Carl Ludwig Siegel

How to use this course

Siegel’s name is attached to more objects than almost any other twentieth-century mathematician: Siegel’s lemma, Siegel zeros, the Riemann–Siegel formula, the Siegel mass formula, Siegel modular forms, the Siegel upper half-space, Siegel disks. These are not variations on one idea. They come from at least five different fields, and in each he proved a theorem that defined the field for decades. This course follows the work roughly in the order he did it.

Each technical lesson opens with a paragraph in italics for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Readers who want only the life can read Parts I and III together with those opening paragraphs.

Timeline

Berlin, 1896–1919

Carl Ludwig Siegel was born in Berlin on 31 December 1896. His father, Peter Siegel, was a postman. The family had no academic background, and Siegel grew up in modest circumstances in the imperial capital.

He entered the University of Berlin in the autumn of 1915, in the second year of the First World War. He chose astronomy, by his own later account because it seemed as far as possible from earthly affairs. He also took mathematics and theoretical physics, and heard Max Planck. The decisive teacher was Georg Frobenius, whose lectures on number theory in his first semester turned him from the stars to the integers. At the end of that semester, on Frobenius’s proposal, he received the university’s Eisenstein Prize.

Refusing the war

Siegel was a convinced opponent of the war. When he was called up in 1917 he could not, or would not, adapt to military life, and he spent a period in a psychiatric institution before being discharged. He later said that he endured it mainly thanks to the support of Edmund Landau, whose father ran a clinic nearby. The episode left him with a lifelong hatred of militarism and of authority exercised for its own sake. It also made Landau, the Göttingen number theorist, his natural next teacher.

Siegel never married. Colleagues remembered him as a loner, wholly absorbed in mathematics, with a sharp tongue and a strong sense of independence.

Göttingen and Hamburg, 1919–1922

In 1919 Siegel moved to Göttingen. In his third semester he had already written a very short paper on how badly algebraic numbers can be approximated by fractions, sharpening a theorem of the Norwegian Axel Thue. Landau saw its importance. Expanded, it became Siegel’s dissertation, Approximation algebraischer Zahlen, defended on 9 June 1920 and published in 1921. Lesson 04 explains it; it remains one of the most influential theses in number theory.

In the winter of 1920–21 he held a teaching assignment at the new University of Hamburg, where Erich Hecke was professor. He then returned to Göttingen as assistant to Richard Courant and habilitated on 10 December 1921. The habilitation thesis extended the circle method of G. H. Hardy and J. E. Littlewood to Waring’s problem in totally real number fields, the question of writing algebraic integers as sums of a bounded number of \(k\)-th powers. In the same years he gave a new proof of the functional equation of Dedekind zeta functions.

Within three years of his doctorate he had published fundamental work in Diophantine approximation, additive number theory and analytic number theory. Frankfurt offered him a chair at the age of 25.

Frankfurt, 1922–1937

On 1 August 1922 Siegel became full professor at the young University of Frankfurt, succeeding Arthur Schoenflies. His colleagues were Max Dehn, Ernst Hellinger, Paul Epstein and Otto Szász. Siegel later called the years that followed the best of his life. The small department was unusually close. From 1922 Dehn ran a seminar on the history of mathematics, which lasted thirteen years, in which the participants read the classical works in their original languages. Siegel took part throughout. The same respect for original sources shows in his reconstruction of Riemann’s unpublished work (Lesson 07).

These fifteen years produced most of his deepest work in number theory: the 1929 theorem on integral points, the 1932 Riemann–Siegel formula, the 1935 class number theorem, and the analytic theory of quadratic forms (1935–37). His Frankfurt doctoral students included Kurt Mahler and Wilhelm Maier (both 1927), Theodor Schneider (1934), and Helene (Hel) Braun and Walter Wagner (1937). Schneider’s thesis solved Hilbert’s seventh problem, independently of Aleksandr Gelfond.

In 1930 he spent a semester in Göttingen standing in for Hilbert, while Emmy Noether covered his Frankfurt teaching. In 1935 he visited the Institute for Advanced Study in Princeton from January to the summer.

