Atle Selberg

How to use this course

Atle Selberg published relatively little, and almost every paper opened a field. Analytic number theorists use his sieve and his mollifiers. Specialists in automorphic forms use his trace formula, his Eisenstein series and the Rankin–Selberg method. Geometers and group theorists use his lemma on torsion-free subgroups and his rigidity theorem. Random-matrix theorists use his integral. This course presents these ideas alongside the story of a self-taught student in occupied Norway who became a permanent fixture of the Institute for Advanced Study.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow the chronology of his work: Rankin–Selberg (1940), the critical line (1942), the integral (1944), the sieve (1947), the elementary proof (1948), the trace formula (1956), rigidity (1960), the eigenvalue bound (1965) and the Selberg class (1989).

Timeline

Langesund and Gjøvik, 1917–1935

Atle Selberg was born on 14 June 1917 in Langesund, a small coastal town in southern Norway. He was the youngest of nine children. His father, Ole Michael Ludvigsen Selberg, was a mathematics teacher; his mother, Anna Kristina Skeie, was also a teacher. Three of his brothers became mathematicians or engineers: Henrik Selberg, later a professor at the University of Oslo, and the twins Sigmund and Arne, who taught at the Norwegian Institute of Technology in Trondheim. Sigmund was a pure mathematician; Arne worked in engineering mathematics.

Selberg attended secondary school in Gjøvik. At about thirteen he began to read the mathematics books in his father’s library. One early experience he later recalled was meeting Leibniz’s series \(1-\tfrac13+\tfrac15-\cdots=\tfrac\pi4\), a link between whole numbers and the circle that made a lasting impression on him.

Ramanujan

In 1934, aged seventeen, he came across the collected papers of Srinivasa Ramanujan. He described the encounter for the rest of his life as the decisive event of his youth: the formulas were strange and beautiful, and the identities seemed to come from nowhere. The same year he published his first paper. He found, independently, an exact convergent series for the partition function \(p(n)\) suggested by the Hardy–Ramanujan expansion, but Hans Rademacher published such a formula first, in 1937. Selberg finished school in Gjøvik in 1935 and entered the University of Oslo that autumn.

Oslo under occupation, 1935–1947

At Oslo the number theorist Carl Størmer took an interest in him and helped him as an undergraduate. Selberg took his master’s degree in the spring of 1939 and spent the autumn at Uppsala, where he heard Trygve Nagell. In April 1940 Germany invaded Norway. Selberg took part in resisting the invasion and was held prisoner for a time.

For the next five years he worked almost entirely alone. Foreign journals stopped arriving, and the university was closed by the occupation authorities late in 1943. He later said that the isolation helped him: he did not know what others were doing, so he was not discouraged from trying hard problems. In this period he wrote his paper on the mean square of Ramanujan’s τ-function (1940), found independently of Robert Rankin, proved his theorem on zeros of the zeta function (1942), and found his multiple integral (1944). From 1942 he held a research fellowship at the university.

The doctorate

He defended his thesis, On the zeros of Riemann’s zeta-function, on 22 October 1943. The examiners were Harald Bohr and Thoralf Skolem. Bohr, in Copenhagen, could not travel, and Størmer read out his report. When the war ended in 1945, his results became known abroad and caused astonishment.

Princeton, 1947–1950

In 1947 Carl Ludwig Siegel invited Selberg to the Institute for Advanced Study in Princeton. That summer, on 13 August in Stockholm, he married Hedvig Liebermann. He spent 1947–48 at the Institute and the following year as associate professor at Syracuse University. In 1949 he returned to the Institute as a permanent member. In 1951 he was appointed professor in the School of Mathematics, a position he held until he became emeritus in 1987.

The 1947 sieve and the 1948 elementary proof of the prime number theorem (Lessons 05 and 06) came in these years. The second led to a painful priority dispute with Paul Erdős, which Lesson 06 describes.

