### Carl Ludwig Siegel is widely regarded as one of the twentieth century’s top mathematicians. Prior to his enrollment at Humboldt University at the age of nineteen, little is known about his life. Although he planned to study astronomy at first, his lecturers at Humboldt encouraged him to study number theory instead. Later, he completed his Ph.D. dissertation on Diophantine approximations at the University of Göttingen, and two years later, he began his work at Frankfurt am Main’s Johann Wolfgang Goethe-Universität. With his ‘Thue–Siegel–Roth theorem,’ he was able to establish himself as a famous mathematician even before that. He fled his homeland at the outbreak of WWII as an anti-Nazi and anti-militarist, teaching in the United States from 1940 to 1951. Otherwise, he spent much of his time in Germany, teaching and conducting research in mathematics. He is best known today for his contributions to number theory and celestial mechanics. He had produced multiple textbooks on these subjects in addition to publishing a number of groundbreaking studies on them.

## Childhood and Adolescence

Carl Ludwig Siegel was born in Berlin, Germany, on December 31, 1896. His father, whose surname was never revealed, worked as a postal worker. There isn’t much else known about his family or his early years.

Despite the fact that we have no information about his education, he must have received a thorough education because he was later discovered to be able to read the writings of ancient mathematicians in their original language. He must have been a good student as well, as he entered Humboldt University in 1915 with degrees in astronomy, mathematics, and physics.

Professors at Humboldt University ran the beginner’s classes themselves back then. In this manner, they could identify bright children from the start and guide their careers accordingly.

Although Siegel intended to study astronomy at Humboldt University, he was quickly recruited by theoretical physicist Max Karl Ernst Ludwig Planck and mathematician Ferdinand Georg Frobenius. Soon after, under the influence of Frobenius, Siegel abandoned astronomy in favor of number theory.

Siegel was conscripted into the army in 1917 as the First World War advanced. He struggled to acclimate to army life as an antimilitarist. He was also sent to a psychiatric facility for a period, but nothing could help him. He was eventually discharged from the army.

During his stay at the psychiatric facility, Siegel met Edmund Georg Hermann Landau, a University of Göttingen professor who specializes in number theory and complex analysis. Siegel later stated that it was only due to Landau that he was able to survive his time at the institute.

Siegel did not return to Humboldt University after being discharged. In 1919, he began working for Edmund Landau at Georg-August University of Göttingen as a teaching and research assistant. He wrote his dissertation on Diophantine approximations under Landau’s supervision.

He earned his doctorate degree in 1920. His dissertation was hailed as “a watershed moment in Diophantine approximation history.” He stayed at the University of Göttingen after that, researching a variety of issues and producing numerous groundbreaking publications.

One of Carl Ludwig Siegel’s greatest successes during this time period was his work on Roth’s theorem, which he completed in 1921. It cemented his reputation as a world-class mathematician. As a result, when Arthur Moritz Schönflies departed from the Johann Wolfgang Goethe-Universität in Frankfurt am Main in 1922, Siegel was asked to take his place.

## Early on in your career

Carl Ludwig Siegel began his career as a Professor of Mathematics at Johann Wolfgang Goethe-Universität in Frankfurt is Main in 1922. Many prominent mathematicians, like Ernst Hellinger, Otto Szász, Paul Epstein, and Max Dehn, were already working in the same department at the time, providing a lively environment.

Siegel built tight bonds with his new coworkers soon after starting his new job, working together without regard for personal gain. They would get together every Thursday afternoon from 4 to 6 p.m. to discuss various topics.

Siegel, Hellinger, Epstein, and Dehn began working together on several projects soon after. One such event was the history of mathematics seminar, which began in 1922. It lasted thirteen years, and Siegel referred to them as the happiest recollections of his life in later years.

The seminar attendees were forced to read ancient mathematicians’ writings in their native language. Nonetheless, the number of participants was always at least six, and they studied Euclid, Archimedes, Fibonacci, Cardan, Stevin, Viète, Kepler, Desargues, Descartes, Fermat, Huygens, Barrow, and Gregory together.

Siegel was also a committed educator. He had a small number of students at first, and only two advanced students. They were both late for class one day, and when they arrived, Siegel had already begun teaching, having filled up an entire portion of the blackboard.

He had 143 students enrolled in his differential and integral calculus classes by 1928, and he had to spend a lot of time editing their papers as a result. Despite this, he kept working on his research.