The end of the Frankfurt school

The Nazi seizure of power destroyed the department. Szász emigrated in 1933. Dehn, Hellinger and Epstein, all of Jewish descent, were forced out in 1935; Siegel returned from Princeton to find them gone. Siegel refused to open his lectures with the Hitler salute and refused to collect for the party’s winter relief fund. A rector’s note of 1935 described him as certainly no National Socialist. He dedicated a 1936 paper to Epstein and openly criticized the politically motivated appointment of a replacement teacher. Unable to bear Frankfurt any longer, he accepted a chair at Göttingen and moved there on 1 January 1938. Schneider and Braun followed him. After the November 1938 pogrom Hellinger was held for weeks in Dachau and then emigrated; Dehn escaped to Norway; Epstein took his own life in August 1939.

Approximating algebraic numbers

For the general reader. Every real number can be approximated by fractions, and some fractions do surprisingly well: \(355/113\) agrees with \(\pi\) to six decimals. A natural question is how well an irrational root of a polynomial, such as \(\sqrt[3]{2}\), can be approximated. Siegel’s thesis showed that algebraic numbers resist good approximation much more than had been known. That fact turns out to control whether polynomial equations have finitely or infinitely many whole-number solutions.

From Liouville to Roth

Let \(\alpha\) be a real algebraic number of degree \(n\ge2\). Say that \(\alpha\) has approximation exponent at most \(\mu\) if for every \(\varepsilon>0\) there is \(c=c(\alpha,\varepsilon)>0\) with

Joseph Liouville (1844) showed \(\mu=n\) works. The successive improvements are a history of the subject:

All these bounds are ineffective beyond Liouville’s: the proofs show that there are only finitely many very good approximations, but they do not bound how large those approximations could be. This feature reappears in Lessons 05 and 06.

The method

Thue’s idea, which Siegel made flexible, has three steps. First, construct a polynomial \(P(x,y)\) with integer coefficients that are not too large, vanishing to high order at \((\alpha,\alpha)\). Second, suppose there are two exceptionally good approximations \(p_1/q_1\) and \(p_2/q_2\) with \(q_2\) much larger than \(q_1\). Then \(P(p_1/q_1,p_2/q_2)\), or one of its low-order derivatives, is a nonzero rational with denominator at most \(q_1^{a}q_2^{b}\), so its absolute value is at least \(q_1^{-a}q_2^{-b}\). Third, Taylor expansion around \((\alpha,\alpha)\) shows the same quantity is extremely small. The two estimates contradict each other. The first step needs a polynomial with small integer coefficients satisfying many linear conditions, and Siegel isolated the tool for it.

Proof sketch. Let \(H\) be a positive integer and consider the \((H+1)^N\) vectors \(x\in\{0,\dots,H\}^N\). Each row form \(L_i(x)=\sum_j a_{ij}x_j\) takes values in an interval of length at most \(NBH\), so the vector \((L_1(x),\dots,L_M(x))\) takes at most \((NBH+1)^M\) values. If \((H+1)^N>(NBH+1)^M\), two vectors \(x\ne x'\) give the same values, and \(x-x'\) is a nonzero solution with entries bounded by \(H\). Choosing \(H\) of size about \((NB)^{M/(N-M)}\) makes the inequality hold. The more unknowns you have relative to equations, the smaller the solution.

Integral points and transcendence

For the general reader. In a single long paper of 1929, Siegel proved two theorems that each founded a field. The first says that most polynomial equations in two variables have only finitely many solutions in whole numbers. The second gives a method to prove that values of functions like the Bessel functions of physics are transcendental, that is, not roots of any polynomial with integer coefficients.

Integral points on curves

Genus 0 with fewer points at infinity can fail: the Pell equation \(x^2-2y^2=1\) has infinitely many integer solutions. Kurt Mahler extended the theorem to \(S\)-integers for elliptic curves, and the general \(S\)-integer version is now usually called the Siegel–Mahler theorem.