At the International Congress of Mathematicians held at Harvard in 1950, Selberg received the Fields Medal together with Laurent Schwartz. The citation recognized his generalization of Viggo Brun’s sieve methods and his results on the zeros of the zeta function. Siegel remarked that by 31 Selberg had already secured his place in the history of science.

Zeros on the critical line

For the general reader. The Riemann hypothesis says that every non-trivial zero of the zeta function lies on one vertical line in the complex plane. It is still unproved. Hardy showed in 1914 that infinitely many zeros lie on the line. In 1942 Selberg showed that a fixed positive proportion of them do, a much stronger statement, by inventing a device for damping the wild oscillations of the zeta function.

Counting zeros

Let \(N(T)\) be the number of zeros \(\rho=\beta+i\gamma\) of \(\zeta\) with \(0<\gamma\le T\), and \(N_0(T)\) the number of these with \(\beta=\tfrac12\). The Riemann–von Mangoldt formula gives

Define \(\theta(t)=\arg\big(\pi^{-it/2}\,\Gamma(\tfrac14+\tfrac{it}{2})\big)\), chosen continuous with \(\theta(0)=0\), and Hardy’s function \(Z(t)=e^{i\theta(t)}\zeta(\tfrac12+it)\). The functional equation makes \(Z(t)\) real for real \(t\), and \(|Z(t)|=|\zeta(\tfrac12+it)|\). Each sign change of \(Z\) is a zero on the critical line.

Hardy and John Littlewood had shown \(N_0(T)\ge cT\) in 1921, a vanishing proportion.

Proof idea: mollification

To detect sign changes of \(Z\) in an interval \((t,t+h)\), compare \(\big|\int_t^{t+h}Z(u)\,du\big|\) with \(\int_t^{t+h}|Z(u)|\,du\). If the first is smaller, \(Z\) changes sign. Hardy and Littlewood carried out this comparison on average over \(t\le T\) with \(h\) of constant size. The trouble is that \(|\zeta(\tfrac12+it)|\) has large, rare peaks that dominate the second moment. Selberg multiplied \(Z\) by \(|\varphi(\tfrac12+it)|^2\), where

with \(X=T^{\varepsilon}\) for a small \(\varepsilon\). The product \(Z|\varphi|^2\) has the same sign as \(Z\) but behaves like a function of constant size on average. With it, the comparison works on intervals of length \(h\asymp1/\log T\), and a positive proportion of the \(\asymp T\log T\) such intervals in \([0,T]\) contain a sign change. The mollifier is now a standard tool. Norman Levinson’s 1974 refinement showed that more than a third of the zeros are on the line, and Brian Conrey raised the proportion to more than two fifths in 1989.

The Selberg sieve

For the general reader. The sieve of Eratosthenes finds primes by crossing out multiples. Counting exactly what survives is hopeless for large numbers, because the error terms pile up. Viggo Brun found in 1919 how to get useful bounds anyway. In 1947 Selberg found a far simpler method: choose weights to minimize a quadratic form. The resulting upper bounds are often the best known.

The setting

Let \(\mathcal A\) be a finite sequence of integers, \(\mathcal P\) a set of primes, \(z\ge2\) and \(P(z)=\prod_{p\in\mathcal P,\,p<z}p\). The sifted sum is \(S(\mathcal A,z)=\#\{a\in\mathcal A:\gcd(a,P(z))=1\}\). Assume that for squarefree \(d\mid P(z)\),

where \(g\) is multiplicative with \(0\le g(p)<1\) and \(r_d\) is an error term. Selberg’s idea: for any real \(\lambda_d\) with \(\lambda_1=1\), the indicator of \(\gcd(a,P(z))=1\) is at most \(\big(\sum_{d\mid\gcd(a,P(z))}\lambda_d\big)^2\). Restrict to \(\lambda_d=0\) for \(d\ge\sqrt D\), expand the square and sum over \(\mathcal A\).

