Russell Sherman Lehman

Professor Russell Sherman Lehman was born in Ames, Iowa on January 25, 1930. Lehman grew up on a small family farm in Oregon and attended a one-room schoolhouse until high school. His appearance on a radio program called “Quiz Kid” led to his receiving a four-year Pepsi-Cola scholarship that could be used at any university in the United States.

Lehman chose Stanford University for college and received bachelor's, master's and doctoral degrees from Stanford. His 1954 PhD dissertation, “Developments in the Neighborhood of the Beach of Surface Waves over an Inclined Bottom,” was written under the supervision of Hans Lewy, a UC Berkeley professor who was serving as a visiting professor at Stanford while Lehman was a graduate student. As the dissertation title suggests, Lehman's work was in applied mathematics. Nonetheless, the content of the thesis shows that the author was engaged with number theory early in his career: Lehman discovered when a surface is sloped with angle απ, the properties of waves flowing over the surface depend on the extent to which α is approximated by rational numbers.

After leaving Stanford in 1954, Lehman spent a short time at the RAND Corporation before being drafted into the US Army. Perhaps because of his work at RAND, Lehman published an article in 1955 entitled “On confirmation and rational betting.” Here are the first two paragraphs of the article:

According to Lehman's son Andrew Lehman, this 1955 article was the first appearance in print of the phrase “dutch book,” a well-known term in probability that now has its own Wikipedia page.

In the Army, Lehman worked at the Ballistic Research Laboratory at the Aberdeen Proving Ground in Maryland. Once his military service had concluded, Lehman continued his research in Göttingen on a Fulbright grant.

Around this time, Berkeley began to recruit Sherman Lehman. The department chair who led the recruitment effort was the number theorist D. H. Lehmer. Sherman Lehman's PhD advisor Hans Lewy and other colleagues at Berkeley aided the recruitment effort. Ultimately, Lehman agreed to join the Berkeley mathematics department at the start of the 1958 calendar year. (He wished to remain in Europe until the end of his Fulbright grant.) Sherman Lehman was promoted to associate professor in January, 1962 and then to full professor in 1966. His first publications as a Berkeley faculty member were articles on dynamic programming, two of them written jointly with Richard Bellman, the inventor of dynamic programming and a former colleague at the RAND Corporation. As the title of his 1963 article “Algebraic properties of the composition of solutions of partial differential equations” suggests, Lehman continued to combine algebraic interests into his analytic work around this time.

Lehman's first article on number theory was his 1966 publication “On the difference π(x)-li(x),” which compared the number of prime numbers ≤x with the logarithmic integral that was known to estimate this number closely. This article melds together theoretical insights with targeted computations; for example, Lehman shows that there are more than 10⁵⁰⁰ integers between 1.53×10¹¹⁶⁵ and 1.65×10¹¹⁶⁵ for which the integral is smaller than the number that it estimates. (For small values of x, the integral is an over-estimate.)

In the 1960s, the Berkeley Mathematics Department led the world in the emerging application of computers to calculation in number theory. Berkeley mathematicians D. H. Lehmer and Emma Lehmer were pioneers in this endeavor. Lehman worked with his colleague René De Vogelaere to develop Berkeley's course in numerical analysis and to ensure that there were adequate computing facilities in place for students to explore examples while in the course. Lehman was an enthusiastic instructor in upper-division courses who frequently taught Berkeley's course in number theory (Math 115). In addition, Lehman served frequently as an applied mathematics major advisor. At this time, Lehman was an avid mountaineer and thus enjoyed a sport that is a favorite of many mathematicians and scientists.

After 1966, Lehman was a number theorist as well as an applied mathematician. Lehman's subsequent articles concerned the zeros of the Riemann ζ-function and the factorization of large integers. Lehman's articles in this area continue to be important in current research. His 1974 article “Factoring large integers” was cited at least twice in 2023.

Lehman's research career was cut short by an automobile accident in 1968 and subsequent stroke in 1969. Even after these setbacks, however, Lehman continued to teach courses in number theory and related topics and to supervise graduate students. Lehman had 11 PhD students at Berkeley, the first in 1963 and the last in 1995. Probably Lehman's most notable student is Richard W. Cottle, a professor of Management Science and Engineering at Stanford University, who was advised jointly by Lehman and George Dantzig (known for the simplex algorithm).

Lehman retired in 1994 but continued to be an active member of the Berkeley mathematics community for years after his retirement. According to Andrew Lehman, his father Sherman had a deep love for UC Berkeley. Sherman Lehman passed away peacefully on July 28, 2023, at the age of 93. He is survived by his six children — Clifford, Anne, John, Helen, Andrew, and Veli — eight grandchildren, and two great grandchildren. He is remembered fondly by his colleagues, neighbors and family.

Acknowledgements: We are indebted to Lehman's son Andrew Lehman for information about his father's early life. Another source for this memorial has been C. Moore's book Mathematics at Berkeley: a history.

Kenneth A. Ribet