Proof idea

Siegel combined two deep inputs. The Mordell–Weil theorem says the group of rational points of the Jacobian \(J\) of \(C\) is finitely generated. Embed \(C\) in \(J\). If there were infinitely many integral points \(P\), write \(P=mQ+R\) with \(R\) from a finite set of coset representatives of \(J(K)/mJ(K)\). Pulling back by multiplication by \(m\) gives a cover \(C'\to C\), unramified, on which the points become ‘smaller’ by a factor about \(m^2\) in height. Integral points on \(C\) approach the points at infinity very closely in the \(p\)-adic or real sense, and on the cover this closeness translates into approximations of an algebraic number of large degree by numbers of smaller height, too good for Siegel’s improved Thue inequality. Taking \(m\) large produces a contradiction. Because Thue–Siegel bounds are ineffective, the theorem gives no bound on the size of the solutions. Effective versions for many curves came only with Alan Baker’s work on linear forms in logarithms in the late 1960s. For curves of genus at least 2, Gerd Faltings’ 1983 proof of Mordell’s conjecture gives finiteness of all rational points, not only integral ones.

E-functions

Siegel called an entire function \(f(z)=\sum_{k\ge0}a_k\,z^k/k!\) an E-function if the \(a_k\) lie in a fixed number field, if the house and the common denominator of \(a_0,\dots,a_k\) grow at most like \(C^k\) up to a factor \(k^{\varepsilon k}\) for every \(\varepsilon\), and if \(f\) satisfies a linear differential equation with rational function coefficients. Examples are \(e^z\) and the Bessel function \(J_0(z)=\sum_k (-1)^k (z/2)^{2k}/(k!)^2\).

The proof again begins with Siegel’s lemma: an auxiliary function \(\sum_i P_i(z)f_i(z)\) with polynomial coefficients is built to vanish to high order at 0. His student Theodor Schneider used auxiliary-function methods of this kind in 1934 to prove that \(\alpha^\beta\) is transcendental for algebraic \(\alpha\ne0,1\) and algebraic irrational \(\beta\) (the Gelfond–Schneider theorem, Hilbert’s seventh problem).

Class numbers and the Siegel zero

For the general reader. Gauss conjectured that the class number of imaginary quadratic fields, which measures how badly unique factorization fails, grows without bound. Siegel proved it grows almost as fast as the square root of the discriminant. His proof has a strange feature: it works by contradiction against a hypothetical zero of an \(L\)-function that probably does not exist, so it cannot tell you how large the constants are. Such a hypothetical zero is now called a Siegel zero.

The class number formula

Let \(-d<0\) be a fundamental discriminant, \(\chi_{-d}=\big(\tfrac{-d}{\cdot}\big)\) the associated real character and \(h(-d)\) the class number of \(\mathbb Q(\sqrt{-d})\). For \(d>4\), Dirichlet’s formula reads

Upper bounds \(L(1,\chi)\ll\log d\) are easy. The difficulty is a lower bound, which is the same as keeping \(L(s,\chi)\) from having a real zero very close to \(s=1\).

Why the constant is ineffective

A standard modern form of the proof runs as follows. Fix \(\varepsilon\). Case 1: no real character has a zero in \((1-\varepsilon/10,1)\); then the bound follows for every \(\chi\) with an explicit constant. Case 2: some \(\chi_1\) of conductor \(q_1\) has such a zero \(\beta_1\). For any other \(\chi\) of conductor \(q\), study \(F(s)=\zeta(s)L(s,\chi_1)L(s,\chi)L(s,\chi\chi_1)\), the Dedekind zeta function of a biquadratic field. Its Dirichlet coefficients are non-negative, and \(F(\beta_1)\le0\). Comparing the residue at \(s=1\), which contains \(L(1,\chi)\), with the vanishing at \(\beta_1\) gives \(L(1,\chi)\gg q^{-\varepsilon}\), with a constant depending on \(q_1\) and \(\beta_1\). Since nobody knows whether Case 2 occurs, or for which \(\chi_1\), the constant cannot be computed. The hypothetical \(\beta_1\) is the Siegel zero; the generalized Riemann hypothesis would rule it out.