Proof sketch

The main term is the quadratic form \(Q=\sum_{d_1,d_2}\lambda_{d_1}\lambda_{d_2}\,g([d_1,d_2])\). Write \(g([d_1,d_2])=g(d_1)g(d_2)/g((d_1,d_2))\) and \(1/g(m)=\sum_{e\mid m}h(e)\) with \(h(p)=(1-g(p))/g(p)\). Then

Möbius inversion turns the constraint \(\lambda_1=1\) into the linear condition \(\sum_e\mu(e)y_e=1\). By Cauchy–Schwarz the minimum of \(\sum h(e)y_e^2\) under that condition is \(1/\sum_e 1/h(e)=1/G\). The optimal weights satisfy \(|\lambda_d|\le1\), and the remainder involves pairs \(d_1,d_2\) with \([d_1,d_2]=d\), at most \(3^{\omega(d)}\) of them.

The parity barrier

Selberg also showed the limits of the method. Sieve hypotheses cannot tell apart integers with an even and an odd number of prime factors, so no sieve of this kind can, by itself, give a lower bound for primes. This is the parity problem. It explains why Jingrun Chen’s 1973 theorem, that every large even number is a prime plus a number with at most two prime factors, is the limit of pure sieve methods, and why later advances needed extra information beyond the sieve axioms.

The elementary prime number theorem

For the general reader. The prime number theorem says that the number of primes up to \(x\) is about \(x/\log x\). It was first proved in 1896 with complex analysis, and many believed that a proof without complex numbers was impossible. In 1948 Selberg found an identity about primes that needs only elementary estimates, and from it he and Paul Erdős each obtained elementary proofs. The episode made Selberg famous and left a lasting quarrel.

Proof sketch

The arithmetic identity behind the formula is

Sum the right side over \(n\le x\): it equals \(\sum_{d\le x}\mu(d)\sum_{m\le x/d}\log^2m\). Replace the inner sum by its Euler–Maclaurin expansion \(F(y)=y\log^2y-2y\log y+2y+O(\log^2 y)\). The main terms then reduce, through the elementary estimates \(\sum_{d\le y}\mu(d)/d=O(1)\), \(\sum_{d\le y}\mu(d)\log(y/d)/d=O(1)\) and their relatives, to \(2x\log x+O(x)\). No information about the zeros of \(\zeta\) is used.

From the formula to the theorem

Write \(\psi(x)=x+R(x)\). The fundamental formula gives an integral inequality of the form \(|R(x)|\log x\le\sum_{n\le x}|R(x/n)|\log n\cdot(\text{weights})+O(x)\), which says that \(|R|\) is controlled by its own averages. Selberg first showed that \(\liminf\theta(x)/x+\limsup\theta(x)/x=2\). An extra input then forces both limits to equal 1. Erdős supplied one such input in July 1948, an elementary proof that every interval \((x,(1+\varepsilon)x]\) contains at least \(\delta(\varepsilon)x/\log x\) primes. Selberg used it to complete a first proof on 18 July 1948, and soon found a second proof that avoided Erdős’s result.

The dispute

The two men disagreed over how to publish. Selberg proposed separate papers, each with its author’s own contribution; Erdős wanted a joint paper. In the end Selberg’s An elementary proof of the prime-number theorem appeared in Annals of Mathematics in 1949, and Erdős published his own version in the Proceedings of the National Academy of Sciences. Hermann Weyl was drawn into the correspondence. Selberg’s account, with documents, was later published by Dorian Goldfeld. Selberg also gave an elementary proof of Dirichlet’s theorem on primes in arithmetic progressions (1949).

Rankin–Selberg and the eigenvalue conjecture

For the general reader. Modular forms are highly symmetric functions on the upper half-plane whose Fourier coefficients carry arithmetic information. Ramanujan conjectured that the coefficients of his τ-function are small. Rankin and Selberg found, independently around 1939–1940, a way to compute their average size by integrating against a second kind of symmetric function, an Eisenstein series. Twenty-five years later Selberg turned the method into a lower bound for the vibrations of a hyperbolic surface, and stated a conjecture that is still open.

