Atle Selberg

Atle Selberg (14 June 1917 – 6 August 2007) was a prominent Norwegian-American mathematician renowned for his foundational contributions to analytic number theory and the theory of automorphic forms, most notably for providing an elementary proof of the prime number theorem and developing the Selberg trace formula.

Early life and education

Atle Selberg was born in Langesund, Norway, into a family with a strong mathematical tradition. His father, Ole Michael Ludvigsen Selberg, was a high school teacher and mathematician, and two of his older brothers, Sigmund and Arne, also became mathematicians. Selberg's interest in mathematics was sparked early on when he discovered the works of the self-taught Indian mathematician Srinivasa Ramanujan. He pursued his higher education at the University of Oslo, where he completed his Ph.D. in 1943. His doctoral research was conducted under the challenging conditions of the German occupation of Norway during World War II.

Mathematical contributions

Selberg's work profoundly shaped modern number theory and spectral geometry. His contributions are characterized by deep analytical insights and the creation of powerful new mathematical tools.

Elementary proof of the prime number theorem: In 1948, Selberg achieved a major breakthrough by discovering an elementary proof of the prime number theorem, a result that had previously only been proven using complex analysis by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896. Selberg's proof relied on a fundamental identity now known as Selberg's identity. The Hungarian mathematician Paul Erdős independently and almost simultaneously developed a similar elementary proof based on Selberg's identity. A subsequent dispute over the publication and attribution of the proof led to a lifelong estrangement between the two mathematicians.

Selberg trace formula: Introduced in 1956, the Selberg trace formula is one of his most profound achievements. It establishes a deep relationship between the length spectrum of closed geodesics on a Riemann surface and the eigenvalues of the Laplace-Beltrami operator on that surface. This formula serves as a non-commutative analogue of the Poisson summation formula and has become a central tool in the theory of automorphic forms, representation theory, and spectral geometry.

Analytic number theory and the Riemann zeta function: In 1942, Selberg proved the critical line theorem, demonstrating that a positive proportion of the non-trivial zeros of the Riemann zeta function lie on the critical line (where the real part is 1/2). This was a significant step toward the unproven Riemann hypothesis. He also developed the Selberg sieve, a versatile combinatorial method used to estimate the size of sifted sets of positive integers, which has found wide applications in prime number theory. Furthermore, his work on the Rankin-Selberg method provided a way to construct and study L-functions associated with automorphic forms.

Selberg zeta function: In connection with his trace formula, Selberg introduced the Selberg zeta function, an analogue of the Riemann zeta function for Riemann surfaces. The zeros of this function are related to the lengths of closed geodesics on the surface, providing a geometric counterpart to the distribution of prime numbers.

Career

Following World War II, Selberg's growing reputation led to an invitation to the United States. In 1947, he joined the Institute for Advanced Study (IAS) in Princeton, New Jersey. He became a permanent member of the faculty in 1949 and remained affiliated with the IAS for the rest of his career, eventually becoming a professor emeritus. At the IAS, he worked alongside other mathematical luminaries such as John von Neumann, Hermann Weyl, and Carl Ludwig Siegel, fostering an environment of rigorous and independent research.

Awards and honors

Throughout his life, Selberg received numerous prestigious accolades in recognition of his mathematical achievements. He was awarded the Fields Medal in 1950 at the International Congress of Mathematicians in Cambridge, Massachusetts, primarily for his work on the zeros of the Riemann zeta function and his elementary proof of the prime number theorem. In 1986, he received the Wolf Prize in Mathematics. In 2002, Selberg was awarded a special honorary Abel Prize to commemorate the bicentennial of Niels Henrik Abel's birth, prior to the establishment of the regular annual Abel Prize in 2003. He was also a member of several scientific academies, including the Norwegian Academy of Science and Letters, the United States National Academy of Sciences, and the Royal Society of London.

Personal life and death

Selberg married Hedvig Lie in 1947, and the couple had two children. He became a naturalized citizen of the United States in 1953. Known for his independent working style, Selberg often preferred to work alone and was highly meticulous in his research. He died of heart failure at his home in Princeton, New Jersey, on 6 August 2007, at the age of 90. His collected papers were published in two volumes by Springer-Verlag in 1989 and 1991, preserving his legacy for future generations of mathematicians.