He published an important article on linear equations in 1929. It’s known as ‘Siegel’s lemma,’ and it’s a pure existence theorem that refers to the bounds on the solutions of the equations derived by the building of auxiliary functions. He also proved ‘Bourget’s hypothesis’ in the same year.

Siegel uncovered an unpublished text by Bernhard Riemann written in the 1850s in 1932. He deduced an asymptotic formula from this work, which became known as the Riemann-Siegel formula.

## Later in Life

On 30 January 1933, Hitler was elected Chancellor of Germany, and on 7 April 1933, the Civil Service Law was passed, prohibiting Jewish professors from teaching at German universities. Although Siegel was unaffected, his friend Otto Szász was discharged from the army, which Siegel found deeply troubling.

He took a six-month sabbatical leave at Princeton’s Institute for Advanced Study from January to June 1935. When he returned, he discovered Epstein, Hellinger, and Dehn had been fired from their positions. In the same year, he made a correction to the Smith-Minkowski formula.

At the invitation of the International Mathematical Union, he traveled to Oslo, Norway in 1936 to attend the International Congress of Mathematics. Getting invited to speak at ICM is almost like being inducted into the hall of fame, so it was a huge honor for him.

He was invited to attend the University of Göttingen in 1937. He moved to Göttingen in early 1938 after accepting the appointment near the close of the year. Here, too, he discovered that Nazi policies had a strong influence on life, both on and off-campus.

Because of the political climate, Siegel lived in Göttingen in a semi-retired state. It did not, however, deter him from pursuing his intellectual interests. He began work on what became known as the ‘Siegel Modular Form’ in 1939. In the same year, he invented the concept of ‘Siegel upper half-space.’

In September 1939, when the Second World War broke out as a result of the German invasion of Poland, Siegel felt he could no longer reside there. He sailed for Denmark in early 1940 and then traveled to the United States through Norway.

In the United States, he joined the Institute for Advanced Study at Princeton, where he worked as a member of the mathematics department from September 1940 until June 1945. His professorship was appointed permanently in September 1946. However, considering his stay as a self-imposed exile in the United States, he was dissatisfied there.

After obtaining an offer from the University of Göttingen in June 1951, Siegel came home after resigning from the Institute for Advanced Study. He stayed at the University of Göttingen for the next eight years, continuing to write a number of groundbreaking works in mathematics.

Siegel left the University of Göttingen in 1959. He did not, however, cease working, publishing major works far into his eighties. He also worked at Princeton’s Institute for Advanced Study for a short time in the fall of 1960.

He proposed that e1/2, or around 60.65%, of all prime numbers, are regular in the asymptotic sense of natural density in 1964 when he was almost seventy years old. Siegel’s Conjecture was later named after him.

Siegel also enjoyed instructing, not just sophisticated theory but also basic courses. However, because he required excellence and thoroughness, he only had a few research students working for him. Kurt Mahler, Christian Pommerenke, Theodor Schneider, and Jürgen Moser were among his students who went on to become outstanding mathematicians.

## Major Projects of Carl Ludwig Siegel

Carl Ludwig Siegel is well recognized for his contributions to Diophantine approximation’s ‘Thue–Siegel–Roth theorem.’ “A given algebraic number (alpha) may not have too many good rational number approximations,” according to Roth’s original statement. Siegel refined the concept of very good’ in 1921 while working on the theorem.

Siegel’s contribution to the ‘Smith-Minkowski-Siegel’ formula is likewise well-known. In 1935, he discovered a flaw in the Smith-Minkowski formula, as it was then known. He was able to remedy the issue by working on it. The formula became known as the ‘Smith-Minkowski-Siegel’ formula after that.

## Achievements & Awards

Carl Ludwig Siegel and Israel Gelfand of Soviet Russia shared the first Wolf Prize in Mathematics in 1978. “For his contributions to the theory of numbers, theory of multiple complex variables, and celestial mechanics,” Siegel received this prestigious honor.

## Death and the Afterlife

Carl Ludwig Siegel married only once in his life and devoted his entire life to mathematics. His mental power remained unabated even as he became older, and he published a number of publications in his seventies. He also went on lecture trips around the world.

He died on April 4, 1981, at the age of 84, in Göttingen, West Germany.

## Estimated Net worth

The estimated net worth of Carl Ludwig Siegel is about **$1 million**.