Unfolding

Let \(\Gamma=\mathrm{SL}_2(\mathbb Z)\), \(\Gamma_\infty\) its subgroup of translations \(\pm\begin{pmatrix}1&n\\0&1\end{pmatrix}\), and \(d\mu=dx\,dy/y^2\). The non-holomorphic Eisenstein series is

It continues meromorphically to \(\mathbb C\), with a simple pole at \(s=1\) of residue \(3/\pi\). Let \(f(z)=\sum_{n\ge1}a(n)e^{2\pi inz}\) be a cusp form of weight \(k\). Since \(y^k|f(z)|^2\) is \(\Gamma\)-invariant, the integral over \(\Gamma\backslash\mathbb H\) unfolds to the strip \(\Gamma_\infty\backslash\mathbb H\):

For Ramanujan’s \(\Delta=\sum\tau(n)q^n\) of weight 12, this shows that \(\tau(n)\) is of size \(n^{11/2}\) on average, in line with Ramanujan’s conjecture \(|\tau(p)|\le2p^{11/2}\), which Pierre Deligne proved in 1974. Rankin–Selberg convolutions \(L(s,f\times g)\) are now basic objects of the Langlands programme.

The eigenvalue conjecture

For the congruence subgroup \(\Gamma_0(N)\), the Laplacian \(\Delta=-y^2(\partial_x^2+\partial_y^2)\) on \(L^2(\Gamma_0(N)\backslash\mathbb H)\) has eigenvalue \(\lambda_0=0\) (constants) and further discrete eigenvalues \(\lambda_1\le\lambda_2\le\cdots\). Write \(\lambda=s(1-s)\).

The proof studies the Kloosterman zeta function \(\sum_c S(m,n;c)c^{-2s}\), whose poles are governed by the exceptional eigenvalues, and bounds it with André Weil’s estimate for Kloosterman sums. The conjecture is the archimedean analogue of the Ramanujan conjecture. The best bound known, due to Henry Kim and Peter Sarnak (2003), is \(\lambda_1\ge\tfrac{975}{4096}\approx0.238\).

The trace formula

For the general reader. A drum’s sound is determined by its shape. On a curved surface of negative curvature, Selberg found an exact equation relating the frequencies at which the surface can vibrate to the lengths of its closed geodesics, the loops a string would form if pulled tight. The equation has the same shape as the explicit formula that links the zeros of the zeta function to the primes. It is the template for much of modern number theory.

Setting

Let \(\Gamma\subset\mathrm{PSL}_2(\mathbb R)\) be discrete, cocompact and torsion-free, so \(X=\Gamma\backslash\mathbb H\) is a compact surface of genus \(g\ge2\) with area \(4\pi(g-1)\). Each non-identity \(\gamma\in\Gamma\) is hyperbolic and is conjugate to \(z\mapsto e^{\ell}z\), where \(\ell=\ell(\gamma)>0\) is the length of the corresponding closed geodesic and \(|\operatorname{tr}\gamma|=2\cosh(\ell/2)\). Conjugacy classes correspond to closed geodesics; a class is primitive if it is not a power. Let \(0=\lambda_0<\lambda_1\le\cdots\) be the Laplace eigenvalues, \(\lambda_j=\tfrac14+r_j^2\).

Proof idea

An integral operator \(Lf(z)=\int_{\mathbb H}k(z,w)f(w)\,d\mu(w)\) with a point-pair invariant kernel \(k(z,w)=\Phi(d(z,w))\) commutes with isometries, so every Laplace eigenfunction is an eigenfunction of \(L\), with eigenvalue \(h(r)\) given by the Selberg transform of \(\Phi\). On \(X\) the kernel is \(K(z,w)=\sum_{\gamma\in\Gamma}k(z,\gamma w)\). Compute \(\int_XK(z,z)\,d\mu\) twice: spectrally, it is \(\sum_jh(r_j)\); geometrically, grouping \(\gamma\) by conjugacy class, the identity gives the area term and each hyperbolic class gives an orbital integral, which evaluates to the length term.

