: 160–188. (Presented 1737.)
1792 – 1798CONJECTUREContested
The prime number theorem is conjectured
Studying tables of primes, Carl Friedrich Gauss, as a teenager, and Adrien-Marie Legendre both notice that primes thin out like $1/\ln x$ , and conjecture how many primes lie below $x$ . Legendre publishes his formula in 1798. Proving it would take another century.
What is disputed
Legendre published first (1798). Gauss's claim to have found the law in 1792–93, at about fifteen, comes from a letter he wrote to Johann Encke in 1849, more than fifty years later, and he published nothing on it at the time. Gauss's version, the logarithmic integral, is the better approximation.
Carl Friedrich Gauss, Adrien-Marie Legendre
› Sources (1)
- Goldstein, L. J. (1973). A history of the prime number theorem. American Mathematical Monthly 80(6): 599–615.
1837PROOF
Dirichlet's primes in arithmetic progressions
Peter Gustav Lejeune Dirichlet proves that any sequence $a, a + d, a + 2d, \ldots$ with $a$ and $d$ sharing no factor contains infinitely many primes. He does it by inventing L-functions and characters, and using calculus to control them. This is usually taken as the birth of analytic number theory as a method.
Peter Gustav Lejeune Dirichlet
› Sources (1)
- Dirichlet, P. G. L. (1837). Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften: 45–81.
1859CONJECTURE
Riemann's paper on the number of primes
In an eight-page paper, his only one on number theory, Bernhard Riemann extends the zeta function to complex numbers and gives an exact formula for counting primes in terms of its zeros. In passing he remarks that it is "very probable" all the nontrivial zeros have real part $\tfrac12$ : the Riemann hypothesis.
Bernhard Riemann
› Sources (1)
- Riemann, B. (1859). Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Königlichen Preussischen Akademie der Wissenschaften zu Berlin: 671–680.
1896PROOF
The prime number theorem is proved
Jacques Hadamard and Charles-Jean de la Vallée Poussin independently prove that $\pi(x) \sim x/\ln x$ , by showing that the zeta function has no zeros on the line with real part 1. It is exactly the kind of fact about zeros Riemann's programme required. The two proofs appeared in the same year, and credit has always been shared.
Jacques Hadamard, Charles-Jean de la Vallée Poussin
› Sources (2)
- Hadamard, J. (1896). Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques. Bulletin de la Société Mathématique de France 24: 199–220.
- de la Vallée Poussin, C.-J. (1896). Recherches analytiques sur la théorie des nombres premiers. Annales de la Société Scientifique de Bruxelles 20.
1948 – 1949PROOFContested
An "elementary" proof of the prime number theorem
Atle Selberg and Paul Erdős find a proof of the prime number theorem that avoids complex analysis entirely. Many had believed that impossible, since the theorem seemed to be really about the zeros of $\zeta$ . Selberg received a Fields Medal in 1950, partly for this work.
What is disputed
The proof came out of a brief, intense collaboration that ended in one of the bitterest priority disputes in modern mathematics. Selberg had a key inequality. Erdős used it to take a crucial step, and Selberg then completed a proof. The two could not agree on joint publication, and Selberg published alone. Accounts from each side, and from colleagues who took sides, differ on who contributed what.
Atle Selberg, Paul Erdős
› Sources (2)
- Selberg, A. (1949). An elementary proof of the prime-number theorem. Annals of Mathematics 50(2): 305–313.
- Goldfeld, D. (2004). The elementary proof of the prime number theorem: an historical perspective. In D. Chudnovsky et al. (eds.), Number Theory: New York Seminar 2003: 179–192. Springer.
2013PROOF
Bounded gaps between primes
Yitang Zhang, then an almost unknown lecturer, proves that infinitely many pairs of primes differ by less than 70 million, the first finite bound ever. Within months James Maynard (and independently Terence Tao) found a simpler method, and a collaborative Polymath project pushed the bound down to 246. The twin prime conjecture asks for a gap of 2.
Yitang Zhang, James Maynard
› Sources (2)
- Zhang, Y. (2014). Bounded gaps between primes. Annals of Mathematics 179(3): 1121–1174.
- Maynard, J. (2015). Small gaps between primes. Annals of Mathematics 181(1): 383–413.

Applications

Where it is used

Quantum physics↗ Physics · Nuclear Structure
Zeta zeros behave like quantum energy levels
In 1972 Hugh Montgomery found that the spacings between zeta zeros follow a statistical law. Freeman Dyson recognised it at once as the law governing the energy levels of heavy atomic nuclei, as modelled by random matrices. Why the primes should behave like a quantum chaotic system is unexplained, and it inspires hopes of a physical route to the Riemann hypothesis.
› Sources (1)
- Montgomery, H. L. (1973). The pair correlation of zeros of the zeta function. Proceedings of Symposia in Pure Mathematics 24: 181–193.
Cryptography
Finding large primes quickly
Encryption keys need primes hundreds of digits long. The prime number theorem guarantees they are plentiful: about one in every 710 numbers near $2^{1024}$ is prime, so testing random candidates finds one fast.

Open problems

Where the map runs out

Open

The Riemann hypothesis

Open since 1859; a Clay Millennium Prize Problem and one of Hilbert's 1900 problems.

All nontrivial zeros of the Riemann zeta function lie on the critical line, real part $\tfrac12$ . Trillions of zeros have been checked by computer and every one lies on the line. Equivalently, the primes are distributed as regularly as they possibly could be: $\pi(x)$ stays within about $\sqrt{x}\,\ln x$ of its predicted value.

Why it is hard: The zeta function is defined by an infinite series that stops converging exactly in the region where the zeros live. Known methods push the zero-free region only slightly beyond Hadamard's line. Hardy proved in 1914 that infinitely many zeros are on the critical line, and later work showed that at least 40% are, but not all. Many proposed approaches, such as finding the zeros as eigenvalues of some operator, have never been made to work.
What resolving it unlocks: Hundreds of published theorems are proved assuming it. A proof would upgrade them all and would sharpen nearly every estimate about how primes are spaced.

› Sources (1)

Bombieri, E. (2006). The Riemann hypothesis. In J. Carlson, A. Jaffe & A. Wiles (eds.), The Millennium Prize Problems: 107–124. Clay Mathematics Institute / AMS.

Open

The twin prime conjecture

Open as of 2026. Bounded gaps proved in 2013; the best unconditional gap is 246.

Are there infinitely many pairs of primes that differ by 2, like 11 and 13, or 101 and 103? They grow rarer but never seem to stop, and the largest known pair has hundreds of thousands of digits.

Why it is hard: Sieve methods, which count primes by crossing out multiples, run into the "parity problem": they cannot tell numbers with an even number of prime factors from those with an odd number, and a gap of exactly 2 falls on the wrong side of that barrier. Zhang and Maynard's methods get to 246, and, assuming stronger conjectures, to 6, but not to 2.
What resolving it unlocks: Breaking the parity barrier would be a methodological revolution, opening many other additive problems about primes, including Goldbach's.