Gauss’s list of imaginary quadratic fields with class number 1 was proved complete only later, by Kurt Heegner (1952), Alan Baker (1966) and Harold Stark (1967), by entirely different means.

Two consequences

Siegel–Walfisz is the standard input for the distribution of primes in progressions with small modulus, for example in Vinogradov’s three-primes theorem. The implied constant inherits the ineffectivity.

Riemann’s Nachlass: the Riemann–Siegel formula

For the general reader. Bernhard Riemann published only one paper on prime numbers, in 1859, and it contains no numerical work. Seventy years later Siegel deciphered Riemann’s private notes in the Göttingen library and found that Riemann had a powerful method for computing the zeta function, which he had never published. Siegel’s 1932 reconstruction is still the basis of every large-scale computation of zeros of the zeta function.

The function \(Z(t)\)

Define the Riemann–Siegel theta function and \(Z\) by

The functional equation of \(\zeta\) makes \(Z(t)\) real for real \(t\), with \(|Z(t)|=|\zeta(\tfrac12+it)|\). So zeros on the critical line are sign changes of a real function, which can be detected rigorously. Stirling’s formula gives

Where it comes from

The main sum is the approximate functional equation: \(\zeta(s)\approx\sum_{n\le N}n^{-s}+\chi(s)\sum_{n\le N}n^{s-1}\), which on the critical line pairs into the cosines above. Riemann had an exact integral representation of the remainder, a contour integral of \(e^{i\pi z^2}\)-type kernels against \(1/(e^{2\pi i z}-1)\). Its asymptotic evaluation by the saddle-point method produces the correction terms. Siegel found these computations scattered across Riemann’s notes, including numerical values for the first zeros, and wrote them up with complete proofs. The episode overturned the view that Riemann’s 1859 conjectures were inspired guesses: he had computed.

The formula needs \(N\approx\sqrt{t/2\pi}\) terms, whereas Euler–Maclaurin summation needs about \(t/2\pi\). Turing, Lehmer, and later Odlyzko and others used it to verify the Riemann hypothesis for billions of zeros.

Quadratic forms and the mass formula

For the general reader. In how many ways can a number be written as a sum of four squares, or of eight? Formulas for such counts go back to Jacobi. Siegel found the general principle behind them. A single quadratic form can be hard to understand, but the average over its whole family, its genus, is given exactly by a product of simple local counts, one for each prime. His three papers of 1935–37 created the analytic theory of quadratic forms.

Class and genus

A positive definite integral quadratic form in \(m\) variables is \(Q(x)=x^{\mathsf T}Sx\) with \(S\) a symmetric positive definite integer matrix (or a lattice \(L\subset\mathbb R^m\) with integral inner products). Two forms are in the same class if \(S'=U^{\mathsf T}SU\) for some \(U\in\mathrm{GL}_m(\mathbb Z)\), and in the same genus if they are equivalent over \(\mathbb R\) and over every \(p\)-adic ring \(\mathbb Z_p\). A genus contains finitely many classes \(S_1,\dots,S_h\), each with a finite automorphism group \(O(S_i)\). Let \(r(S_i,T)\) be the number of representations \(X^{\mathsf T}S_iX=T\) of a form \(T\) of size \(n\le m\).

The case \(T=S\) gives the mass formula: the mass \(\sum_i 1/|O(S_i)|\) of a genus is an explicit product of local factors. Henry Smith and Hermann Minkowski had stated versions in the nineteenth century; Siegel gave the first complete proof. When the genus has one class, the theorem gives exact formulas, recovering Jacobi’s four-square formula. André Weil later recast it, in the language of adeles, as an identity between a theta series averaged over the genus and an Eisenstein series: the Siegel–Weil formula.

The Siegel upper half-space

For the general reader. Classical modular forms live on the upper half-plane, a space of complex numbers. They organize elliptic curves and encode arithmetic. Siegel, led there by quadratic forms, built the same theory on a space of matrices. The resulting Siegel modular forms describe abelian varieties, the higher-dimensional cousins of elliptic curves, and are central in modern number theory.