The Selberg zeta function

Applying the trace formula to \(h(r)=\frac1{r^2+(s-\frac12)^2}-\frac1{r^2+(\beta-\frac12)^2}\) shows that \(Z'/Z\) is a sum over the spectrum. Hence \(Z\) extends to an entire function whose non-trivial zeros are exactly the \(s_j=\tfrac12\pm ir_j\), with \(\lambda_j=s_j(1-s_j)\). Apart from the finitely many \(\lambda_j<\tfrac14\), these lie on \(\operatorname{Re}s=\tfrac12\): the Riemann hypothesis holds for \(Z\), because the Laplacian is self-adjoint. Selberg also developed the theory for non-compact quotients such as \(\mathrm{SL}_2(\mathbb Z)\backslash\mathbb H\), where Eisenstein series supply a continuous spectrum.

Discrete groups and rigidity

For the general reader. A hyperbolic surface of genus two can be bent continuously into infinitely many different shapes. Selberg discovered in 1960 that the higher-dimensional analogues built from larger matrix groups cannot be bent at all: their shape is locked by their symmetry group. This rigidity became one of the central themes of geometry, leading to the work of Mostow and Margulis.

Proof for \(\mathrm{SL}_n(\mathbb Z)\). Let \(\Gamma(p)\) be the kernel of reduction mod a prime \(p\ge3\); it has finite index. Suppose \(g=I+p^aB\in\Gamma(p)\) with \(a\ge1\), \(B\) not divisible by \(p\), and \(g^m=I\). Replacing \(g\) by a power, we may assume \(m\) is prime. Then \(0=g^m-I=mp^aB+\binom m2p^{2a}B^2+\cdots\). If \(m\ne p\), reduce mod \(p^{a+1}\) to get \(mB\equiv0\), a contradiction. If \(m=p\), the first term is \(p^{a+1}B\), and since \(p\ge3\) all others are divisible by \(p^{2a+1}\), again a contradiction. For a general finitely generated group, the entries lie in a finitely generated ring, and one reduces modulo a suitable maximal ideal instead.

Rigidity

A lattice in a Lie group \(G\) is a discrete subgroup \(\Gamma\) with \(\Gamma\backslash G\) of finite volume. \(\Gamma\) is locally rigid if every homomorphism \(\Gamma\to G\) close enough to the inclusion is conjugate to it.

Selberg’s argument studied how the Weyl chambers of the symmetric space \(\mathrm{SL}_n(\mathbb R)/\mathrm{SO}(n)\) behave under deformation. Eugenio Calabi and Edoardo Vesentini, and then André Weil, proved local rigidity for cocompact lattices in general semisimple groups without \(\mathrm{SL}_2(\mathbb R)\) factors, by cohomological methods. George Mostow proved global, strong rigidity in 1968, and Grigory Margulis proved superrigidity and arithmeticity in higher rank in 1974–75, answering a question Selberg had raised: lattices in higher-rank groups are arithmetic, built from matrices with integer entries.

The Selberg integral and the Selberg class

For the general reader. Two of Selberg’s ideas waited decades for their audience. In 1944 he evaluated a many-dimensional integral in a short Norwegian paper that few read. In the 1970s and 1980s physicists and combinatorialists rediscovered it in random matrices and conformal field theory. In 1989 he proposed axioms for the L-functions of number theory and stated conjectures that would explain why they all seem to come from automorphic forms.

For \(n=1\) it is Euler’s beta integral. Selberg proved it first for positive integer \(\gamma\) and extended it to general \(\gamma\) by Carlson’s theorem on analytic functions. Its limits give the normalizing constants of the Gaussian and Laguerre ensembles of random matrix theory, and Mehta’s integral. Peter Forrester and Ole Warnaar have surveyed its many descendants.

