mathematics

Hugh Lowell Montgomery

Hugh Lowell Montgomery is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hugh Lowell Montgomery rather than just read about it. In short: Hugh Lowell Montgomery (born 1944) is an American mathematician, working in the fields of analytic number theory and mathematical analysis. He is the namesake of Montgomery's pair correlation conjecture on the zeros of the Riemann zeta function, is known for his development of large sieve methods, and is the author of multiple books on number theory and analysis.

Key takeaways

Hugh Lowell Montgomery belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hugh Lowell Montgomery to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hugh Lowell Montgomery from memory before moving on to harder problems.

Reference excerpt

Hugh Lowell Montgomery (born 1944) is an American mathematician, working in the fields of analytic number theory and mathematical analysis. He is the namesake of Montgomery's pair correlation conjecture on the zeros of the Riemann zeta function, is known for his development of large sieve methods, and is the author of multiple books on number theory and analysis. He is a professor emeritus at the University of Michigan.

Education and career Montgomery was born on August 26, 1944, in Muncie, Indiana. He was an undergraduate at the University of Illinois Urbana-Champaign. On graduating in 1966, he became a Marshall scholar at the University of Cambridge in England. There, he became a Fellow of Trinity College, Cambridge in 1969, and completed his Ph.D. in 1972. His dissertation, Topics in Multiplicative Number Theory, was supervised by Harold Davenport. He became an assistant professor of mathematics at the University of Michigan in 1972. He was quickly promoted, to associate professor in 1973 and full professor in 1975. At the University of Michigan, he advised 19 doctoral students, including Sidney Graham in 1977, Brian Conrey in 1980, and Russell Lyons in 1983. He retired as a professor emeritus in 2020.

Recognition Montgomery was a 1972 recipient of the Adams Prize, and the 1974 recipient of the Salem Prize. In 1974, Montgomery was an invited speaker of the International Congress of Mathematicians (ICM) in Vancouver. In 2012, he became a fellow of the American Mathematical Society.

Selected publications

Books Montgomery, Hugh L. (1971). Topics in Multiplicative Number Theory. Lecture Notes in Mathematics. Vol. 227. Berlin & New York: Springer-Verlag. MR 0337847. Zbl 0216.03501. Niven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L. (1991). An Introduction to the Theory of Numbers (5th ed.). New York: John Wiley & Sons. ISBN 0-471-62546-9. MR 1083765. Zbl 0742.11001. Montgomery, Hugh L. (1994). Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis. CBMS Regional Conference Series in Mathematics. Vol. 84. Washington, DC and Providence, Rhode Island: Conference Board of the Mathematical Sciences and American Mathematical Society. doi:10.1090/cbms/084. ISBN 0-8218-0737-4. MR 1297543. Zbl 0814.11001. Montgomery, Hugh L.; Vaughan, Robert C. (2007). Multiplicative Number Theory. I. Classical Theory. Cambridge Studies in Advanced Mathematics. Vol. 97. Cambridge, UK: Cambridge University Press. ISBN 978-0-521-84903-6. MR 2378655. Zbl 1142.11001. Montgomery, Hugh L. (2014). Early Fourier Analysis. The Sally Series: Pure and Applied Undergraduate Texts. Vol. 22. American Mathematical Society. ISBN 9781470415600. MR 3243762. Zbl 1316.42001.

Research articles Montgomery, H. L.; Vaughan, R. C. (1973). "The large sieve". Mathematika. 20 (2): 119–134. doi:10.1112/S0025579300004708. MR 0374060. Montgomery, H. L. (1973). "The pair correlation of zeros of the zeta function". In Diamond, Harold G. (ed.). Analytic Number Theory: Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society, held at St. Louis University, St. Louis, Mo., March 27–30, 1972. Proceedings of Symposia in Pure Mathematics. Vol. 24. Providence, Rhode Island: American Mathematical Society. pp. 181–193. MR 0337821. Levinson, Norman; Montgomery, Hugh L. (1974). "Zeros of the derivatives of the Riemann zeta-function". Acta Mathematica. 133: 49–65. doi:10.1007/BF02392141. MR 0417074. Beauzamy, Bernard; Bombieri, Enrico; Enflo, Per; Montgomery, Hugh L. (1990). "Products of polynomials in many variables". Journal of Number Theory. 36 (2): 219–245. doi:10.1016/0022-314X(90)90075-3. hdl:2027.42/28840. MR 1072467.

References

External links

Official Page An Introduction to the Theory of Numbers, Fifth Edition page

Illustrations

Worked examples

Example 1 — a first encounter with Hugh Lowell Montgomery

Start with the simplest possible case. Write down what Hugh Lowell Montgomery claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hugh Lowell Montgomery before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hugh Lowell Montgomery ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hugh Lowell Montgomery

In research: Hugh Lowell Montgomery appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hugh Lowell Montgomery in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hugh Lowell Montgomery is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1944 births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Hugh Lowell Montgomery outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Hugh Lowell Montgomery” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Hugh Lowell Montgomery in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hugh Lowell Montgomery means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hugh Lowell Montgomery out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hugh Lowell Montgomery in simple terms?

Why does Hugh Lowell Montgomery matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hugh Lowell Montgomery?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hugh Lowell Montgomery.