The space and the group

The Siegel upper half-space of degree \(g\) is

an open set of complex dimension \(g(g+1)/2\). The symplectic group \(\mathrm{Sp}_{2g}(\mathbb R)=\{M\in\mathrm{GL}_{2g}(\mathbb R): M^{\mathsf T}JM=J\}\), with \(J=\begin{pmatrix}0&I\\-I&0\end{pmatrix}\), acts by

For \(g=1\) this is the action of \(\mathrm{SL}_2(\mathbb R)\) on the upper half-plane by Möbius transformations. Siegel showed in his 1943 paper Symplectic Geometry that \(\mathcal H_g\) carries an invariant metric \(ds^2=\operatorname{tr}(Y^{-1}dZ\,Y^{-1}d\bar Z)\), \(Y=\operatorname{Im}Z\), making it a symmetric space in Élie Cartan’s sense, and he constructed a fundamental domain for \(\Gamma_g=\mathrm{Sp}_{2g}(\mathbb Z)\).

The link to Lesson 08: if \(S\) is an even unimodular positive definite form of rank \(m\), its theta series of degree \(g\), \(\vartheta_S^{(g)}(Z)=\sum_{X\in\mathbb Z^{m\times g}}e^{\pi i\operatorname{tr}(X^{\mathsf T}SXZ)}\), is a Siegel modular form of weight \(m/2\), and the coefficient \(a(T)\) counts representations of \(2T\) by \(S\). Siegel’s main theorem says the genus average of these theta series is a Siegel Eisenstein series. The quotient \(\Gamma_g\backslash\mathcal H_g\) is the moduli space of principally polarized abelian varieties of dimension \(g\), now called the Siegel modular variety.

Small divisors and celestial mechanics

For the general reader. Is the solar system stable? Perturbation series for planetary orbits contain denominators that can be arbitrarily small, and for a century nobody could show the series converge. In 1942 Siegel solved the simplest model of this ‘small divisor’ problem. He showed that a rotation disturbed by a nonlinear term remains a rotation in disguise, provided the rotation angle is badly approximable by fractions. The same number theory as his thesis decided a question in dynamics. His student Jürgen Moser carried the method into Kolmogorov–Arnold–Moser (KAM) theory.

The problem

Let \(f(z)=\lambda z+\sum_{k\ge2}a_kz^k\) be holomorphic near 0 with \(|\lambda|=1\), \(\lambda=e^{2\pi i\alpha}\), \(\alpha\) irrational. We seek a holomorphic change of coordinates \(h(z)=z+\sum_{k\ge2}b_kz^k\) with

Comparing coefficients of \(z^k\) gives \((\lambda^k-\lambda)b_k=P_k(a_2,\dots,a_k,b_2,\dots,b_{k-1})\), a polynomial. So \(h\) exists as a formal series, but each step divides by \(\lambda^k-\lambda=\lambda(\lambda^{k-1}-1)\), and \(|\lambda^{n}-1|=2|\sin\pi n\alpha|\) is small whenever \(n\alpha\) is close to an integer. Hubert Cremer had shown that for some \(\alpha\) the divisors are so small that no convergent \(h\) exists.

Proof idea

A direct bound on the recursion loses a factor of about \(|\lambda^{n}-1|^{-1}\ge c'n^{-(\mu-1)}\) at each step, and the products of such factors along nested chains of the recursion could grow like \((k!)^{\mu}\). Siegel’s insight was combinatorial. In any product of divisors \(|\lambda^{n_1}-1|\cdots|\lambda^{n_r}-1|\) arising from the recursion, the very small ones cannot all occur together. Two indices with \(n_i\alpha\) and \(n_j\alpha\) both very close to integers force \((n_i-n_j)\alpha\) close to an integer as well, which the Diophantine condition forbids unless \(|n_i-n_j|\) is large. A careful count shows the product is at most \(C^k\), so the coefficients \(b_k\) grow at most geometrically and \(h\) has positive radius of convergence. Alexander Brjuno later relaxed the hypothesis to the condition \(\sum_n q_n^{-1}\log q_{n+1}<\infty\) on the continued-fraction denominators, and Jean-Christophe Yoccoz proved this sharp for quadratic polynomials (Fields Medal, 1994).