The Selberg class

A Dirichlet series \(F(s)=\sum_{n\ge1}a(n)n^{-s}\) belongs to the Selberg class \(\mathcal S\) if:

1. it converges absolutely for \(\operatorname{Re}s>1\), and \(a(1)=1\);
2. \((s-1)^mF(s)\) extends to an entire function of finite order for some \(m\ge0\);
3. there are \(Q>0\), \(\lambda_j>0\), \(\operatorname{Re}\mu_j\ge0\) and \(|\omega|=1\) such that \(\Phi(s)=Q^s\prod_j\Gamma(\lambda_js+\mu_j)F(s)\) satisfies \(\Phi(s)=\omega\,\overline{\Phi(1-\bar s)}\);
4. (Ramanujan) \(a(n)=O(n^{\varepsilon})\) for every \(\varepsilon>0\);
5. (Euler product) \(\log F(s)=\sum_nb(n)n^{-s}\) with \(b(n)=0\) unless \(n\) is a prime power, and \(b(n)=O(n^{\vartheta})\) for some \(\vartheta<\tfrac12\).

The degree \(d_F=2\sum_j\lambda_j\) does not depend on the choice of gamma factors. Selberg conjectured that \(\sum_{p\le x}|a_F(p)|^2/p=n_F\log\log x+O(1)\) with \(n_F\) a positive integer, and that for distinct primitive \(F,G\) the corresponding sum of \(a_F(p)\overline{a_G(p)}/p\) is bounded (Conjecture B). These would imply unique factorization in \(\mathcal S\) into primitive functions, and, with the Langlands programme, that every element of \(\mathcal S\) is an automorphic L-function. Jerzy Kaczorowski and Alberto Perelli proved that there are no elements of degree strictly between 0 and 1, or between 1 and 2.

Forty years at the Institute, 1949–1987

Selberg spent his whole career after 1949 at the Institute for Advanced Study, as permanent member and, from 1951, professor. His colleagues included Siegel, Weyl, Weil and, later, Robert Langlands and Enrico Bombieri. He had no regular teaching duties and did not build a school of doctoral students. His influence came through lectures, conversation and a small number of papers of great depth.

Much of what he knew he did not publish. He lectured on the trace formula before the 1956 paper appeared, and his notes on Eisenstein series and on the analytic theory of automorphic forms for groups of higher rank circulated for years in mimeographed form. They shaped the work of Harish-Chandra and Langlands on the general theory of Eisenstein series, and the later development of the Arthur–Selberg trace formula. Dennis Hejhal’s volumes on the trace formula for \(\mathrm{PSL}_2(\mathbb R)\) brought much of the theory into print.

His only co-author was Sarvadaman Chowla. Their papers on Epstein zeta functions (1949 and 1967) contain the Chowla–Selberg formula expressing periods of elliptic curves with complex multiplication through values of the gamma function. He gave an invited address at the 1962 Congress in Stockholm. Bombieri summarized his style as simplicity and elegance of method joined to powerful results. His wife Hedvig worked at the Institute in the 1950s and later at the Princeton Plasma Physics Laboratory.

Death and legacy

Selberg became professor emeritus in 1987 and continued to think about mathematics and to lecture well into his eighties. In 1987 a symposium in Oslo marked his seventieth birthday, and his Collected Papers appeared in two volumes in 1989 and 1991. Hedvig Selberg died in 1995. In 2003 he married Betty Frances Compton. He died of heart failure at his home in Princeton on 6 August 2007, aged 90. Peter Sarnak called him a mathematician’s mathematician.

Honours

• Prizes: Fields Medal (1950); Wolf Prize in Mathematics (1986, with Samuel Eilenberg), for profound and original work on number theory, discrete groups and automorphic forms; honorary Abel Prize for the Abel bicentenary (2002); Gunnerus Medal of the Royal Norwegian Society of Sciences and Letters (2002).
• Orders and degrees: Commander with Star of the Royal Norwegian Order of St Olav (1987); honorary doctorate of the Norwegian Institute of Technology (1972).
• Academies: Norwegian Academy of Science and Letters, Royal Danish and Royal Swedish Academies, American Academy of Arts and Sciences, Indian National Science Academy; honorary member of the London Mathematical Society (1985).

