Atle Selberg

Early Life and Education

Atle Selberg was born on June 14, 1917, in Langesund, Norway. He was the son of Anna Kristina Selberg and Ole Michael Ludvigsen Selberg, a professor of mathematics. Growing up in a family deeply rooted in academia, Selberg was exposed to mathematics from an early age. His brothers, Henrik and Sigmund, also became mathematicians, fostering an environment where mathematical discussions were commonplace.

Selberg attended the University of Oslo, where he completed his undergraduate studies in mathematics. His early work was influenced by the mathematical traditions of his family and the academic environment of the university. He received his Ph.D. in 1943 under the supervision of the renowned mathematician Viggo Brun. His dissertation focused on the sieve methods in number theory, a field that would become central to his later work.

Contributions to Mathematics

Sieve Theory

Selberg's early contributions to sieve theory were groundbreaking. He developed the Selberg sieve, a method that provided a new approach to problems in analytic number theory. The Selberg sieve is a combinatorial tool used to estimate the size of sets of integers that satisfy certain conditions, such as being prime numbers. This method has applications in various areas of number theory, including the distribution of prime numbers and the study of Diophantine equations.

Selberg Trace Formula

One of Selberg's most significant contributions is the Selberg trace formula, which he introduced in the 1950s. This formula is a powerful tool in spectral theory and automorphic forms. It establishes a relationship between the eigenvalues of the Laplace operator on a Riemannian manifold and the lengths of closed geodesics on the manifold. The Selberg trace formula has far-reaching implications in mathematical physics, representation theory, and algebraic geometry.

Prime Number Theorem

Selberg made significant contributions to the proof of the prime number theorem. In 1948, he provided an "elementary" proof of the theorem, which states that the number of prime numbers less than a given number \( x \) is asymptotically equal to \( \frac{x}{\log x} \). This proof, developed independently and simultaneously with Paul Erdős, was notable for not relying on complex analysis, which was a departure from previous proofs by Jacques Hadamard and Charles-Jean de la Vallée Poussin.

Selberg Class

Selberg introduced the concept of the Selberg class, a set of Dirichlet series that generalize the properties of the Riemann zeta function. The Selberg class is defined by a set of axioms that include analytic continuation, functional equation, and Euler product. This framework has been instrumental in the study of L-functions and their applications in number theory.

Academic Career

Early Career

After completing his Ph.D., Selberg continued his research in Norway during the difficult years of World War II. Despite the challenging circumstances, he made significant progress in his work on sieve theory and the distribution of prime numbers. In 1947, he moved to the United States, where he joined the Institute for Advanced Study (IAS) in Princeton, New Jersey.

Institute for Advanced Study

At the IAS, Selberg worked alongside some of the most prominent mathematicians of the time, including John von Neumann, Albert Einstein, and Hermann Weyl. His tenure at the IAS was marked by prolific research and numerous collaborations. He remained at the IAS for the rest of his career, becoming a professor in 1951 and later a professor emeritus.

Honors and Awards

Selberg's contributions to mathematics were widely recognized. He received numerous awards, including the Fields Medal in 1950, the Wolf Prize in Mathematics in 1986, and the Abel Prize in 2002. He was also elected to several prestigious academies, including the Norwegian Academy of Science and Letters, the American Academy of Arts and Sciences, and the Royal Society of London.

Legacy

Atle Selberg's work has had a profound impact on modern mathematics. His contributions to sieve theory, the Selberg trace formula, and the elementary proof of the prime number theorem have influenced a wide range of mathematical disciplines. The Selberg class continues to be a central object of study in analytic number theory, and his methods and ideas are still being developed and applied in new contexts.

Selberg's legacy is also reflected in the many mathematicians he mentored and influenced throughout his career. His work has inspired generations of researchers and continues to be a source of deep mathematical insight.

References

[1] Biography of Atle Selberg, Institute for Advanced Study
[2] "Atle Selberg and the Development of Sieve Theory," Journal of Number Theory
[3] "The Selberg Trace Formula and Its Applications," Annals of Mathematics
[4] "Elementary Proof of the Prime Number Theorem," Proceedings of the National Academy of Sciences