Siegel disks and KAM

For the quadratic polynomial \(P(z)=\lambda z+z^2\) with Diophantine \(\alpha\), the largest domain on which \(P\) is conjugate to the rotation \(z\mapsto\lambda z\) is a Siegel disk, the shape drawn at the top of this page. Its interior is foliated by invariant curves on which orbits rotate. Siegel turned to Hamiltonian systems in the same spirit. He proved convergence results for normal forms near equilibria and wrote Vorlesungen über Himmelsmechanik (1956), which Moser revised into Lectures on Celestial Mechanics (1971). Moser’s 1962 twist theorem, Kolmogorov’s 1954 announcement and Arnold’s 1963 proofs together showed that most quasi-periodic motions of nearly integrable systems survive perturbation.

Escape, Princeton, return, 1938–1951

Göttingen in 1938 was a shadow of the institute that Hilbert, Klein and Courant had built. Siegel taught there for two years with Schneider and Braun. His refusal to accommodate the regime had already cost him; his support for Dehn and Hellinger is reported to have blocked his appointment to Constantin Carathéodory’s chair in Munich.

In early 1940 he obtained permission to lecture in Denmark and Norway. In Norway, in March 1940, he met Max Dehn, who had fled there. Siegel did not return. He left Norway for the United States a few days before the German invasion of Norway on 9 April 1940, and joined the Institute for Advanced Study in Princeton.

The Princeton decade was extraordinarily productive. It gave the 1942 paper on iteration of analytic functions, the 1943 Symplectic Geometry, work on automorphic functions in several variables and Lie groups, the 1944 return to Waring’s problem in number fields, and lectures on transcendental numbers and on several complex variables that became books in 1949. He became a permanent professor in 1946. He was close to André Weil and Atle Selberg, who both left vivid recollections of him. Selberg remarked that Siegel’s results seemed almost impossible, even after they were done.

Siegel visited Göttingen in the winter of 1946–47 and, despite everything, chose to go back. In 1951 he returned to his Göttingen chair and stayed for the rest of his life.

Later years and legacy, 1951–1981

In Göttingen Siegel taught with care at every level, including first-year courses, and produced most of his doctoral students: among them Jürgen Moser (1952), Helmut Klingen (1955), Helmut Rüssmann (1958), and Christian Pommerenke and Erhard Gottschling (1959). Jean Dieudonné observed that Siegel had relatively few students because his papers left so little to improve by the same methods. He retired on 1 April 1959 and continued to work and lecture for many years. His Gesammelte Abhandlungen appeared in three volumes in 1966.

He was a sharp critic of fashions he thought empty. In the early 1960s he wrote to Louis Mordell attacking Serge Lang’s Diophantine Geometry for replacing concrete mathematics with abstract formalism, a letter that still circulates in discussions of mathematical style.

Honours

• Plenary speaker, International Congress of Mathematicians, Oslo 1936.
• Member of the Göttingen Academy (1951), the Bavarian Academy (1958), and the Danish, Norwegian and Swedish academies, the Académie des sciences and the Accademia dei Lincei; honorary member of the London Mathematical Society (1956); foreign associate of the US National Academy of Sciences (1968).
• Pour le Mérite for Sciences and Arts (1963); Great Cross of Merit with Star of the Federal Republic of Germany (1964).
• Honorary doctorates including Frankfurt (1964).
• The first Wolf Prize in Mathematics (1978), shared with Israel Gelfand.

Siegel died in Göttingen on 4 April 1981, aged 84, and is buried in the city cemetery.