Influence

• Analytic number theory: the Λ² sieve and the parity principle frame every modern sieve result, from Chen’s theorem to the bounded-gaps work of Zhang and Maynard, which uses Selberg-type weights. Mollifiers remain the main tool for proportions of zeros.
• Automorphic forms: the trace formula, Eisenstein series and Rankin–Selberg convolutions are foundations of the Langlands programme.
• Geometry and dynamics: rigidity and Selberg’s lemma began the line leading to Mostow and Margulis; the Selberg zeta function and the eigenvalue conjecture link spectral geometry, quantum chaos and expander graphs.
• Mathematical physics: the Selberg integral underlies exact computations in random matrix theory and conformal field theory.

Connections

People linked to Atle Selberg across this collection. A name without a link has no page here yet. See Atle Selberg in the network of all 337 lives →

Influenced by

• Eratosthenes
• G. H. Hardy
• Carl Størmer
• Harald Bohr
• Srinivasa Ramanujan
• Viggo Brun

Influenced

• Grigory Margulis
• Harish-Chandra
• Robert Langlands

Colleagues and collaborators

• André Weil
• Carl Ludwig Siegel 1947-1951
• Harish-Chandra
• Hermann Weyl
• Robert Langlands
• Enrico Bombieri
• Robert Rankin
• Sarvadaman Chowla 1949-1967

Rivals

• Paul Erdős 1948-1949

Family

• Arne Selberg
• Henrik Selberg
• Sigmund Selberg

Also mentioned in this life

• Niels Henrik Abel
• Augustin-Louis Cauchy
• Arthur Compton
• Pierre Deligne
• Peter Gustav Lejeune Dirichlet
• Leonhard Euler
• Joseph Fourier
• Pierre-Simon Laplace
• Gottfried Wilhelm Leibniz
• Sophus Lie
• Colin Maclaurin
• August Ferdinand Möbius
• Bernhard Riemann
• Oswald Teichmüller

Fields

• analytic number theory 4 lives
• automorphic forms 4 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Atle Selberg, MacTutor History of Mathematics.
2. Atle Selberg, Wikipedia.
3. Institute for Advanced Study, Atle Selberg 1917–2007, press release, August 2007.
4. D. R. Heath-Brown, Atle Selberg (1917–2007), Bull. London Math. Soc. (2010).
5. N. A. Baas and C. F. Skau, The lord of the numbers, Atle Selberg: on his life and mathematics, Bull. AMS 45 (2008), 617–649.
6. D. Goldfeld, The elementary proof of the prime number theorem: an historical perspective.
7. Obituaries in The Times and The Guardian, 2007.

Mathematics

1. A. Selberg, Collected Papers, vols. I–II, Springer, 1989 and 1991.
2. A. Selberg, On the zeros of Riemann’s zeta-function, Skr. Norske Vid.-Akad. Oslo (1942), no. 10.
3. A. Selberg, Bemerkninger om et multipelt integral, Norsk Mat. Tidsskr. 26 (1944), 71–78.
4. A. Selberg, On an elementary method in the theory of primes, Norske Vid. Selsk. Forh. 19 (1947).
5. A. Selberg, An elementary proof of the prime-number theorem, Ann. of Math. 50 (1949), 305–313.
6. A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 47–87.
7. A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces, in Contributions to Function Theory, Bombay, 1960.
8. A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math. 8 (1965).
9. A. Selberg, Old and new conjectures and results about a class of Dirichlet series, Proc. Amalfi Conf. Analytic Number Theory (1992).
10. D. A. Hejhal, The Selberg Trace Formula for PSL(2,ℝ), Lecture Notes in Math. 548 and 1001, Springer, 1976 and 1983.
11. H. Iwaniec, Spectral Methods of Automorphic Forms, 2nd ed., AMS, 2002.
12. J. Friedlander and H. Iwaniec, Opera de Cribro, AMS Colloquium Publications 57, 2010.
13. P. J. Forrester and S. O. Warnaar, The importance of the Selberg integral, Bull. AMS 45 (2008), 489–534.
14. J. Kaczorowski, Axiomatic theory of L-functions: the Selberg class, in Analytic Number Theory, Lecture Notes in Math. 1891, Springer, 2006.