Jeffrey Clark Lagarias (born November 16, 1949, in

Pittsburgh Pittsburgh ( ) is a city in Allegheny County, Pennsylvania, United States, and its county seat. It is the List of municipalities in Pennsylvania#Municipalities, second-most populous city in Pennsylvania (after Philadelphia) and the List of Un ...

Pennsylvania Pennsylvania, officially the Commonwealth of Pennsylvania, is a U.S. state, state spanning the Mid-Atlantic (United States), Mid-Atlantic, Northeastern United States, Northeastern, Appalachian, and Great Lakes region, Great Lakes regions o ...

, United States) is a

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

and professor at the

University of Michigan The University of Michigan (U-M, U of M, or Michigan) is a public university, public research university in Ann Arbor, Michigan, United States. Founded in 1817, it is the oldest institution of higher education in the state. The University of Mi ...

Education

While in high school in 1966, Lagarias studied

astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...

at the Summer Science Program. He completed an S.B. and S.M. in Mathematics at the

Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a Private university, private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of moder ...

in 1972. The title of his thesis was "Evaluation of certain character sums". He was a

Putnam Fellow The William Lowell Putnam Mathematical Competition, often abbreviated to Putnam Competition, is an annual mathematics competition for undergraduate college students enrolled at institutions of higher learning in the United States and Canada (regar ...

at MIT in 1970. He received his Ph.D. in Mathematics from MIT for his thesis "The 4-part of the class group of a quadratic field", in 1974. His advisor for both his masters and Ph.D. was

Harold Stark Harold Mead Stark (born August 6, 1939) is an Americans, American mathematician, specializing in number theory. He is best known for his solution of the Carl Friedrich Gauss, Gauss class number 1 problem, in effect Stark–Heegner theorem, corre ...

Career

In 1974, he joined

AT&T Bell Laboratories Nokia Bell Labs, commonly referred to as ''Bell Labs'', is an American industrial research and development company owned by Finnish technology company Nokia. With headquarters located in Murray Hill, New Jersey, Murray Hill, New Jersey, the compa ...

and eventually became a member of technical staff. From 1995 to 2004, he was a Technology Consultant at AT&T Research Laboratories. In 2004, he moved to the

as a professor of mathematics.

Research

Lagarias originally worked in analytic

algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...

. His later work has been in

theoretical computer science Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Associati ...

. Lagarias discovered an elementary problem that is equivalent to the

Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...

, namely whether for all ''n'' > 0, we have :

\sigma(n) \le H_n + e^ \ln H_n

with equality only when ''n'' = 1. Here ''H''_''n'' is the ''n''th

harmonic number In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dot ...

, the sum of the reciprocals of the first

n

positive integers, and ''σ''(''n'') is the

divisor function In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (includi ...

, the sum of the positive divisors of ''n''. He disproved Keller's conjecture in dimensions at least 10. Lagarias has also done work on the

Collatz conjecture The Collatz conjecture is one of the most famous List of unsolved problems in mathematics, unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer ...

and Li's criterion and has written several highly cited papers in symbolic computation with

Dave Bayer David Allen Bayer (born November 29, 1955) is an American mathematician known for his contributions in algebra and symbolic computation and for his consulting work in the movie industry. He is a professor of mathematics at Barnard College, Columbi ...

Awards and honors

Lagarias received in 1986 a Lester R. Ford award from the

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university A university () is an educational institution, institution of tertiary edu ...

and again in 2007. In 2012, he became a fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

. In 2024 he was elected as a member of the

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...

References

External links

*
Jeffrey Clark Lagarias homepage
University of Michigan {{DEFAULTSORT:Lagarias, Jeffrey 1949 births Living people Scientists from Pittsburgh Massachusetts Institute of Technology School of Science alumni Scientists at Bell Labs 20th-century American mathematicians 21st-century American mathematicians American number theorists American theoretical computer scientists University of Michigan faculty Putnam Fellows Summer Science Program Fellows of the American Mathematical Society Fellows of the Society for Industrial and Applied Mathematics Members of the United States National Academy of Sciences