Legacy

• Diophantine geometry: his finiteness theorem and Siegel’s lemma are the starting points of the subject that led to Faltings’ theorem and the subspace theorem.
• Transcendence: E-functions and the auxiliary-function method run through the work of Schneider, Shidlovsky, Baker and their successors.
• Analytic number theory: the Siegel zero remains the central obstruction to effective results about primes in progressions and class numbers.
• Automorphic forms: Siegel modular forms and the Siegel–Weil formula are foundations of the Langlands programme and of the arithmetic of abelian varieties.
• Dynamics: the solution of the first small-divisor problem opened the way to KAM theory and to the study of Siegel disks in complex dynamics.

Connections

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

Teachers and advisors

• Edmund Landau 1920

Students

• Ferdinand Braun 1937
• Christian Pommerenke 1959
• Erhard Gottschling 1959
• Helmut Klingen 1955
• Helmut Rüssmann 1958
• Jürgen Moser 1952
• Kurt Mahler 1927
• Theodor Schneider 1934
• Wilhelm Maier 1927

Influenced by

• Bernhard Riemann
• Georg Frobenius 1915-1917
• Hermann Minkowski
• Axel Thue

Influenced

• Richard Brauer

Colleagues and collaborators

• André Weil
• Atle Selberg 1947-1951
• Emmy Noether 1930
• Erich Hecke 1920-1921
• Helmut Hasse
• Israel Gelfand 1978
• Richard Courant 1921-1922
• Teiji Takagi 1922
• Ernst Hellinger 1922-1935

Rivals

• Serge Lang

Friends

• André Weil 1927-1981
• Max Dehn 1922-1952

Also mentioned in this life

• Friedrich Wilhelm Bessel
• Richard Dedekind
• Peter Gustav Lejeune Dirichlet
• Élie Cartan
• Leonhard Euler
• Joseph Fourier
• Évariste Galois
• Carl Friedrich Gauss
• G. H. Hardy
• David Hilbert
• Carl Gustav Jacob Jacobi
• Felix Klein
• Andrey Kolmogorov
• Robert Langlands
• Pierre-Simon Laplace
• Henri Lebesgue
• Sophus Lie
• Joseph Liouville
• Colin Maclaurin
• John Milnor
• August Ferdinand Möbius
• Max Planck
• Alan Turing
• Hermann Weyl

Fields

• number theory 45 lives
• automorphic forms 4 lives
• celestial mechanics 16 lives

On CoursesHub

• Classical Mechanics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Carl Ludwig Siegel, MacTutor History of Mathematics.
2. Carl Ludwig Siegel, Wikipedia.
3. Siegel, Carl Ludwig, Neue Deutsche Biographie.
4. Forschung Frankfurt 2/2008, Der Mathematiker Carl Ludwig Siegel in Frankfurt (PDF, in German).
5. Siegel, Dictionary of Scientific Biography entry, via MacTutor (PDF).
6. Carl Ludwig Siegel, Mathematics Genealogy Project.
7. C. L. Siegel, Zur Geschichte des Frankfurter Mathematischen Seminars, lecture of 1964, in Gesammelte Abhandlungen III.

Mathematics

1. C. L. Siegel, Approximation algebraischer Zahlen, Math. Z. 10 (1921).
2. C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. (1929).
3. C. L. Siegel, Über Riemanns Nachlaß zur analytischen Zahlentheorie, Quellen und Studien zur Geschichte der Mathematik B 2 (1932).
4. C. L. Siegel, Über die Classenzahl quadratischer Zahlkörper, Acta Arith. 1 (1935).
5. C. L. Siegel, Über die analytische Theorie der quadratischen Formen I–III, Ann. of Math. 36–38 (1935–37).
6. C. L. Siegel, Einführung in die Theorie der Modulfunktionen n-ten Grades, Math. Ann. 116 (1939).
7. C. L. Siegel, Iteration of analytic functions, Ann. of Math. 43 (1942).
8. C. L. Siegel, Symplectic geometry, Amer. J. Math. 65 (1943).
9. C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Springer, 1971.
10. H. M. Edwards, Riemann’s Zeta Function, Academic Press, 1974 (chapter 7 on the Riemann–Siegel formula).
11. Siegel’s theorem on integral points, Siegel zero, Riemann–Siegel formula, Wikipedia.
12. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer, chapter 16 (mass formulas).