John Friedlander of the University of Toronto and Henryk Iwaniec of Rutgers University will receive the 2017 AMS Joseph L. Doob Prize. The two are honored for their book Opera de Cribro (AMS, 2010).

The prime numbers, the building blocks of the whole numbers, have fascinated humankind for millennia. While it has been known since the time of Euclid that the number of primes is infinite, exactly how they are distributed among the whole numbers is still not understood. The Latin title of the prizewinning book by Friedlander and Iwaniec could be translated as A Laborious Work Around the Sieve, where in this context a "sieve" is a mathematical tool for sifting prime numbers out of sets of whole numbers.

The Sieve of Eratosthenes, dating from the third century BC, is a simple, efficient method to produce a table of prime numbers. For a long time, it was the only way to study the mysterious sequence of the primes. In the early 20th century, improvements came through the work of Norwegian mathematician Viggo Brun, who combined the Sieve of Eratosthenes with ideas from combinatorics. Tools from another branch of mathematics, complex analysis, came into play through the work of English mathematicians G.H. Hardy and J.E. Littlewood, and of the iconic Indian mathematician Srinivasa Ramanujan (protagonist of the 2016 film The Man Who Knew Infinity).

For 30 years, Brun's method and its refinements were the main tools in sieve theory. Then, in 1950, another Norwegian mathematician, Atle Selberg, put forward a new, simple, and elegant method. As his method was independent of that of Brun, the combination of the two gave rise to deep new results.

The latter part of the 20th century saw the proof of many profound results on classical prime-number questions that had previously been considered inaccessible. Among these was a formula for the number of primes representable as the sum of a square and of a fourth power, obtained by Friedlander and Iwaniec in 1998.

With these developments, the time was ripe for a new book dealing with prime-number sieves and the techniques needed for their applications. Written by two of the top masters of the subject, Opera de Cribro is an insightful and comprehensive presentation of the theory and application of sieves. In addition to providing the latest technical background and results, the book looks to the future by raising new questions, giving partial answers, and indicating new ways of approaching the problems.

With high-quality writing, clear explanations, and numerous examples, the book helps readers understand the subject in depth. "These features distinguish this unique monograph from anything that had been written before on the subject and lift it to the level of a true masterpiece," the prize citation says.

The two prizewinners collaborated on an expository article on number sieves, "What is the Parity Phenomenon?", which appeared in the August 2009 issue of the AMS Notices.

Born in Toronto, John Friedlander received his BSc from the University of Toronto and his MA from the University of Waterloo. In 1972, he earned his PhD at Pennsylvania State University under the supervision of S. Chowla. His first position was that of assistant to Atle Selberg at the Institute for Advanced Study. After further positions at IAS, the Massachusetts Institute of Technology, the Scuola Normale Superiore in Pisa, and the University of Illinois at Urbana-Champaign, he returned to the University of Toronto as a faculty member in 1980. He was Mathematics Department Chair from 1987 to 1991 and since 2002 has been University Professor of Mathematics. He was awarded the Jeffery-Williams Prize of the Canadian Mathematical Society (1999) and the CRM-Fields (currently CRM-Fields-PIMS) Prize of the Canadian Mathematical Institutes (2002). He gave an invited lecture at the International Congress of Mathematicians in Zurich in 1994. He is a Fellow of the Royal Society of Canada, a Founding Fellow of the Fields Institute, and a Fellow of the AMS.

Born in Elblag, Poland, Henryk Iwaniec graduated from Warsaw University in 1971 and received his PhD in 1972. In 1976 he defended his habilitation thesis at the Institute of Mathematics of the Polish Academy of Sciences and was elected to member correspondent. He left Poland in 1983 to take visiting positions in the USA, including long stays at the Institute for Advanced Study in Princeton. In 1987, he was appointed to his present position as New Jersey State Professor of Mathematics at Rutgers University. He was elected to the American Academy of Arts and Sciences (1995), the US National Academy of Sciences (2006), and the Polska Akademia Umiejetnosci (2006, foreign member). He has received numerous prizes including the Sierpinski Medal (1996), the Ostrowski Prize (2001, shared with Richard Taylor and Peter Sarnak), the AMS Cole Prize in Number Theory (2002, shared with Richard Taylor), the AMS Steele Prize for Mathematical Exposition (2011), the Banach Medal of the Polish Academy of Sciences (2015), and the Shaw Prize in Mathematical Sciences (2015, shared with Gerd Faltings). He was an invited speaker at the International Congress of Mathematicians in Helsinki (1978), Berkeley (1986), and Madrid (2006).

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Presented every three years, the AMS Doob Prize recognizes a single, relatively recent, outstanding research book that makes a seminal contribution to the research literature, reflects the highest standards of research exposition, and promises to have a deep and long-term impact in its area. The prize will be awarded Thursday, January 5, 2017, at the Joint Mathematics Meetings in Atlanta.

Find out more about AMS prizes and awards at http://www.

Founded in 1888 to further mathematical research and scholarship, today the American Mathematical Society fulfills its mission through programs and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and to everyday life.